1 Documenta Mathematica Journal der Deutschen Mathematiker-Vereinigung Gegründet 1996 Extra Volume Optimization Sto...

Documenta Mathematica Journal der Deutschen Mathematiker-Vereinigung ¨ ndet 1996 Gegru Extra Volume

Optimization Stories 21st International Symposium on Mathematical Programming Berlin, August 19–24, 2012

Editor: ¨ tschel Martin Gro

Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, ver¨ offentlicht Forschungsarbeiten aus allen mathematischen Gebieten und wird in traditioneller Weise referiert. Es wird indiziert durch Mathematical Reviews, Science Citation Index Expanded, Zentralblatt f¨ ur Mathematik. Artikel k¨ onnen als TEX-Dateien per E-Mail bei einem der Herausgeber eingereicht werden. Hinweise f¨ ur die Vorbereitung der Artikel k¨ onnen unter der unten angegebenen WWW-Adresse gefunden werden.

Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, publishes research manuscripts out of all mathematical fields and is refereed in the traditional manner. It is indexed in Mathematical Reviews, Science Citation Index Expanded, Zentralblatt f¨ ur Mathematik. Manuscripts should be submitted as TEX -files by e-mail to one of the editors. Hints for manuscript preparation can be found under the following web address. http://www.math.uni-bielefeld.de/documenta

¨ ftsfu ¨ hrende Herausgeber / Managing Editors: Gescha Alfred K. Louis, Saarbr¨ ucken Ulf Rehmann (techn.), Bielefeld Peter Schneider, M¨ unster

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Herausgeber / Editors: Christian B¨ ar, Potsdam Don Blasius, Los Angeles Joachim c*ntz, M¨ unster Patrick Delorme, Marseille Gavril Farkas, Berlin (HU) Edward Frenkel, Berkeley Friedrich G¨ otze, Bielefeld Ursula Hamenst¨ adt, Bonn Lars Hesselholt, Cambridge, MA (MIT) Max Karoubi, Paris Stephen Lichtenbaum Eckhard Meinrenken, Toronto Alexander S. Merkurjev, Los Angeles Anil Nerode, Ithaca Thomas Peternell, Bayreuth Takeshi Saito, Tokyo Stefan Schwede, Bonn Heinz Siedentop, M¨ unchen (LMU) Wolfgang Soergel, Freiburg

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ISBN 978-3-936609-58-5 ISSN 1431-0635 (Print) ISSN 1431-0643 (Internet)

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Documenta Mathematica Extra Volume: Optimization Stories, 2012

Preface

1

Introduction

3

Stories about the Old Masters of Optimization

7

Ya-xiang Yuan Jiu Zhang Suan Shu and the Gauss Algorithm for Linear Equations

9–14

Eberhard Knobloch Leibniz and the Brachistochrone

15–18

Eberhard Knobloch Leibniz and the Infinite

19–23

Peter Deuflhard A Short History of Newton’s Method

25–30

Eberhard Knobloch Euler and Infinite Speed

31–35

Eberhard Knobloch Euler and Variations ¨ tschel and Ya-xiang Yuan Martin Gro ¨ nigsberg, Euler, Mei-Ko Kwan, Ko and a Chinese Postman

Linear Programming Stories David Shanno Who Invented the Interior-Point Method? George L. Nemhauser Column Generation for Linear and Integer Programming ¨ nter M. Ziegler Gu Who Solved the Hirsch Conjecture? Friedrich Eisenbrand Pope Gregory, the Calendar, and Continued Fractions

37–42

43–50 51 55–64 65–73 75–85

87–93 iii

Martin Henk ¨ wner–John Ellipsoids Lo

95–106

Robert E. Bixby A Brief History of Linear and Mixed-Integer Programming Computation

Discrete Optimization Stories

107–121 123

ˇil and Helena Neˇ ˇilova ´ Jaroslav Neˇ setr setr The Origins of Minimal Spanning Tree Algorithms – Bor˚ uvka and Jarn´ik

127–141

William H. Cunningham The Coming of the Matroids

143–153

Alexander Schrijver On the History of the Shortest Path Problem

155–167

Alexander Schrijver On the History of the Transportation and Maximum Flow Problems

169–180

William R. Pulleyblank Edmonds, Matching and the Birth of Polyhedral Combinatorics ¨ tschel Thomas L. Gertzen and Martin Gro Flinders Petrie, the Travelling Salesman Problem, and the Beginning of Mathematical Modeling in Archaeology ¨ hring Rolf H. Mo D. Ray Fulkerson and Project Scheduling ´rard Cornue ´jols Ge The Ongoing Story of Gomory Cuts

181–197

199–210 211–219 221–226

William Cook Markowitz and Manne + Eastman + Land and Doig = Branch and Bound

227–238

Susanne Albers Ronald Graham: Laying the Foundations of Online Optimization

239–245

Continuous Optimization Stories ´chal Claude Lemare Cauchy and the Gradient Method iv

247 251–254

Richard W. Cottle William Karush and the KKT Theorem Margaret H. Wright Nelder, Mead, and the Other Simplex Method Jean-Louis Goffin Subgradient Optimization in Nonsmooth Optimization (including the Soviet Revolution) ´ bal Robert Mifflin and Claudia Sagastiza A Science Fiction Story in Nonsmooth Optimization Originating at IIASA Andreas Griewank Broyden Updating, the Good and the Bad! Hans Josef Pesch ´odory on the Road to the Maximum Principle Carathe Hans Josef Pesch and Michael Plail The Cold War and the Maximum Principle of Optimal Control Hans Josef Pesch The Princess and Infinite-Dimensional Optimization

Computing Stories

255–269 271–276

277–290

291–300 301–315 317–329

331–343 345–356 357

David S. Johnson A Brief History of NP-Completeness, 1954–2012 Robert Fourer On the Evolution of Optimization Modeling Systems Andreas Griewank Who Invented the Reverse Mode of Differentiation? ´ l Rojas Rau Gordon Moore and His Law: Numerical Methods to the Rescue

More Optimization Stories

359–376 377–388 389–400

401–415 417

Thomas M. Liebling and Lionel Pournin Voronoi Diagrams and Delaunay Triangulations: Ubiquitous Siamese Twins ¨ dgen Konrad Schmu Around Hilbert’s 17th Problem Michael Joswig From Kepler to Hales, and Back to Hilbert v

419–431 433–438 439–446

Matthias Ehrgott Vilfredo Pareto and Multi-objective Optimization

447–453

Walter Schachermayer Optimisation and Utility Functions

455–460

vi

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Preface

When in danger of turning this preface into an essay about why it is important to know the history of optimization, I remembered my favorite Antoine de Saint-Exupery quote: “If you want to build a ship, don’t drum up the men to gather wood, divide the work and give orders. Instead, teach them to yearn for the vast and endless sea.” Optimization history is not just important; it is simply fascinating, thrilling, funny, and full of surprises. This book makes an attempt to get this view of history across by asking questions such as: • Did Newton create the Newton method? • Has Gauss imported Gauss elimination from China? • Who invented interior point methods? • Was the Kuhn-Tucker theorem of 1951 already proved in 1939? • Did the Hungarian algorithm originate in Budapest, Princeton or Berlin? • Who built the first program-controlled computing machine in the world? • Was the term NP-complete created by a vote initiated by Don Knuth? • Did the Cold War have an influence on the maximum principle? • Was the discovery of the max-flow min-cut theorem a result of the Second World War? • Did Voronoi invent Voronoi diagrams? • Were regular matroids characterized by a code breaking chemist? • Did an archaeologist invent the Hamming distance and the TSP? • What has the Kepler conjecture to do with “mathematical philosophy”? • Have you ever heard of an Italian named Wilfried Fritz, born in France and deceased in Switzerland? • What does the electrification of South-Moravia have to do with spanning trees? • Did Euler cheat Russia and Prussia concerning stolen horses? • And why did Omar Khayyam compute the third convergent of a continued fraction? Interested? How many of these questions can you answer? Some of them touch fundamental issues of optimization, others appear anecdotal or even somewhat obscure, but there may be more behind them than you think. The forty-one articles in this book and my introductions to the sections provide some full and some partial answers. Just glance through the book, and I hope you will get stuck and start reading. Documenta Mathematica

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Why is the book not entitled Optimization History? Well, this would have put in a serious claim that I did not want to meet. This book is not intended to compete with scholarly historical research. A few articles, though, get close to that. No article is fiction; all are based on solid historical information. But I have asked the authors to present also their particular views, and if something is historically not clear, to present their own opinions. Most of all, I requested to write in an entertaining way addressing the casual reader. The articles in this book are not meant for the rare quiet moments in a study. You can read them on a train or plane ride; and I do hope that you get excited about some of the topics presented and start investigating their history by digging deeper into the subject. The references in the articles show you how to do that. Berlin, August 2012

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Martin Gr¨otschel

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Introduction

When an International Symposium on Mathematical Programming is hosted in Berlin and when Leonhard Euler is one of the local (and global) mathematical heroes, one cannot resist the temptation to begin the introduction by quoting an Euler statement from 1744 that every optimizer loves: Cum enim mundi universi fabrica sit perfectissima atque a Creatore sapientissimo absoluta, nihil omnino in mundo contingit, in quo non maximi minimive ratio quaepiam eluceat; quamobrem dubium prorsus est nullum, quin omnes mundi effectus ex causis finalibus ope methodi maximorum et minimorum aeque feliciter determinari queant, atque ex ipsis causis efficientibus. Briefly and very freely translated: Nothing in the world takes place without optimization, and there is no doubt that all aspects of the world that have a rational basis can be explained by optimization methods. It is not so bad to hear such a statement from one the greatest mathematicians of all time. Optimization is a mathematical discipline that differs considerably from other areas of mathematics. Practical problems, more generally, classes of problems, usually arising in fields outside of mathematics, are in the center, and mathematical models are invented that somehow grasp the essence of the problems. Then mathematical theory is developed to understand the structure of the models. And here, every branch of mathematics that helps provide insight is welcome to support the investigations. Optimization is, thus, influenced in many ways from many sources and has no unified theory, although there exist “core technologies” such as linear, nonlinear, combinatorial and stochastic optimization, each with a rich body of results. But it is not unusual that all of a sudden, methods, appearing far removed at first sight, start playing important roles. The ultimate goal of optimization is not just a good understanding of models; the research has to yield algorithms that efficiently solve the problems one has started from. And this ties optimization with the computational sciences. One can infer from these introductory remarks that the historic roots of optimization are manifold and widespread and that there is no straight line of development. And this makes the history of optimization even more interesting. Most optimizers I know are not so keen on really thorough and scholarly Documenta Mathematica

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historical articles. That is why I thought that the best way to popularize the history of optimization is by presenting brief entertaining and easy to read articles with a clear and narrow focus. The articles in this book are of three types. The first type, and the majority of articles belongs to this group, is about a person (usually a famous mathematician, or sometimes a not so well-known person who deserves to come to the fore) and about one major achievement (e.g., Cauchy and the gradient method, Flinders Petrie and the TSP, or Karush and the KKT theorem). Such articles contain a brief CV of the person (unless he is too well known like Euler or Leibniz) and then discuss the particular result, algorithm, or achievement that is important for the history of optimization. I have asked the authors to also add “personal flavor”, for instance, in cases where the authors had personal encounters with or have private information about the colleague portrayed. The second type of articles is of the sort “Who invented . . . ?”. In many cases it is not really obvious who did what first, and thus, the task of this article type is to explore the contributions and come to a conclusion. And a few articles survey certain developments such as Moore’s Law, the history of column generation or of NP-completeness. I wrote to the authors on February 22, 2012 when the serious work on this book began: I am not requesting a completely thorough account of the history of a certain optimization subject or a perfect CV of a great optimizer. I would like the articles to be appetizers. They should show, in particular the younger colleagues, that optimization is a fascinating human endeavor and that there are lots of interesting stories that happen in the development of our field. There can be surprises, funny and even tragic stories. There has to be serious and correct information, of course, but some human touch and/or humor should show. In my opinion almost all authors have achieved this goal. My initial favorite for the book title was “Short Optimization Histories”. I wanted to have short articles on focused aspects of the history of optimization that should present good stories and should have the flavor of a short story in fiction literature. I considered this title a nice play with words but was defeated by my colleagues. After long discussions, even including a vote, the current title was selected. I hope it carries the desired connotations. I am happy to mention that this book has a predecessor. For the ISMP in Amsterdam in 1991, J. K. Lenstra, A. Rinnoy Kan, and A. Schrijver edited the book History of Mathematical Programming: A Collection of Personal Reminiscences (CWI and North-Holland, 1991). This book contains an outstanding collection of articles by the pioneers of optimization themselves on their own achievements. Great reading, try to get a copy of this book! This present book complements the ISMP 1991 volume; it is broader in scope and provides an outside view. Documenta Mathematica

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Introduction

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Finally, I would like to thank Christoph Eyrich for all the (in the end very hectic) typesetting work and Ulf Rehmann for his help in editing the book in Documenta Mathematica style and his efficient handling of the publishing process. Believe it or not, the last article and the last requests for corrections arrived on July 24, 2012. I am confident that the printed volume is ready for distribution on August 20. Another final remark which occurred to me while proof-reading this introduction: Did you notice that Euler used in the text quoted, the words maxima and minima, but not optimization (as I did in my rough translation)? Where is the first appearance of the term optimization (in any language) – in the mathematical sense? One can easily find a quote from 1857, but is this the first? I do not know. If you have a clue, please, send me an email. And the final final remark: Some authors suggested opening Wikis (or something like that) on some of the topics discussed in this book. This issue will be explored in the near future. The history of the usage of the term optimization could, in fact, be a good “starting Wiki”. Martin Gr¨otschel

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Stories about the Old Masters of Optimization

I believe that optimization is in some way “built into nature”. In many of their approaches to understand nature, physicists, chemists, biologists, and others assume that the systems they try to comprehend tend to reach a state that is characterized by the optimality of some function. In statistical mechanics, e.g., the consensus is that the systems considered develop in the direction of an energy minimal configuration, called ground state. I also think that, in many of their activities, humans have the desire to be efficient and save resources. I therefore reckon that, beginning with the origin of our species, humans have attempted to be unwasteful whenever strenuous efforts lay ahead. I am very sure that our very ancient forefathers planned travel routes along short or safe paths, organized their hunting endeavors carefully, tried to reduce the work involved in ploughing and harvesting, and meticulously designed the logistics needed for the construction of buildings. There are no traces that these desires to be efficient were considered a mathematical endeavor. If one looks back at the development of our field, it is the middle of the 20th century when optimization (or mathematical programming, which is the term mostly used until recently) took off. But some of the great old masters have, of course, investigated optimization questions and laid the foundations of several of the subfields of today’s optimization theory. It is beyond the scope of this book to survey these contributions in great detail. Instead, I decided to cover only a few historically interesting cases and to mix these with some anecdotes. The problem of solving linear equations comes up almost everywhere in mathematics; many optimization algorithms need fast subroutines for this task. It is hence not surprising that many algorithms for solving linear equations have been designed throughout history; and it is not so clear who invented what first and which algorithm version should carry which name. The most prominent algorithm is often called Gaussian elimination, although Gauss never claimed to have invented this method. One article in this section highlights the appearance of Gaussian elimination in China more than 2000 years ago. Another important algorithm is the Newton method. Many algorithms in optimization try to mimic this method in some way with the aim to avoid its unwanted properties and to maintain its quadratic convergence speed. One Documenta Mathematica

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article tries to clarify whether Newton really invented the algorithm named after him. It is impossible to omit the birth of the calculus of variations in a book like this. And therefore, the interesting story around the invention of the brachistochrone is outlined. All this happened in 1696 and was induced by a challenge put by Johann Bernoulli to his fellow mathematicians. Similarly, the birth of graph theory in 1736 cannot by skipped. Euler, though, missed to view the K¨ onigsberg bridges problem as an optimization problem and thus did not become the father of combinatorial optimization. It is somewhat surprising to learn that it took more than 200 years until an optimization version of Euler’s graph problem was considered. This happened in China. It is outside the scope of this book to sketch the monumental contributions of giants such as Euler and Leibniz. Many voluminous books cover aspects of their work. Three more articles, two on Euler and one on Leibniz, of this section on the old masters are of somewhat anecdotal nature. Two articles discuss the struggle of Euler and Leibniz with “infinity” and one displays a slight human weakness of Euler. Did he cheat a bit in dealing with state authorities? Martin Gr¨otschel

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Jiu Zhang Suan Shu and the Gauss Algorithm for Linear Equations Ya-xiang Yuan

2010 Mathematics Subject Classification: 01A25, 65F05 Keywords and Phrases: Linear equations, elimination, mathematics history, ancient China Jiu Zhang Suan Shu, or The Nine Chapters on the Mathematical Art, is an ancient Chinese mathematics book, which was composed by several generations of scholars from the tenth to the second century BC. Liu Hui (225–295), one of the greatest mathematicians of ancient China, edited and published The Nine Chapters on the Mathematical Art (Jiu Zhang Suan Shu) in the year 263. In the preface of that book [5], Liu Hui gave a detailed account of the history of the book, including the following sentences: When Zhou Gong1 set up the rules for ceremonies, nine branches of mathematics were emerged, which eventually developed to the Nine Chapters of the Mathematical Art. Brutal Emperor Qin Shi Huang2 burnt books, damaging many classical books, including the Nine Chapters. Later, in Han Dynasty, Zhang Cang3 and Geng Shou Chang were famous for their mathematical skills. Zhang Cang and others re-arranged and edited the Nine Chapters of Mathematical Art based on the damaged original texts. From what Liu Hui recorded, we can clearly infer that Zhang Cang played an important role in composing The Nine Chapters of Mathematical Art, and that the current version of the book remains more or less the same as it was in the 2nd century BC, but may not be the same as it had been before the Qin Dynasty. The contents of The Nine Chapters of Mathematical Art are the followings: 1 Zhou Gong, whose real name was Ji Dan, was the fourth son of the founding King of the Zhou Dynasty, Zhou Wen Wang (C. 1152BC – 1056BC). 2 Qin Shi Huang (259BC – 210BC) was the first emperor of China, whose tomb in XiAn is famous for its annex Terracotta Army. 3 Zhang Cang (256BC – 152BC), was a politician, mathematician and astronomer. He was once the prime minister of Western Han.

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Ya-xiang Yuan

Figure 1: Liu Hui (225–295) • Chapter 1, Fang Tian (Rectangular field). • Chapter 2, Su Mi (Millet and rice). • Chapter 3, Cui Fen (Proportional distribution). • Chapter 4 Shao Guang (Lesser breadth). • Chapter 5, Shang Gong (Measuring works). • Chapter 6, Jun Shu (Equitable transportation). • Chapter 7, Ying Bu Zu (Surplus and deficit). • Chapter 8, Fang Cheng (Rectangular arrays). • Chapter 9, Gou Gu (Base and altitude). Many elegant mathematical techniques are discussed in The Nine Chapters on the Mathematical Art. For example, Chapter 9 is about problems of measuring length or height of objects by using properties of right-angled triangles. The main theorem of Chapter 9 is the Gou Gu theorem, which is known in the West as the Pythagorean theorem. Chapter 8 of the book, Fang Cheng, is dedicated to solve real-life problems such as calculating yields of grain, numbers of domestic animals, and prices of different products by solving linear equations. There are 18 problems in the chapter. Problem 13 is essentially an under-determined linear system (6 variables and 5 equations), the other 17 problems are problems which can be formulated as well-defined linear equations with variables ranging from 2 to 5. Documenta Mathematica

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Figure 2: Problem 1, Chapter 8 of Jiu Zhang Suan Shu

The technique given in the chapter for solving these problems is elimination, which is exactly the same as the so-called Gauss elimination in the West. For example, Problem 1 in the chapter states as follows: Problem I. There are three grades of grain: top, medium and low. Three sheaves of top-grade, two sheaves of medium-grade and one sheaf of low-grade are 39 Dous4 . Two sheaves of top-grade, three sheaves of medium-grade and one sheaf of low-grade are 34 Dous. One sheaf of top-grade, two sheaves of medium-grade and three sheaves of low-grade are 26 Dous. How many Dous does one sheaf of top-grade, medium-grade and low-grade grain yield respectively? In the book, the solution is given right after the problem is stated. Afterwards, Liu Hui gave a detailed commentary about the algorithm for solving the problem. The algorithm described in the book is as follows. Putting three sheaves of top-grade grain, two sheaves of mediumgrade grain, and one sheaf of low-grade grain and the total 39 Dous in a column on the right, then putting the other two columns in the middle and on the left. 4 Dou,

a unit of dry measurement for grain in ancient China, is one deciliter. Documenta Mathematica

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This gives the following array: 1 2 3 26

2 3 1 34

3 2 1 39

Then, the algorithm continues as follows. Multiplying the middle column by top-grade grain of the right column, then eliminating top-grade grain from the middle column by repeated subtracting the right column. This gives the following tabular: 1 2 3 26

2×3 3×3 1×3 34 × 3

3 2 1 39

=⇒

1 2 3 26

6−3−3 9−2−2 3−1−1 102 − 39 − 39

3 2 1 39

1 2 3 26

=⇒

5 1 24

3 2 1 39

From the above tabular, we can see that the top position in the middle column is already eliminated. Calculations in ancient China were done by moving small wood or bamboo sticks (actually, the Chinese translation of operational research is Yun Chou which means moving sticks), namely addition is done by adding sticks, and subtraction is done by taking away sticks. Thus, when no sticks are left in a position (indicating a zero element), this place is eliminated. The algorithm goes on as follows. Similarly, multiplying the right column and also doing the subtraction. The above sentence yields the following tabular: 1×3−3 2×3−2 3×3−1 26 × 3 − 39

5 1 24

3 2 1 39

=⇒

4 8 39

5 1 24

3 2 1 39

Then, multiplying the left column by medium-grade grain of the middle column, and carrying out the repeated subtraction. 4×5−5×4 8×5−1×4 39 × 5 − 24 × 4

5 1 24

3 2 1 39

=⇒

36 99

5 1 24

3 2 1 39

Now the remaining two numbers in the left column decides the yield of low-grade grain: the upper one is the denominator, the lower one is the numerator. Documenta Mathematica

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Figure 3: Algorithm descriptions, Chapter 8 of Jiu Zhang Suan Shu Thus, the yield of low-grade grain = 99/36 = 2 34 Dous. The algorithm continues as follows. Now, to obtain the yield of medium-grade grain from the middle column, the denominator is the top number, and the numerator is the bottom number minus the middle number times the yield of lowgrade grain. Therefore, the yield of medium-grade grain = [24 − 1 × 2 43 ]/5 = 4 14 Dous. To calculate the yield of top-grade grain by the right column, the denominator is the top number, and the numerator is the bottom number minus the second number times the yield of medium-grade grain and the third number times the yield of low-grade grain. Consequently, the yield of top-grade grain = [39 − 2 × 4 41 − 1 × 2 34 ]/3 = 9 14 Dous. It is easy to see that the above procedure is exactly the same as the Gauss elimination [2] for the following linear equations: 3x + 2y + z = 39 2x + 3y + z = 34 x + 2y + 3z = 26 Documenta Mathematica

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The only difference is the way in which the numbers are arranged in the arrays. To be more exact, if we rotate all the above rectangular arrays anti-clockwise 90 degree, we obtain the corresponding matrices of the Gauss elimination. This is not unexpected, as in ancient China, people wrote from top to bottom, and then from right to left, while in the West, people write from left to right and then from top to bottom. Thus, from the algorithm description in Chapter 8 of The Nine Chapters on the Mathematical Art, we conclude that the Gauss elimination was discovered at least over 2200 years ago in ancient China. Recently, more and more western scholars [1, 6] credit this simple yet elegant elimination algorithm to ancient Chinese mathematicians. For detailed history of the Gauss elimination, there are two very good review papers [3, 4], where many interesting stories are told. Acknowledgement. I would like to my colleague, Professor Wenlin Li for providing all the pictures used in this article. References [1] P. Gabriel, Matrizen Geometrie Lineare Algebra, Birkh¨auser, 1997. [2] G.H. Golub and C.F. Van Loan, Matrix Computations (3rd ed.), Johns Hopkins, 1996. [3] J.F. Grcar, How ordinary elimination became Gaussian elimination, Historia Mathematica 38:(2), 163–218, 2011. [4] J.F. Grcar, Mathematicians of Gaussian elimination, Notices of the American Mathematical Society, 58:(6), 782–792, 2011. [5] H. Liu, Jiu Zhang Suan Shu Zhu, (in Chinese), 263. [6] C.D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.

Ya-xiang Yuan Academy of Mathematics and Systems Science Chinese Academy of Sciences Zhong Guan Cun Donglu 55 Beijing 100190 China [emailprotected]

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Leibniz and the Brachistochrone Eberhard Knobloch

2010 Mathematics Subject Classification: 01A45, 49-03 Keywords and Phrases: Leibniz, Johann Bernoulli, Galileo, cycloid, calculus squabble 1696 was the year of birth of the calculus of variations. As usual in those days, the Swiss mathematician Johann Bernoulli, one of Leibniz’s closest friends and followers, issued a provocative mathematical challenge in the scholarly journal Acta Eruditorum (Transactions of scholars) in June 1696 inviting the mathematicians to solve this new problem: Given two points A and B in a vertical plane, find the path AMB down which a movable point M must by virtue of its weight fall from A to B in the shortest possible time. In order to encourage “the enthusiasts of such things” (harum rerum amatores) Bernoulli emphasized the usefulness of the problem not only in mechanics but also in other sciences and added that the curve being sought is not the straight line but a curve well-known to geometers. He would publicize it by the end of the year if nobody should publicize it within this period. When Bernoulli published his challenge he did not know that Galileo had dealt with a related problem without having in mind Bernoulli’s generality. And he could not know that his challenge would lead to one of the most famous priority disputes in the history of mathematics. He communicated the problem to Leibniz in a private letter, dated June 19, 1696 and dispatched from Groningen in the Netherlands, asking him to occupy himself with it. Leibniz sent him his answer, together with the correct solution, just one week later on June 26 from Hannover. He proposed the name tachystoptota (curve of quickest descent), avowing that the problem is indeed most beautiful and that it had attracted him against his will and that he hesitated because of its beauty like Eve before the apple. He deduced the correct differential equation but failed to recognize that the curve was a cycloid until Bernoulli informed him in his answer dating from July 31. He took up Leibniz’s biblical reference adding that he was very happy about this comparison provided that he was not regarded as the snake that had offered the apple. Leibniz must certainly have been happy to hear that the curve is the Documenta Mathematica

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Figure 1: Bernoulli’s figure of the brachistochrone (Die Streitschriften von Jacob und Johann Bernoulli, Variationsrechnung. Bearbeitet und kommentiert von Herman H. Goldstine, mit historischen Anmerkungen von Patricia Radeletde Grave. Basel-Boston-Berlin 1991, 212)

cycloid, for which Huygens had shown the property of isochronism. For that reason he, Bernoulli, had given it the name brachystochrona. Leibniz adopted Bernoulli’s description. On June 28 he had already communicated the problem to Rudolf Christian von Bodenhausen in Florence, again praising its extraordinary beauty in order to encourage the Italian mathematicians to solve it. In Switzerland Jacob Bernoulli, and in France Pierre Varignon, had been informed. He asked Johann Bernoulli to extend the deadline until June 1697 because in late autumn 1696 the existence of only three solutions, by Johann and his elder brother Jacob Bernoulli and by himself, were known. Bernoulli agreed insofar as he published a new announcement in the December issue of the Acta Eruditorum that he would suppress his own solution until Easter 1697. In addition to that he wrote a printed leaflet that appeared in January 1697. The May 1697 issue of the Acta Eruditorum contained an introductory historical paper by Leibniz on the catenary and on the brachistochrone. He renounced the publication of his own solution of the brachistochrone problem because it corresponded, he said, with the other solutions (cum caeteris consentiat). Then the five known solutions by Johann, Jacob Bernoulli, the Marquis de l’Hospital, Ehrenfried Walther von Tschirnhaus, and Isaac Newton were published or reprinted (Newton). Newton had not revealed his name. But Johann Bernoulli recognized the author, “from the claw of the lion” (ex ungue leonem), as he said. Documenta Mathematica

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Figure 2: Galileo’s figure regarding the fall of a particle along a circular polygon (Galileo Galilei: Le opere, vol. VIII, Firenze 1965, 262)

Leibniz made some statements in his paper that are worth discussing. First of all he maintained that Galileo had already dealt with the catenary and with the brachistochrone as well, without being able to find the correct solution. He had falsely identified the catenary with a parabola and the brachistochrone with a circular arc. Unfortunately Johann Bernoulli relied on Leibniz’s false statement and repeated it in June 1697, and later so did many other authors up to the present time. Neither the one nor the other assertion is in reality true. What had Galileo really said in his Discorsi? He had rightly emphasized the similarity between the catenary and a parabola. He did not and could not look for the curve of quickest descent, that is, for the brachistochrone. Such a general problem was still beyond the mathematical horizon of the mathematicians of his time. He had considered an arc of a circle CBD of not more than 90° in a vertical plane with C the lowest point on the circle, D the highest point and B any other point on the arc of the circle. He proved the correct theorem that the time for a particle to fall along the broken line DBC is less than the time for it to descend along the line DC. Let us enlarge the number of points on the circle between D and C. The larger the number of points is, the less is the time for the particle to descend along the broken line DEF G . . . C. For Galileo a circle was a polygon with infinitely many, infinitely small sides. Hence he rightly concluded that the swiftest time of fall from D to C is along a portion of the circle. Galileo only compared the times of fall along the sides of circular polygons the circle being the limit case of them. Secondly, Leibniz said that the only mathematicians to have solved the problem are those he had guessed would be capable of solving it; in other words, Documenta Mathematica

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only those who had suffiently penetrated in the mysteries of his differential calculus. This he had predicted for the brother of Johann Bernoulli and the Marquis de l’Hospital, for Huygens if he were alive, for Hudde if he had not given up such pursuits, for Newton if he would take the trouble. The words were carelessly written because their obvious meaning was that Newton was indebted to the differential calculus for his solution. Even if Leibniz did not want to make such a claim, and this is certain in 1697, his words could be interpreted in such a way. There was indeed a reader who chose this interpretation: the French emigrant Nicolas Fatio de Duillier, one of Newton’s closest followers. Fatio was deeply offended at not having been mentioned by Leibniz among those authors who could master the brachistochrone problem. In 1699 he published a lengthy analysis of the brachistochrone. Therein he praised his own mathematical originality and sharply accused Leibniz of being only the second inventor of the calculus. Fatio’s publication was the beginning of the calculus squabble. But this is another story. References [1] H. H. Goldstine, Introduction, in: Die Streitschriften von Jacob und Johann Bernoulli, Variationsrechnung, bearbeitet und kommentiert von H. H. Goldstine mit historischen Anmerkungen von P. Radelet-de Grave, Birkh¨auser, Basel-Boston-Berlin 1991, pp. 1–113. [2] E. Knobloch, Le calcul leibnizien dans la correspondance entre Leibniz et Jean Bernoulli. in: G. Abel, H.-J. Engfer, C. Hubig (eds.), Neuzeitliches Denken, Festschrift f¨ ur Hans Poser zum 65. Geburtstag, W. de Gruyter, Berlin-New York 2002, pp. 173–193. [3] Eberhard Knobloch, Galilei und Leibniz, Wehrhahn, Hannover 2012. [4] Jeanne Peiffer, Le probl`eme de la brachystochrone `a travers les relations de Jean I. Bernoulli avec l’Hospital et Varignon, in: H.-J. Hess, F. Nagel (eds.), Der Ausbau des Calculus durch Leibniz und die Br¨ uder Bernoulli. Steiner, Wiesbaden 1989, pp. 59–81 (= Studia Leibnitiana Sonderheft 17).

Eberhard Knobloch Berlin-Brandenburg Academy of Sciences and Humanities Technische Universit¨at Berlin H 72 Straße des 17. Juni 135 10623 Berlin [emailprotected]

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Leibniz and the Infinite Eberhard Knobloch

2010 Mathematics Subject Classification: 01A45, 28-03 Keywords and Phrases: Leibniz, infinite, infinitely small, mathematical rigour, integration theory The German universal genius Gottfried Wilhelm Leibniz was born in Leipzig on the 21st of June according to the Julian calendar (on the 1st of July according to the Gregorian calendar) 1646. From 1661 he studied at the universities of Leipzig and Jena. On February 22, 1667 he became Doctor of Laws at the university of N¨ urnberg-Altdorf. He declined the professorship that was offered to him at this university. For a short time he accepted a position at the court of appeal of Mainz. From 1672 to 1676 he spent four years in Paris where he invented his differential and integral calculus in autumn 1675. From 1676 up to the end of his life he earned his living as librarian at the court of the duke, then elector, of Hannover. In 1700 he was appointed president of the newly founded Electoral Academy of Sciences of Berlin. He contributed to nearly all scientific disciplines and left the incredibly huge amount of about 200 000 sheets of paper. Less than one half of them have been published up to now. In Paris he became one of the best mathematicians of his time within a few years. He was especially interested in the infinite. But what did he mean by this notion? His comments on Galileo’s Discorsi give the answer. Therein Galileo had demonstrated that there is a one-to-one correspondence between the set of the natural numbers and the set of the square numbers. Hence in his eyes the Euclidean axiom “The whole is greater than a part” was invalidated in the sense that it could not be applied there: infinite sets cannot be compared with each other with regard to their size. Leibniz contradicted him. For him it was impossible that this axiom failed. This only seemed to be the case because Galileo had presupposed the existence of actually infinite sets. For him the universal validity of rules was more important than the existence of objects, in this case of actually infinite numbers or actually infinite sets. Hence Leibniz did not admit actual infinity in mathematics. “Infinite” meant “larger than any given quantity”. He used the mode of possibility in order to characterize the mathematical infinite: it is always possible to find a quantity that is larger than any given quantity. Documenta Mathematica

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Figure 1: Portrait of Leibniz by A. Scheit, 1703 (By courtesy of the Gottfried Wilhelm Leibniz Library, Hannover)

By using the mode of possibility he consciously imitated ancient models like Aristotle, Archimedes, and Euclid. Aristotle had defined the notion of quantity in his Metaphysics: quantity is what can be divided into parts being in it. Something (a division) can be done in this case. If a division of a certain object is not possible, the object cannot be a quantity. In the 17th and 18th centuries mathematics was the science of quantities. Hence it could not handle non-quantities. Hence Leibniz avoided non-quantities in mathematics by all means. Indivisibles were non-quantities by definition: they cannot be divided. Yet they occurred even in the title of Bonaventura Cavalieri’s main work Geometry developed by a new method by means of the indivisibles of continua. Cavalieri’s indivisibles were points of a line, straight lines of a plane, planes of a solid. Leibniz amply used this notion, for example in the title of the first publication of his integral calculus Analysis of indivisibles and infinites that appeared in 1686. But according to his mathematical convictions he had to look for a suitable, new interpretation of the notion. From 1673 he tried different possibilities like smallest, unassignable magnitude, smaller than any assignable quantity. He rightly rejected all of them because there are no smallest quantities and because a quantity that is smaller than any assignable quantity is equal to zero or nothing. In spring 1673 he Documenta Mathematica

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finally stated that indivisibles have to be defined as infinitely small quantities or the ratio of which to a perceivable quantity is infinite. Thus he had shifted the problem. Now he had to answer the question: What does it mean to be infinitely small? Still in 1673 he gave an excellent answer: infinitely small means smaller than any given quantity. He again used the mode of possibility and introduced a consistent notion. Its if-then structure – if somebody proposes a quantity, then there will be a smaller quantity – rightly reminds the modern reader of Weierstraß’s ǫ-δ-language. Leibniz’s language can be translated into Weierstraß’s language. Leibniz used this well-defined notion throughout the longest mathematical treatise he ever wrote, in his Arithmetical quadrature of the circle, of the ellipse, and of the hyperbola. Unfortunately it remained unpublished during his lifetime though he wrote it already in the years 1675/76. Only in 1993 did the first printed version appear in G¨ottingen. For this reason Leibniz has been falsely accused of neglecting mathematical rigour again and again up to the present day. His Arithmetical quadrature contains the counterdemonstration of that false criticism. Therein theorem 6 gives a completely rigorous foundation of infinitesimal geometry by means of Riemannian sums. Leibniz foresaw its deterrent effect saying: The reading of this proposition can be omitted if somebody does not want supreme rigour in demonstrating proposition 7. And it will be better that it be disregarded at the beginning and that it be read only after the whole subject has been understood, in order that its excessive exactness does not discourage the mind from the other, far more agreeable, things by making it become weary prematurely. For it achieves only this: that two spaces of which one passes into the other if we progress infinitely, approach each other with a difference that is smaller than any arbitrary assigned difference, even if the number of steps remains finite. This is usually taken for granted, even by those who claim to give rigorous demonstrations. Leibniz referred to the ancients like Archimedes who was still the model of mathematical rigour. After the demonstration Leibniz stated: “Hence the method of indivisibles which finds the areas of spaces by means of sums of lines can be regarded as proven.” He explicitly formulated the fundamental idea of the differential calculus, that is, the linearization of curves: The readers will notice what a large field of discovery is opened up once they have well understood only this: that every curvilinear figure is nothing but a polygon with infinitely many infinitely small sides. When he published his differential calculus for the first time in 1684 he repeated this crucial idea. From that publication he had to justify his invention. In 1701 he rightly explained: Documenta Mathematica

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Figure 2: First page of Leibniz’s treatise Arithmetical quadrature of the circle etc. (By courtesy of the Gottfried Wilhelm Leibniz Library, Hannover. Shelf mark LH XXXV 2,1 folio 7r)

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Because instead of the infinite and the infinitely small one takes quantities that are as large or as small as it is necessary so that the error is smaller than the given error so that one differs from the style of Archimedes only by the expressions which are more direct in our method and more suitable for the art of invention. The story convincingly demonstrates the correctness of his saying: “Those who know me only by my publications don’t know me.” References [1] E. Knobloch, Leibniz’s rigorous foundation of infinitesimal geometry by means of Riemannian sums, Synthese 133 (2002), 59–73. [2] [2] E. Knobloch, Galileo and German thinkers: Leibniz, in: L. Pepe (ed.), Galileo e la scuola galileiana nelle Universit`a del Seicento, Cooperativa Libraria Universitaria Bologna, Bologna 2011, pp. 127–139.

Eberhard Knobloch Berlin-Brandenburg Academy of Sciences and Humanities Technische Universit¨at Berlin H 72 Straße des 17. Juni 135 10623 Berlin [emailprotected]

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A Short History of Newton’s Method Peter Deuflhard

2010 Mathematics Subject Classification: 01A45, 65-03, 65H05, 65H10, 65J15, 65K99 Keywords and Phrases: History of Newton’s method, Simpson, Raphson, Kantorovich, Mysoskikh, geometric approach, algebraic approach

If an algorithm converges unreasonably fast, it must be Newton’s method. John Dennis (private communication)

It is an old dream in the design of optimization algorithms, to mimic Newton’s method due to its enticing quadratic convergence. But: Is Newton’s method really Newton’s method? Linear perturbation approach Assume that we have to solve a scalar equation in one variable, say f (x) = 0 with an appropriate guess x0 of the unknown solution x∗ at hand. Upon introducing the perturbation ∆x = x∗ − x0 , Taylor’s expansion dropping terms of order higher than linear in the perturbation, yields the approximate equation . f ′ (x0 )∆x = −f (x0 ) , which may lead to an iterative equation of the kind xk+1 = xk − Documenta Mathematica

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assuming the denominator to be non-zero. This is usually named Newton’s method. The perturbation theory carries over to rather general nonlinear operator equations, say F (x) = 0,

x ∈ D ⊂ X,

F : D → Y,

where X, Y are Banach spaces. The corresponding Newton iteration is then typically written in the form F ′ (xk )∆xk = −F (xk ),

xk+1 = xk + ∆xk ,

k = 0, 1, . . .

For more details and extensions see, e.g., the textbook [1] and references therein. Convergence From the linear perturbation approach, local quadratic convergence will be clearly expected for the scalar case. For the general case of operator equations F (x) = 0, the convergence of the generalized Newton scheme has first been proven by two Russian mathematicians: In 1939, L. Kantorovich [5] was merely able to show local linear convergence, which he improved in 1948/49 to local quadratic convergence, see [6, 7]. Also in 1949, I. Mysovskikh [9] gave a much simpler independent proof of local quadratic convergence under slightly different theoretical assumptions, which are exploited in modern Newton algorithms, see again [1]. Among later convergence theorems the ones due to J. Ortega and W.C. Rheinboldt [11] and the affine invariant theorems given in [2, 3] may be worth mentioning. Geometric approach The standard approach to Newton’s method in elementary textbooks is given in Figure 1. It starts from the fact that any root of f may be interpreted as the intersection of the graph of f (x) with the real axis. In Newton’s method, this graph is replaced by its tangent in x0 ; the first iterate x1 is then defined as the intersection of the tangent with the real axis. Upon repeating this geometric process, a close-by solution point x∗ can be constructed to any desired accuracy. On the basis of this geometric approach, this iteration will converge globally for convex (or concave) f . At first glance, this geometric derivation seems to be restricted to the scalar case, since the graph of f (x) is a typically one-dimensional concept. A careful examination of the subject in more than one dimension, however, naturally leads to a topological path called Newton path, which can be used for the construction of modern adaptive Newton algorithms, see again [1]. Documenta Mathematica

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f

x0

x∗

f (x0 )

x

Figure 1: Newton’s method for a scalar equation Historical road The long way of Newton’s method to become Newton’s method has been well studied, see, e.g., N. Kollerstrom [8] or T.J. Ypma [13]. According to these articles, the following facts seem to be agreed upon among the experts: • In 1600, Francois Vieta (1540–1603) had designed a perturbation technique for the solution of the scalar polynomial equations, which supplied one decimal place of the unknown solution per step via the explicit calculation of successive polynomials of the successive perturbations. In modern terms, the method converged linearly. It seems that this method had also been published in 1427 by the Persian astronomer and mathematician al-K¯ ash¯ı (1380–1429) in his The Key to Arithmetic based on much earlier work by al-Biruni (973–1048); it is not clear to which extent this work was known in Europe. Around 1647, Vieta’s method was simplified by the English mathematician Oughtred (1574–1660). • In 1664, Isaac Newton (1643–1727) got to know Vieta’s method. Up to 1669 he had improved it by linearizing the successively arising polynomials. As an example, he discussed the numerical solution of the cubic polynomial f (x) := x3 − 2x − 5 = 0 . Newton first noted that the integer part of the root is 2 setting x0 = 2. Next, by means of x = 2 + p, he obtained the polynomial equation p3 + 6p2 + 10p − 1 = 0 . He neglected terms higher than first order setting p ≈ 0.1. Next, he inserted p = 0.1 + q and constructed the polynomial equation q 3 + 6.3q 2 + 11.23q + 0.061 = 0 . Again neglecting higher order terms he found q ≈ −0.0054. Continuation of the process one further step led him to r ≈ 0.00004853 and therefore to the third iterate x3 = x0 + p + q + r = 2.09455147 . Documenta Mathematica

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Peter Deuflhard Note that the relations 10p − 1 = 0 and 11.23q + 0.061 = 0 given above correspond precisely to p = x1 − x0 = −f (x0 )/f ′ (x0 ) and to q = x2 − x1 = −f (x1 )/f ′ (x1 ) . As the example shows, he had also observed that by keeping all decimal places of the corrections, the number of accurate places would double per each step – i.e., quadratic convergence. In 1687 (Philosophiae Naturalis Principia Mathematica), the first nonpolynomial equation showed up: it is the well-known equation from astronomy x − e sin(x) = M between the mean anomaly M and the eccentric anomaly x. Here Newton used his already developed polynomial techniques via the series expansion of sin and cos. However, no hint on the derivative concept is incorporated! • In 1690, Joseph Raphson (1648–1715) managed to avoid the tedious computation of the successive polynomials, playing the computational scheme back to the original polynomial; in this now fully iterative scheme, he also kept all decimal places of the corrections. He had the feeling that his method differed from Newton’s method at least by its derivation. • In 1740, Thomas Simpson (1710–1761) actually introduced derivatives (‘fluxiones’) in his book ‘Essays on Several Curious and Useful Subjects in Speculative and Mix’d Mathematicks [No typo!], Illustrated by a Variety of Examples’. He wrote down the true iteration for one (nonpolynomial) equation and for a system of two equations in two unknowns thus making the correct extension to systems for the first time. His notation is already quite close to our present one (which seems to date back to J. Fourier).

The interested reader may find more historical details in the book by H. H. Goldstine [4] or even try to read the original work by Newton in Latin [10]; however, even with good knowledge of Latin, this treatise is not readable to modern mathematicians due to the ancient notation. That is why D.T. Whiteside [12] edited a modernized English translation. What is Newton’s method? Under the aspect of historical truth, the following would come out: • For scalar equations, one might speak of the Newton–Raphson method. • For more general equations, the name Newton–Simpson method would be more appropriate. Documenta Mathematica

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Under the convergence aspect, one might be tempted to define Newton’s method via its quadratic convergence. However, this only covers the pure Newton method. There are plenty of variants like the simplified Newton method, Newton-like methods, quasi-Newton methods, inexact Newton methods, global Newton methods etc. Only very few of them exhibit quadratic convergence. In fact, even the Newton–Raphson algorithm for scalar equations as realized in hardware within modern calculators converges only linearly due to finite precision, which means they asymptotically implement some Vieta algorithm. Hence, one will resort to the fact that Newton methods simply exploit derivative information in one way or the other. Acknowledgement The author wishes to thank E. Knobloch for having pointed him to several interesting historical sources. References [1] P. Deuflhard. Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, volume 35 of Computational Mathematics. Springer International, 2004. [2] P. Deuflhard and G. Heindl. Affine Invariant Convergence theorems for Newton’s Method and Extensions to related Methods. SIAM J. Numer. Anal., 16:1–10, 1979. [3] P. Deuflhard and F.A. Potra. Asymptotic Mesh Independence of NewtonGalerkin Methods via a Refined Mysovskii Theorem. SIAM J. Numer. Anal., 29:1395–1412, 1992. [4] H. H. Goldstine. A history of Numerical Analysis from the 16th through the 19th Century. Spri9nger, 1977. [5] L. Kantorovich. The method of successive approximations for functional equations. Acta Math., 71:63–97, 1939. [6] L. Kantorovich. On Newton’s Method for Functional Equations. (Russian). Dokl. Akad. Nauk SSSR, 59:1237–1249, 1948. [7] L. Kantorovich. On Newton’s Method. (Russian). Steklov, 28:104–144, 1949.

Trudy Mat. Inst.

[8] N. Kollerstrom. Thomas Simpson and ‘Newton’s Method of Approximation’: an enduring myth. British Journal for History of Science, 25:347– 354, 1992. [9] I. Mysovskikh. On convergence of Newton’s method. (Russian). Trudy Mat. Inst. Steklov, 28:145–147, 1949. Documenta Mathematica

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[10] I. Newton. Philosophiae naturalis principia mathematica. Colonia Allobrogum: sumptibus Cl. et Ant. Philibert, 1760. [11] J.M. Ortega and W.C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Classics in Appl. Math. SIAM Publications, Philadelphia, 2nd edition, 2000. [12] D.T. Whiteside. The Mathematical Papers of Isaac Newton (7 volumes), 1967–1976. [13] T.J. Ypma. Historical Development of the Newton-Raphson Method. SIAM Rev., 37:531–551, 1995.

Peter Deuflhard Konrad-Zuse-Zentrum f¨ ur Informationstechnik Berlin (ZIB) Takustraße 7 14195 Berlin Germany [emailprotected]

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Euler and Infinite Speed Eberhard Knobloch

2010 Mathematics Subject Classification: 01A50, 70-03 Keywords and Phrases: Euler, Galileo, Maupertuis, tunnel through the earth, damped oscillation The famous Swiss mathematician Leonhard Euler was born in Basel on the 15th of April 1707. Already in 1720 when he was still a thirteen-year-old boy, he enrolled at the University of Basel. One year later, he obtained the Bachelor’s degree. In 1723 when he was sixteen years old, he obtained his Master’s degree (A. L. M. = Master of Liberal Arts). In 1727 without ever having obtained the Ph. D. degree he submitted a short habilitation thesis (consisting of fifteen pages); that is, a thesis in application for the vacant professorship of physics at the University of Basel. At that time he had published two papers, one of them being partially faulty. No wonder that the commission which looked for a suitable candidate for the professorship did not elect him. Yet Euler was very much infuriated by this decision. Still in 1727, he went to St. Petersburg in order to work at the newly founded Academy of Sciences. He never came back to Switzerland. Between 1741 and 1766 he lived and worked in Berlin at the reformed Academy of Sciences and Literature of Berlin. In 1766 he returned to St. Petersburg where he died on the 18th of September 1783. The complete title of his habilitation thesis reads: May it bring you happiness and good fortune – Physical dissertation on sound which Leonhard Euler, Master of the liberal arts submits to the public examination of the learned in the juridical lecture-room on February 18, 1727 at 9 o’clock looking at the free professorship of physics by order of the magnificent and wisest class of philosophers whereby the divine will is nodding assent. The most eminent young man Ernst Ludwig Burchard, candidate of philosophy, is responding. As we know, all imploring was in vain: Euler did not get the position. The thesis is all the more interesting because Euler had added a supplement in which he formulated six statements regarding utterly different subjects. For example he maintained that Leibniz’s theory of preestablished harmony between body Documenta Mathematica

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Figure 1: Leonhard Euler (1707–1783) (L. Euler, Opera omnia, series I, vol. 1, Leipzig – Berlin 1911, Engraving after p. I)

and soul is false, without mentioning the name of his illustrious predecessor. Another statement prescribed the construction of a ship mast. The third statement considered a thought experiment: What would happen at the centre of the earth if a stone were dropped into a straight tunnel drilled to the centre of the earth and beyond to the other side of the planet? Euler distinguished between exactly three possiblities: Either the stone will rest at the centre or will at once proceed beyond it or it will immediately return from the centre to us. There is no mention of speed. Euler just stated that the last case will take place. No justification or explanation is given, though none of these three possibilities had the slightest evidence. What is worse, a far better answer had been already given by Galileo in 1632. In the second day of his Dialogue about the two main world systems Galileo discussed this thought experiment in order to refute Aristotle’s distinction between natural and unnatural motions. The natural motion of heavy bodies is the straight fall to the centre of the earth. But what about a cannon ball Documenta Mathematica

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Figure 2: Title page of Euler’s Physical dissertation on sound (L. Euler, Opera omnia, series III, vol. 1, Leipzig – Berlin 1926, p. 181)

that has dropped into such an earth tunnel? Even the Aristotelian Simplicio avowed that the cannon ball would reach the same height from which it had dropped into the tunnel in the other half of the tunnel. The natural motion would change into an unnatural motion. Galileo erroneously presupposed a constant gravitation. But he rightly deduced an oscillating motion of the cannon ball. Euler did not mention the Italian mathematician. Presumably he did not know his solution of the thought experiment. Nine years later he came back to this question in his Mechanics or the science of motion set forth analytically. Now he explicitly concluded that the speed of the falling stone will become infinitely large in the centre of the earth. Nevertheless it will immediately return to the starting-point. Euler admitted: This seems to differ from truth because hardly any reason is obvious why a body, having infinitely large speed that it has acquired in C, should proceed to any other region than to CB, especially because the direction of the infinite speed turns to this region. However that may be, here we have to be conf´ıdent more in the calculation than in our judgement and to confess that we do not understand at all the jump if it is done from the infinite into the finite. Documenta Mathematica

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Figure 3: L. Euler, Mechanics, vol. 1, 1736, § 272 (Explanation 2) (L. Euler, Opera omnia, series II, vol. 1, Leipzig – Berlin 1912, p. 88)

Euler’s result was the consequence of his mathematical modelling of the situation (an impermissible commutation of limits). When in 1739 Benjamin Robbins wrote his review of Euler’s Mechanics he put as follows: When y, the distance of the body from the center, is made negative, the terms of the distance expressed by yn , where n may be any number affirmative, or negative, whole number or fraction, are sometimes changed with it. The centripetal force being as some power of the fraction; if, when y is supposed negative, yn be still affirmative, the solution gives the velocity of the body in its subsequent ascent from the center; but if yn by this supposition becomes also negative, the solution exhibits the velocity, after the body has passed the center, upon condition, that the centripetal force becomes centrifugal; and when on this supposition yn becomes impossible, the determination of the velocity beyond the center is impossible, the condition being so. The French physicist Pierre-Louis Moreau de Maupertuis was the president of the Academy of Sciences and of Literature in Berlin at the beginning of Euler’s sojourn in Berlin. He unfortunately proposed to construct such an earth tunnel. His proposal was ridiculed by Voltaire on the occasion of the famous quarrel about the principle of least action between Maupertuis, Euler, and the Prussian king Frederick the Great on the one side and the Swiss mathematician Samuel K¨ onig on the other side. Thus Euler’s curious statement about the dropping stone had a satirical aftermath. In 1753 Voltaire published his Lampoon of Documenta Mathematica

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Doctor Akakia. Therein he made Euler regret that he had more conf´ıdence in his calculation than in human judgement. In truth Euler never recanted his solution. References [1] Emil A. Fellmann: Leonhard Euler, translated by Erika Gautschi and Walter Gautschi. Basel – Boston – Berlin 2007. [2] Eberhard Knobloch: Euler – The historical perspective. In: Physica D 237 (2008), 1887–1893. [3] R¨ udiger Thiele: Leonhard Euler. Leipzig 1982.

Eberhard Knobloch Berlin-Brandenburg Academy of Sciences and Humanities Technische Universit¨at Berlin H 72 Straße des 17. Juni 135 10623 Berlin [emailprotected]

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Euler and Variations Eberhard Knobloch

2010 Mathematics Subject Classification: 01A50, 01A70 Keywords and Phrases: Euler, Seven-Years War, Euler’s estate in Lietzow, Euler’s losses When Euler came to Berlin in 1741, accepting the offer of the Prussian king Frederick II. to work at the Berlin academy of sciences, the king himself was at the first Silesian war with Austria. It was only the first of three wars that he waged. Two years later Euler bought a house in the centre of Berlin in the “B¨ arenstraße”, today “Behrenstraße” number 21. There he lived up to 1766 when he left Berlin in order to return to St. Petersburg. Yet already in those days, life was expensive in a city like Berlin. Hence in 1753 he bought an estate outside Berlin in the small village of Lietzow, belonging to the administrative district of Charlottenburg, that is to-day a part of a district of the city of Berlin. He paid 6000 Imperial Taler (Reichsthaler) for it. From then onward his large family lived on this estate, including his widowed mother, while he himself remained in Berlin. Whenever he had Russian students of mathematics they too lived in the house in Berlin: from 1743 to 1744 Kirill Grigorevich Rasumovskii, later president of the Russian Academy of Sciences in St. Petersburg, and Grigorii Nikolaevich Teplov, from 1752 to 1756 sem*n Kirillovich Kotelnikov, in 1754 Michail Sofronov, from 1754 to 1756 Stepan Yakovlevich Rumovskii. It did not happen by chance that 1756 was the year of departure. In 1756 Frederick II. began the Seven-Years War by penetrating into Saxony. His Prussian troops fought against the allied Russian, Saxon, and Austrian troops. Euler carried on sending scientific manuscripts to St. Petersburg – that is, to Russia – and kept his good relations with the academy there. Yet he secretly helped the Prussian king with his knowledge of the Russian language by translating intercepted Russian messages. If the time did not suffice for a diligent translation he offered to summarize the content. For example in September 1758 a courier of the Russian guard was taken captive together with two Cossacks near to Neustettin. They carried seventy-nine letters for the Russian court. Euler’s translation of the report of a Russian agent and of the statements of two Prussian deserters is still kept in the archives of the BerlinBrandenburg Academy of Sciences and Humanities (http://euler.bbaw.de/ euleriana/ansicht.php?seite=216). Documenta Mathematica

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The following years became very difficult for the Prussian king. In 1759 the allied Austrian and Russian troops defeated the troops of Frederick II. in the neighbourhood of Kunersdorf. On October 9, 1760 Russian and Saxon troops temporarily occupied Berlin and plundered the surrounding villages, especially Lietzow, and including Euler’s estate. The command of the Russian Count Chernishef to spare this estate from plunder came too late. Just nine days later, on October 18, 1760 Euler wrote to the historian Gerhard Friedrich M¨ uller in St. Petersburg, since 1754 perpetual secretary of the Russian Academy of Sciences, in order to complain about this robbery and to make a claim for damages. “I have always wished that Berlin should be occupied by Russian troops if it should be ever occupied by foreign troops”, he wrote, “yet the visit of the Russian officers entailed considerable damage.” He told M¨ uller that he had bought an estate for 6000 Imperial Taler in Charlottenburg that was well-known to Mr. Kotelnikov and to Mr. Rumovskii. On the occasion of that visit everything was removed or devastated. Then he enumerated the losses: I have lost four horses, twelve cows, many head of livestock, much oats and hay. All of the furniture of the house has been ruined. This damage is more than 1100 Imperial Taler according to an exact calculation...All in all the damage is at least 1200 roubles. He asked M¨ uller to inform his former student, then president of the Russian Academy, Count Rasumovskii, about his situation and to support his request. He was indeed amply recompensed by the Russian general and by the Russian tsarina Elisabeth. By chance Euler’s statements about his losses can be checked because the mayor of Charlottenburg elaborated a specification of damages for Lietzow and Charlottenburg that has been preserved in the Main Archives of the country Brandenburg of the Federal Republic of Germany in Potsdam. On October 24, 1760, the mayor sent a letter to the responsible Privy Councillor of War and of Domain (Geheimder Krieges und Domainen Rath) saying: As we have been ordered we have added and would like to most obediently submit the specification of money, grain, and cattle that the city of Charlottenburg has lost by the Russian invasion. [Anbefohlener Maßen haben Wir angeschlossen die Specification so wohl an baaren Gelde als an Getreyde und Vieh was die Stadt Charlottenburg durch die Russischen Invasion verlohren haben gehorsamst einreichen sollen.] The list consists of nine columns. They enumerate the names of the twelve families concerned from the village of Lietzow and the robbery of cash currency, rye, barley and oat, hay, horses, cows, pigs, and sheep. The fourth line mentions Euler’s losses reading: Documenta Mathematica

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Figure 1: Letter of the mayor of Charlottenburg dating from October 24, 1760 (By courtesy of the Brandenburgisches Landeshauptarchiv Potsdam, Rep. 2 Kurm¨ arkische Kriegs- und Dom¨anenkammer Nr. S 3498)

Professor Euler: no cash currency; 1 Wispel, 5 Scheffel rye (1 Wispel = 24 Scheffel, 1 Scheffel = 54,73 litres); 1 Wispel, 6 Scheffel barley and oat; 30 metric hundred-weight of hay; two horses; thirteen cows; seven pigs; twelve sheep. Documenta Mathematica

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Figure 2: List of damages regarding the village Lietzow (By courtesy of the Brandenburgisches Landeshauptarchiv Potsdam, Rep. 2 Kurm¨arlische Kriegsund Dom¨ anenkammer Nr. S 3498) The astonished reader notices at once that Euler has doubled the number of stolen horses. In 1763 he had already negotiated with the Russian Academy of Sciences for his return to St. Petersburg, which indeed took place in 1766. For that reason he sold his estate in Charlottenburg for 8500 Imperial Taler, that is, at a profit of more than forty per cent, thus practising again his private calculus of variations. All in all he made a good profit out of his estate. Documenta Mathematica

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Figure 3: Title page of Euler’s book on the calculus of variations (L. Euler, Opera omnia, series I, vol. 24, Bern 1952, p. 1)

Thanks to a letter from Euler to the president Maupertuis of the Berlin Academy of Sciences and Fine Arts from March 14, 1746 we know that Euler had written his official, famous book on the calculus of variations, his Method of finding curves with an extreme property or the solution of the isoperimetric problem understood in the broadest sense, already in St. Petersburg, that is, in spring 1741 at the latest. It appeared in Lausanne in 1744 including the appendix II with Euler’s explanation of the principle of least action. Constantin Carath´eodory called the book one of the most beautiful mathematical works that has ever been written. But that is another story. Documenta Mathematica

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References [1] Eberhard Knobloch: Leonhard Euler 1707–1783, Zum 300. Geburtstag eines langj¨ ahrigen Wahlberliners. In: Mitteilungen der Deutschen MathematikerVereinigung 15 (2007), 276–288.

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¨ nigsberg, Euler, Mei-Ko Kwan, Ko and a Chinese Postman ¨ tschel and Ya-xiang Yuan Martin Gro

2010 Mathematics Subject Classification: 00-02, 01A05, 05C38, 90-03 Keywords and Phrases: Eulerian graphs, Chinese Postman Problem Looking at the world’s history, nothing very important happened in 1736. There was one exception, at least for mathematicians. Leonhard Euler wrote an article [3] with the title “Solutio Problematis ad Geometriam Situs Pertinentis”, a paper of 13 pages with 21 short paragraphs, published in St. Petersburg, Russia. The paper looks like treating a certain puzzle, and it did not receive much attention for a long period of time. Moreover, in his own research Euler never returned to this particular topic. In retrospect, his article on the bridges of K¨ onigsberg laid the foundations of graph theory, a new branch of mathematics, that is today permeating almost every other science, is employed even in daily life, has become a powerful modeling language and a tool that is of particular importance in discrete mathematics and optimization. Euler could have become the father of combinatorial optimization, but he missed this opportunity. A young Chinese mathematician was the first to consider an optimization version of Euler’s bridges problem which was later called the Chinese Postman Problem in his honor. Readers interested in graph theory papers of historic relevance should consult [1] which contains a collection of 37 important articles, translated into English; [3] is the first one in this collection. ¨ nigsberg bridges probLeonhard Euler: When did he solve the Ko lem? We refrain from saying here more than a few words about the life of Leonhard Euler. Almost infinitely many books and papers describe aspects of his work. The article [5] in this book sketches some of the important steps of his career. Clifford Truesdell’s (1919-2000) estimate that Euler produced about one third of all the mathematical literature of the 18th century indicates his distinguished role. But Euler’s interests went far beyond mathematics. He made significant contributions to engineering, cartography, music theory, philosophy, and theology. Documenta Mathematica

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Figure 1: Ehler’s drawing of K¨onigsberg, 1736

There is almost no book in graph theory today that does not show a copy of the map of Regiomonti in Borussia (K¨onigsberg in Prussia, today, Kaliningrad in Russia) that one can find in Euler’s article and that explains how Euler abstracted the concept of a graph from this map. Fig. 1 shows the real original drawing that we obtained from W. Velminski who made a copy for his book [9] in the St. Petersburg archive from the original Ehler letter mentioned below. It is not known for sure when and from whom Euler learned about the K¨ onigsberg bridges for the first time. (Euler, as far as one knows, never visited K¨ onigsberg.) What is known is that he corresponded with Karl Leonhard Gottlieb Ehler about this problem (variations of the name in the literature: Carl instead of Karl and Ehlers instead of Ehler), where Ehler acted as an intermediary between Euler and the mathematician Heinrich K¨ uhn from Danzig. Ehler was a mathematics enthusiast; and he was the mayor of Danzig from 1740 to 1753. A list of 20 letters exchanged in the period 1735 to 1742 between these two can be found at http://eulerarchive.maa.org/correspondence/ correspondents/Ehler.html. The article [8] investigates three letters that Documenta Mathematica

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deal with the K¨ onigsberg bridges and also shows a copy of Fig. 1. This drawing is from the first of these letters, dated March 9, 1736. One may infer from this letter that Euler and Ehler had already discussed the bridges problem, but if Euler had known the details of the problem, it would not have been necessary for Ehler to produce this drawing. And so it is not unreasonable to assume that Euler learned the problem through this letter. This reasoning, though, contradicts the statement in the minutes of the St. Petersburg Academy that Euler presented the K¨ onigsberg bridges problem to the Academy on August 26, 1735. Velminski claims in [9] that this date may be a misprint. Confusion occurs also with respect to the publication date of Euler’s paper. It is contained in the 1736 Academy volume, but the publication was delayed so that the volume only appeared in 1741. What is known, due to still existing letters, see [8], is that Euler outlined his solution of the problem in letters to Giovanni J. Marinoni (March 13, 1736) and to Ehler (April 3, 1736). And so we prefer to regard 1736 as the birth year of graph theory in which the following problem was addressed: ¨ nigsberg Bridges Problem (briefly KBP): The Ko Is it possible for a pedestrian to walk across all seven bridges in K¨ onigsberg without crossing any bridge twice? Euler could have worked hard to solve this particular problem instance by checking cases, but he, and this distinguishes true mathematicians from puzzle solvers, tries to solve this problem type, once and for all, for all possible instances and not just for K¨ onigsberg. He, thus, formulated what we call the Eulerian Path (or Walk) Problem (briefly EPP): Is it possible to traverse a graph passing through every edge exactly once?

Euler’s results Here is a sketch of what Euler did in his paper. Euler mentions the “almost unknown” geometriam situs, a term introduced by Leibniz and today usually translated into topology or graph theory, and says that “this branch is concerned only with the determination of position and its properties; it does not involve distances, nor calculations made with them.” He claims that the bridges problem belongs to this area. He states the EPP verbally, introduces the symbols a, b, c. . . for the bridges (the edges of the graph) and the symbols A, B, C,. . . for the areas of K¨onigsberg linked by the bridges (the nodes of the graph). (The terms graph, node, vertex, and edge did not exist yet.) He also denotes an edge by a pair of nodes, such as a=AB, introduces the notation ABD for a path that links the nodes A and D via the sequence of edges AC and CD, and defines path length. He even discusses notational difficulties with parallel edges. Graph theory notation and notational trouble have not much changed since 1736! Documenta Mathematica

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Euler also states that solving the problem for K¨onigsberg by enumeration is possible but too laborious and hopeless for EPP in general. Euler then argues that a solution of KBP must have a representation by a sequence AB. . . of 8 letters/nodes from the 4 letters A,B,C,D (with side constraints) and counts node degrees along a path. Degree counting for KBP results in: node A must appear 3 times in the sequence (path), nodes B, C, D must appear twice each, but the sequence must have length 8. This is a contradiction, and KBP is solved. There is no such path! Now follows a verbal statement of what we today call Euler’s Theorem: A graph has an Eulerian path if and only if it has 0 or 2 nodes of odd degree. Euler does not mention connectivity, it appears that he assumes that a graph has to be connected. Afterwards Euler discusses various cases and a more general example. And then he states and proves what one can truly call the First Theorem of Graph Theory: In any graph, the sum of node degrees is equal to twice the number of edges. And he continues with the Second Theorem of Graph Theory: In any graph, the number of nodes of odd degree is even. Euler remarks that KBP could be solved if all bridges were doubled, and then states his theorem formally, copied from [3]:

Euler, though, has shown so far only that if a graph has more than two nodes of odd degree then there is no Eulerian path. He then argues: When it has been determined that such a journey can be made, one still has to find how it should be arranged. For this I use the following rule: let those pairs of bridges which lead from one Documenta Mathematica

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area to another mentally be removed ( deletion of pairs of parallel edges), thereby considerably reducing the number of bridges; it is then an easy task to construct the required route across the remaining bridges, and the bridges which have been removed will not significantly alter the route found, as will become clear after a little thought. I do not therefore think it worthwhile to give any further details concerning the finding of the routes. We do not doubt that Euler knew how to construct an Eulerian path, but the text above is not what one could call a proof. Those who have taught Euler’s theorem in class know the problem. It is really difficult to provide a short sequence of convincing arguments. Hand waving in front of the blackboard usually does the trick! The theory of algorithms did not exist in his time, and Euler did not have the concept of recursion, for instance, to describe his thoughts. In a formal sense, thus, Euler did not prove his characterization of Eulerian graphs. It took 140 further years to get it done.

Carl Hierholzer The final step of the proof has an interesting story of its own. The first full proof of Euler’s theorem was given by C. Hierholzer (1840–1871). He outlined his proof in 1871 to friends but passed away before he had written it up. Christian Wiener re-composed the proof from memory with the help of Jacob L¨ uroth. The resulting paper [4] was published in 1873 and contains what is now sometimes called the Hierholzer algorithm for the construction of an Eulerian path or cycle.

Euler and optimization If one glances through Euler’s publications, it looks like one topic seems to have permeated his work: the idea of minima and maxima. Just read the introduction to this book. One could have guessed that, after having characterized the existence of an Eulerian path or cycle in a graph, he would have raised (and tried to answer) one of the questions: How many edges does one have to add to a graph or how many edges does one have to double so that an Eulerian path or cycle exist? More generally, if one considers walking distances in addition, Euler could have asked: What is the shortest walk covering every edge at least once? He came close to this issue, since he mentioned that one can solve KBP by doubling all edges. If he had done this next step, we could call Euler rightfully the “father of combinatorial optimization”. Euler missed this opportunity. It took 224 years until an optimization version of the Eulerian graph problem was considered, and this was in China. Documenta Mathematica

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Figure 2: Mei-Ko Kwan Mei-Ko Kwan and the Chinese Postman Problem Before going to 1960 we take a step back in history. The great Chinese philosopher Confucius (551 BC – 479 BC) was born in the city of Qufu in Shandong Province. As the homeland of Confucius, Shandong has played a major role in Chinese history. During the Great Leap Forward movement (1958-1960), Chinese scientists were encouraged to solve real-world problems to help Chairman Mao’s ambitious campaign to rapidly transform the country from an agrarian economy into a modern communist society. At that time, many mathematicians in China were engaged in real-world applications, and in particular, carried out operations research (OR) activities, focusing on problems such as transportation and production planning. Shandong, one of the few provinces where early Chinese OR application activities took place, is in fact the birthplace of the Chinese Postman Problem. In 1960, the 26 years old Mei-Ko Kwan (modern PinYin spelling: Mei-Gu Guan), a young lecturer at Shandong Normal University, published his paper [6], in which he stated the following problem. Chinese Postman Problem: A postman has to deliver letters to a given neighborhood. He needs to walk through all the streets in the neighborhood and back to the post-office. How can he design his route so that he walks the shortest distance? Due to this paper and other contributions to optimization, Mei-Ko Kwan became one of the leading experts on mathematical programming in China. He was, for instance, the president of Shandong Normal University from 1984 to 1990, and from 1990 to 1995, director of the OR department of Fudan University, the best university in Shanghai. In 1995, Mei-Ko Kwan moved to Australia and has worked at the Royal Melbourne Institute of Technology. By calling a node of a graph odd or even if the number of edges incident to the node is odd or even, Kwan converted the Chinese postman problem into the following optimization problem on a graph: Documenta Mathematica

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PROBLEM Given a connected graph where 2n of the nodes are odd and all other nodes are even. Suppose we need to add some edges to the graph with the following property: the number of edges added to any odd node is odd and that added to any even node is even. We need to minimize the total length of the added edges. The main theoretical result Kwan proved in [6] is the following theorem: Theorem: For a set of added edges it is necessary and sufficient to be an optimal solution for the above problem if the following two conditions hold: (1) Between any two nodes, no more than one edge is added. (2) In any cycle of the extended graph, the total length of the added edges is not greater than half of the total length of the cycle. His proof is constructive; this way Kwan [6] also proposed a method for finding a solution to the Chinese Postman Problem. Fig. 3 shows two drawings copied from his original paper [6]. In the left diagram, the dotted lines are the added edges, while the right diagram shows an optimal solution:

Figure 3 Kwan’s original paper was published in Chinese. Two years later the paper [6] was translated into English [7], which attracted the attention of Jack Edmonds. Edmonds was the one who introduced this interesting problem to the optimization community outside China, and he was also the first person to name it Chinese Postman Problem. Moreover, J. Edmonds and E. L. Johnson proved in a beautiful paper [2] that the Chinese Postman Problem can be reduced to matching, and thus, that it is solvable in polynomial time. This result was out of reach for mathematicians of the 18th century; even for Kwan this was not an issue since modern complexity theory did not yet exist in 1960. But if Euler had known linear programming and complexity theory, who knows? References [1] N. L. Biggs, E. K. Lloyd and R. J. Wilson, Graph theory 1736–1936, Reprint with corrections, Clarendon Press, 1998. Documenta Mathematica

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[2] J. Edmonds and E. L. Johnson, Matching, Euler tours and the Chinese Postman. Mathematical Programming 5 (1973) 88–124. [3] L. Euler, Solutio Problematis ad Geometriam Situs Pertinentis, Commentarii Academiae Scientiarum Imperialis Petropolitanae 8 (1736/1741) 128– 140. ¨ [4] C. Hierholzer, Uber die M¨oglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren, Mathematische Annalen VI (1873) 30–32. [5] E. Knobloch, Euler and infinite speed, this volume. [6] Mei-Ko Kwan, Programming method using odd or even pints, Acta Mathematica Sinica 10 (1960) 263–266 (in Chinese). [7] Mei-Ko Kwan, Graphic programming using odd or even points, Chinese Mathematics 1 (1962) 273–277. [8] H. Sachs, M. Stiebitz and R. J. Wilson, An Historical Note: Euler’s K¨ onigsberg Letters, Journal of Graph Theory 12 (1988) 133–139. [9] W. Velminski, Leonhard Euler: Die Geburt der Graphentheorie, Kadmos, Berlin, 2008.

Martin Gr¨ otschel Konrad-Zuse-Zentrum f¨ ur Informationstechnik Berlin (ZIB) Takustraße 7 14195 Berlin Germany

Ya-xiang Yuan Academy of Mathematics and Systems Science Chinese Academy of Sciences Zhong Guan Cun Donglu 55 Beijing 100190 China

[emailprotected]

[emailprotected]

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Linear Programming Stories

The history of polyhedra, linear inequalities, and linear programming has many diverse origins. Polyhedra have been around since the beginning of mathematics in ancient times. It appears that Fourier was the first to consider linear inequalities seriously. This was in the first half of the 19th century. He invented a method, today often called Fourier-Motzkin elimination, with which linear programs can be solved, although this notion did not exist in his time. If you want to know anything about the history of linear programming, I strongly recommend consulting Schrijver’s book [5]. It covers all developments in deepest possible elaborateness. This section of the book contains some aspects that complement Schrijver’s historical notes. The origins of the interior point method for linear programming are explored as well as column generation, a methodology that has proved of considerable practical importance in linear and integer programming. The solution of the Hirsch conjecture is outlined, and a survey of the development of computer codes for the solution of linear (and mixed-integer) programs is given. And there are two articles related to the ellipsoid method to which I would like to add a few further details. According to the New York Times of November 7, 1979: “A surprise discovery by an obscure Soviet mathematician has rocked the world of mathematics . . . ”. This obscure person was L. G. Khachiyan who ingeniously modified an algorithm, the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. Nemirovskii and proved in a very short paper [3] that this method solves linear programs in polynomial time. This was indeed a sensation. The ellipsoid method is a failure in practical computation but turned out to be a powerful tool to show the polynomial time solvability of many optimization problems, see [2]. One step in the ellipsoid method is the computation of a least volume ellipsoid containing a given convex body. The story of the persons behind the result that this ellipsoid, the L¨ owner-John ellipsoid, is uniquely determined and has very interesting properties, is told in this section. A second important ingredient of Khachiyan’s modification is “clever rounding”. A best possible approximation of a real number by a rational number with a bounded denominator can be Documenta Mathematica

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achieved by computing a continued fraction. The history and some applications of this technique are covered also in a subsequent article. When L. Lov´ asz, A. Schrijver, and I were working on our book [2] we wanted to eliminate some “dirty tricks” that were needed to make the original version of the ellipsoid method work. The ellipsoid method produces successively shrinking ellipsoids containing the given polyhedron. It terminates due to a volume criterion, and thus it can only be applied to full-dimensional polyhedra. Since one usually does not know whether a given polyhedron is full-dimensional, one has to blow it up appropriately. How can one avoid this artificial blow up? If a polyhedron is not full-dimensional (let us assume its dimension is one less than the space dimension), then it must lie in some hyperplane H. One observation is that, in such a case, the ellipsoid method produces shrinking ellipsoids that get very flat in the direction perpendicular to H. This means that, for these flat ellipsoids, the symmetry hyperplane belonging to the shortest axis must be very close to H. Is it possible to identify H by rounding the equation of this symmetry hyperplane? An immediate idea is to round each coefficient of the equation (using continued fractions), but this does not deliver what one wants. Simultaneous rounding, more precisely simultaneous Diophantine approximation, is needed. We searched all number theory books. There are important results of Dirichlet that could be applied, but no polynomial time algorithms. We were stuck. Then I obtained the following letter from Laci Lov´ asz:

Laci’s algorithm is based on an idea for finding short vectors in a lattice. At about the same time, several other mathematicians were addressing completely different problems that lead to the same type of questions Lov´asz answered. Among these were the brothers Arjen and Hendrik Lenstra with whom Laci teamed up and wrote the famous paper [4]. The algorithm described in [4] is now called LLL algorithm; it spurred enormous interest in many different areas of mathematics and computer science and found various extensions and improvements. The LLL algorithm virtually created a very lively subfield of Documenta Mathematica

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mathematics, lattice basis reduction, and is already the subject of textbooks, see [1]. The brief story sketched here is nicely presented, including many other angles of this development and persons involved, in [6] and describes, in particular, the way the brothers Lenstra and some others recognized the importance of algorithmic basis reduction. From my personal point of view, this important development began with the successful attempt to handle an annoying detail of a linear programming algorithm. Martin Gr¨otschel References [1] M. R. Brenner, Lattice Basis Reduction, CRC Press, Boca Raton, 2012. [2] M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, Berlin, 1988/1993. [3] L. G. Khachiyan, A polynomial algorithm in linear programming (in Russian), Doklady Akademii Nauk SSSR 244 (1979), 1093–1096 (English translation: Soviet Mathematics Doklady 20 (1979), 191–194). [4] A. K. Lenstra, H. W. Lenstra, Jr., and L. Lov´asz. Factoring polynomials with rational coefficients, Mathematische Annalen 261 (1982), 515–534. [5] A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chichester, 1986/1998. [6] I. Smeets et al., The History of the LLL-algorithm, in Phong Q. Nguyen (ed.) et al., The LLL algorithm, Survey and applications, Springer, 2010, pp. 1–17.

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Who Invented the Interior-Point Method? David Shanno

2010 Mathematics Subject Classification: 90C51, 90C05, 90C30 Keywords and Phrases: Interior-point methods, linear programming, nonlinear programming

The controversy Thomas Edison is regarded by many as the greatest inventor in American history. While most people know that he invented the first long-burning incandescent light bulb and the phonograph, the claim is based more generally on the 1093 patents he was granted. The assumption is that the person receiving a patent is legally certified as the inventor of the device which is the subject of the patent. The invention of the stored program computer during and in the period immediately following World War II vastly expanded the range of practical mathematical problems which could be solved numerically. A particular form of problem which received great interest is the linear programming problem, which allocates resources optimally subject to constraints. George Dantzig’s development of the simplex method [5], provided the computational tool still prominent in the field today for the solution of these problems. Continuous development of variants of the simplex method has led to contemporary codes that are quite efficient for many very large problems. However, as the simplex method proceeds from one vertex of the feasible region defined by the constraints to a neighboring vertex, the combinatorial analysis indicates it can be quite inefficient for some problems. In [14], Klee and Minty showed that, in the worst case, the method has exponential complexity in the size of the problem. The question that then presented itself is whether there is another algorithm for linear programming which has polynomial complexity. This question was first answered positively in 1979 by Khachian [13], who adapted the ellipsoid method of Shor [18] and showed that the complexity of the resulting algorithm was polynomial of order mn3 + n4 L, where n represents the number of rows in A, m the number of columns, and L the length of the data. This result was an extremely important theoretical advance. It also created intense interest as a possible computational technique, including a wildly misinformed article in the New York Times claiming it solved the traveling salesman problem. Documenta Mathematica

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However, despite numerous attempts by many in the broad math programming community to implement a viable algorithm, it quickly became apparent that it was an extremely inefficient algorithm for computational work. One interpretation of the simplex method is to consider what is purported to be the Norbert Wiener method of negotiating the halls of the massive main building at MIT. Not wishing to be distracted from thinking by watching where he was going, he simply dragged his hand along the wall, never removing it until he reached his destination. This algorithm clearly would eventually get him to where he was going, provided he began on the correct floor (an initial feasible point). I am not sure how he decided he had arrived, but in general this is akin to the simplex algorithm. A better method is to pay attention to where you are and take the best route. Interior-point algorithms attempt to emulate this strategy. In a 1984 paper, Karmarkar [11] considered the linear programming problem in the form minimize cT x subject to Ax = 0, eT x = 1, x ≥ 0. He began with an initial point x0 that satisfied the constraints and used the projective transformation X −1 x T (x) = T 0 −1 e X0 x where X0 is the diagonal matrix xjj = x0j . The current point x0 is transformed to the point n1 e, which is the central point of the constraints eT x = 1, x0 ≥ 0. Then, any vector in the null space of the matrix AX0 eT in particular

−1 δ = −γ I − B T (BB T ) B X0 c,

can be used to reduce the objective function while remaining in the interior of the feasible region. Here, γ is a step length parameter to keep the step in the interior of the feasible region, which is accomplished by letting ξ=

1 e+δ n

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Karmarkar demonstrated the complexity of this method is of order (mn2 +n3 )L, but the proof required that cT x∗ = 0, where x∗ denotes the optimal solution. Todd and Burrell [19] dealt with this restriction by noting that if v ∗ is the optimal value of the objective function then T

cT x = (c − v ∗ e) x is 0 at the optimal point. They then use duality theory to obtain a convergent sequence of estimates to v ∗ . Note that doing so adds a parameter to the sequence of estimates that will emerge in a different context shortly. The originality of the use of projective transformations and the much stronger complexity results justifiably created a great deal of interest in the method. This interest, however, was mild compared to the interest created by a sequence of claims by Karmarkar and supported by Bell Labs, Karmarkar’s employer, that an algorithm implementing the method was vastly superior to the simplex method. A simpler transformation of the current point into the interior of the feasible region is the basis of the affine scaling method where instead of a projective transformation, the simple linear transformation was proposed by Barnes [2] and Vanderbei et al. [20]. Here, the standard form of the linear programming problem defined by minimize cT x subject to Ax = b, x≥0 is used and the transformation becomes ξ = X0−1 x. Here, the sequence of iterates is defined by x1 = x0 + γ∆x, where again γ is chosen to assure that the iterates do not touch the boundary of the feasible region and

where

−1 ∆x = D − DAT (ADAT ) AD c, D = X02 .

It was later discovered that this work was originally published in 1967 by Dikin [6] who in 1974 proved convergence of the method [7]. No strong complexity bound equivalent to Karmarkar’s is known for this algorithm. Both of the above algorithms create room to move entirely in the interior of the feasible region by transforming the space. A more general method for Documenta Mathematica

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Figure 1: Anthony V. Fiacco (left) and Garth McCormick in 1967 in Fiacco’s office at Research Analysis Corporation (RAC) in McLean, VA (Photo printed with the permission of John McCormick). remaining in the interior was studied prior to either of these methods. An alternative method for remaining interior to the feasible region is to add a component to the objective function which penalizes close approaches to the boundary. This method was first suggested in 1955 in an unpublished manuscript by Frisch [9] and developed in both theoretical and computational detail by Fiacco and McCormick [8] in 1968. Applied to the linear programming problem in standard form, the problem is transformed to minimize cT x − µ

n X

?ln(xi ),

i=1

subject to Ax = b. Here, the method is akin to the invisible fence that is used to keep dogs in an unfenced yard. The closer the dog gets to the boundary, the more he feels shock. Here the amount of shock is determined by the parameter µ, and as µ tends to 0, the boundary, in this case where the solution lies, is approached. The above reformulation is a nonlinear programming problem, and the firstorder conditions may be derived by forming the Lagrangian and differentiating. The resulting step directions are ∆x = −

1 X0 P X0 c + X0 P e, µ0

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Figure 2: Garth McCormick at the desk in his office (Photo printed with the permission of John McCormick). where and as before

−1 P = I − X0 AT (AX02 AT ) AX0 , x1 = x0 + γ∆x.

Fiacco and McCormick actually developed this method for the much harder general nonlinear programming problem. They showed that for a sequence of µ’s which decreases monotonically to 0, the sequence of solutions for each value of µ converges to the solution of the problem. Their book noted that it applied as well to the linear programming problem, but did not further study this particular line of development as at the time they developed this work they felt the algorithm would not be competitive with the simplex method. In 1985 at the Boston ISMP meeting, Karmarkar gave a plenary lecture in which he claimed his algorithm would be 50 or 100 times faster than the best simplex codes of that period. This was greeted with a great deal of skepticism and more that a little annoyance by many in the audience. At the same meeting, Margaret Wright presented the results in Gill et al. [8] that showed there existed values for µ and v ∗ that make Karmarkar’s algorithm a special case of the logarithmic barrier method of Fiacco and McCormick. This observation led to a major outpouring of theoretical papers proving order n3 L complexity for a wide variety of choices for the sequence of µ’s and the search parameter γ. It also led to implementation work on numerical algorithms. An early example of this was the implementation of a dual-affine scaling algorithm Documenta Mathematica

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(derived by applying the affine variant to the dual problem) of Adler et al. [1]. I was personally involved, first with Roy Marsten, in creating a dual-affine scaling implementation. We later joined with Irv Lustig to create an implementation of the primal-dual interior-point code [17] based on an algorithm published by Kojima et al. [15] which assumed the knowledge of an initial feasible point. We addressed initial feasibility using the analysis of Lustig [16]. We later discovered that the implemented algorithm can be derived directly by applying the Fiacco and McCormick logarithmic barrier method to the dual of the problem in standard form and applying Newton’s method to the first order conditions. Meanwhile, AT&T had begun development of the KORBX commercial package which included an eight processor supercomputer and an interior point code to be marketed at a multimillion dollar price. AT&T continued to claim (but not publish) strong computational results for their product. In 1988, they announced that they had obtained a patent on Karmarkar’s method to protect their investment [11]. This patent in and of itself created quite a stir in the mathematics community, as up until that time mathematics was considered not patentable. However, the value of mathematical algorithms in the workplace was changing this view, and continues to do so today. Irv, Roy and I meanwhile completed our first implementation of the primaldual method [17], and in the fall of 1989 presented a computational comparison of our code with KORBX on a set of results which had finally appeared in publication [4]. The comparison was not favorable to KORBX. We distributed free of charge source of our OB1 code to researchers, but were marketing it to industry through XMP Software, a company Roy had started. Shortly after the presentation of the comparative results, we received a letter from AT&T informing us that, while they encouraged our promoting research in this area, we were not to market our code as they owned the patent on all such algorithms. This led us to carefully study the patent. The abstract of the patent follows. A method and apparatus for optimizing resource allocations is disclosed which proceeds in the interior of the solution space polytope instead of on the surface (as does the simplex method), and instead of exterior to the polytope (as does the ellipsoid method). Each successive approximation of the solution point, and the polytope, are normalized such that the solution point is at the center of the normalized polytope. The objective function is then projected into the normalized space and the next step is taken in the interior of the polytope, in the direction of steepest-descent of the objective function gradient and of such a magnitude as to remain within the interior of the polytope. The process is repeated until the optimum solution is closely approximated. The optimization method is sufficiently fast to be useful in real time control systems requiring more or less continual allocation optimization in a changing environment, and in allocation systems heretofore too large for practical Documenta Mathematica

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implementation by linear programming methods. While the patent is for the Karmarkar algorithm, consequent discussions with AT&T patent lawyers made it clear that they were claiming that Karmarkar had invented interior point methods and they held the patent more broadly. The claim was obviously ridiculous, as there is a full chapter entitled Interior Point Algorithms in the Fiacco and McCormick book, which was published and won the Lancester prize in 1968. The people we were dealing with at AT&T seemed totally unaware of the existence of this book, despite its prominence in the mathematical programming community. The AT&T patent was granted in 1988, and there is a rule that nothing can be patented that has been in the public domain for a year or more prior to filing an application for the patent. Thus by the Edison criterion, Karmarkar invented the interior point method, but in fact he was well behind the true pioneers. Meanwhile AT&T continued to claim to Roy, Irv and me that their patent applied to our code. After we consulted our own patent lawyer and were told what of the great expense of challenging the patent, we accepted a licensing agreement with AT&T. For a variety of reasons, the agreement proved to be unworkable, and we shut down XMP Optimization. We then joined with CPlex to create the CPlex barrier code. This code was derived by applying Newton’s method to the log-barrier method of Fiacco and Mccormick applied to the dual problem. It is equivalent to an interior-point method, but using the term barrier rather than interior-point did not fall within the linguistic purview of the AT&T patent.It eventually became clear that AT&T had finally understood that the idea of interior-point methods did not originate with Karmarkar, and to the best of my knowledge they have never again tried to enforce the patent. There is a further irony in AT&T receiving the Karmarkar patent. That patent is specifically for the projective transformation algorithm. Yet Bob Vanderbei, who was a member of the AT&T KORBX team, has told me that the method implemented in KORBX was the affine scaling method, which was also not eligible to be patented as Dikin’s paper was published in 1967. AT&T did patent several techniques involved in the implementation of the affine scaling method [21], [22], such as how to incorporate bounds and ranges, but not the affine scaling interior point itself. Thus the only patent granted specifically for an interior point method was granted to the one algorithm that to the best of my knowledge has never been successfully implemented. Who did invent interior-point methods? With any invention that has proved highly successful, there is never a simple single answer to this question. A case can be made that Orville and Wilbur Wright invented the airplane. It is impossible to credit them alone with the creation of the Boeing 787. Further, in building the plane that made the first powered flight, they undoubtedly learned a great deal from others whose attempts had failed. Documenta Mathematica

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In a letter to Robert Hooke on February 15, 1676, Isaac Newton said “If I have seen further it is by standing on ye sholders of Giants.” Personally, I fully credit Fiacco and McCormick with the invention of interior point methods, and as the result of many discussions with them over the years, I know that they fully agreed with Newton. Indeed a prominent giant in the development of interior point methods is clearly Newton himself, for all of the complexity results for linear programming depend on using Newton’s method to solve the first order equations, and current nonlinear programming algorithms depend on Newton’s method to find a search direction. Another such giant is Lagrange. Both are easy choices, as most methods for solving continuous math programming problems are highly reliant on their work. On more recent work, both Frisch [9] and Carrol [3] must be credited with suggesting two different penalty functions to keep the iterates within the feasible region. Fiacco and McCormick certainly credited them. However, only Fiacco and McCormick developed a whole complete theory of interior point methods, including convergence results and a wealth of ideas for numerical implementation. They did not, however, analyze computational complexity. This field was really just beginning at the time of their work. The book contains many hidden gems, and as Hande Benson, a young colleague of mine has recently discovered, is still totally relevant today. In addition, Fiacco and McCormick also developed the SUMT code to implement the general nonlinear programming algorithm documented in the book. Unfortunately, this was not the success that their theoretical work was. The difficulties encountered in attempting to solve many applications led some people to dismiss the practical value of interior point methods. The problem was simply that the theory was well in advance of computational tools developed later. One particular difficulty was devising a good method to compute the decreasing sequence of µ’s. This was greatly improved by the analysis done when applying the algorithm to linear programming. A good sequence is dependent on the measure of complementarity. Another difficulty was nonconvexity of the objective function in nonlinear programming. The vast later research in trust region methods greatly improved the algorithms, and research on this continues today. The algorithm of SUMT was a pure primal algorithm. The use of the interior point theory to derive primal-dual algorithms produced much better estimates of the Lagrange multipliers. Central to applying the method to very large linear programming problems was the development of efficient sparse Cholesky decompositions to solve the linear equations. The computers at the time this research was done had such limited memories that this work had not yet been undertaken. At that time, it was believed that only iterative methods could be used to solve very large linear systems. The development of almost unlimited computer memories and the development of sparsity preserving ordering algorithms has allowed for very rapid solution of large sparse linear systems. These advances have Documenta Mathematica

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also been applied to the solution of large sparse nonlinear programming problems. Interior point algorithms require an initial feasible point x0 . Finding such a point for pure primal methods such as SUMT is often as difficult as solving the optimization problem. Development of primal-dual algorithms led to reformulation of the problem in such a way that a feasible initial point is easily found for the reformulated problems [16], [17]. The resulting algorithm approach feasibility and optimality simultaneously. This approach is now the standard approach in modern interior-point linear programming codes. It has also proved particularly important in improving interior-point algorithms for nonlinear programming, the problem that originally interested Fiacco and McCormick. The salient point is that any great piece of original work is never close to a finished product, but rather a starting point from which improvements can be made continuously. It can also be extended to new areas of application. Certainly the work of Fiacco and McCormick meets that test of time. I know of no even vaguely comparable work on this topic. References [1] Adler, I., Karmarkar, N., Resende, M. and Veiga, G. (1989), An implementation of Karmarkar’s algorithm for linear programming, Mathematical Programming 44, 297–335. [2] Barnes, E. (1986), A variation on Karmarkar’s algorithm for solving linear programming problems, Mathematical Programming 36, 174–182. [3] Carrol, C. (1961), The created response surface technique for optimizing restrained systems, Operations Research 9, 169–184. [4] Cheng Y., Houck D., Liu J., Meketon M., slu*tsman L., Vanderbei R. and Wang P. (1989), The AT&T KORBX system. AT&T Tech. Journal, 68:7–19. [5] Dantzig, G.(1963), Linear Programming and Extensions, Princeton University Press, Princeton, NJ. [6] Dikin, I. (1967), Iterative solution of problems of linear and quadratic programming, Soviet Mathematics Doklady 8, 674–675. [7] Dikin, I. (1974), On the speed of an iterative process, Upravlyaemye Sistemi 12, 54–60. [8] Fiacco, A. and McCormick, G. (1968), Nonlinear programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York. [9] Frisch, K. (1955), The logarithmic potential method of convex programming, Memorandum, University Institute of Economics, Oslo, Norway. Documenta Mathematica

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[10] Gill, P., Murray, W., Saunders, M., Tomlin, J, and Wright, M. (1986), On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method, Mathematical Programming 36, 183–209. [11] Karmarkar, N. (1984), A new polynomial time algorithm for linear programming, Combinatorica 4, 373–395. [12] Karmarkar, N. (1988), Methods and apparatus for efficient resource allocation, United States Patent Number 4744028. [13] Khachian, L. (1979), A polynomial time algorithm in linear programming, Soviet Mathematics Doklady 20, 191–194. [14] Klee, V. and Minty, G. (1972), How good is the simplex algorithm? in O. Shisha, ed. Inequalities – III, Academic Press, New York, 159–175. [15] Kojima, M., Mizuno, S. and Yoshise, A. (1989), A primal-dual interior point method for linear programming, in N. Megiddo, ed. Progress in Mathematical Programming: Interior Points and Related Methods, Springer Verlag, New York, 29–47. [16] Lustig, I. (1990), Feasibility issues in a primal-dual interior-point method for linear programming, Mathematical Programming 49, 145–162. [17] Lustig, I. Marsten, R. and Shanno, D. (1991), Computational experience with a primal-dual interior point method for linear programming, Linear Algebra and its Applications 152, 191–222. [18] Shor, N. (1964), On the structure of algorithms for the numerical solution of optimal planning and design problems, Ph. D. Thesis, Cybernetic Institute, Academy of Sciences of the Ukrainian SSR, Kiev. [19] Todd, M. and Burrell, B. (1986), An extension of Karmarkar’s algorithm for linear programming using dual variables, Algorithmica 1:4, 409–424. [20] Vanderbei, R., Meketon, M. and Freedman, B. (1986), A modification on Karmarkar’s linear programming algorithm, Algorithmica 1:4, 395–407. [21] Vanderbei, R. (1988), Methods and apparatus for efficient resource allocation, United States Patent Number 4744026. [22] Vanderbei, R. (1989), Methods and apparatus for efficient resource allocation, United States Patent Number 4885686. David Shanno, Professor Emeritus RUTCOR – Rutgers Center of Operations Research Rutgers University New Brunswick, NJ 08903-5062 USA [emailprotected] Documenta Mathematica

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Documenta Math.

Column Generation for Linear and Integer Programming George L. Nemhauser

2010 Mathematics Subject Classification: 90 Keywords and Phrases: Column generation, decomposition, linear programming, integer programming, set partitioning, branch-andprice

1

The beginning – linear programming

Column generation refers to linear programming (LP) algorithms designed to solve problems in which there are a huge number of variables compared to the number of constraints and the simplex algorithm step of determining whether the current basic solution is optimal or finding a variable to enter the basis is done by solving an optimization problem rather than by enumeration. To the best of my knowledge, the idea of using column generation to solve linear programs was first proposed by Ford and Fulkerson [16]. However, I couldn’t find the term column generation in that paper or the subsequent two seminal papers by Dantzig and Wolfe [8] and Gilmore and Gomory [17,18]. The first use of the term that I could find was in [3], a paper with the title “A column generation algorithm for a ship scheduling problem”. Ford and Fulkerson [16] gave a formulation for a multicommodity maximum flow problem in which the variables represented path flows for each commodity. The commodities represent distinct origin-destination pairs and integrality of the flows is not required. This formulation needs a number of variables exponential in the size of the underlying network since the number of paths in a graph is exponential in the size of the network. What motivated them to propose this formulation? A more natural and smaller formulation in terms of the number of constraints plus the numbers of variables is easily obtained by using arc variables rather than path variables. Ford and Fulkerson observed that even with an exponential number of variables in the path formulation, the minimum reduced cost for each commodity could be calculated by solving a shortest path problem, which was already known to be an easy problem. Moreover the number of constraints in the path formulation is the number of arcs, while in the arc formulation it is roughly the (number of nodes)x(number of commodities) + number of arcs. Therefore the size of the basis in the path formulation is independent of the number of commodities and is significantly Documenta Mathematica

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smaller when the number of commodities is large. This advantage in size they claimed might make it possible to solve instances with a large number of commodities with the simplex method. Modestly, they stated that they really had no idea whether the method would be practical since they had only solved a few small instances by hand. It must have been so frustrating to try to do algorithmic research when it was so difficult to test if your ideas could yield practical algorithms. The value of some of these brilliant ideas proposed in the infancy of mathematical programming would not be proven for decades. Much of this early work was done at the RAND Corporation with its ‘all star’ team of applied mathematicians including Bellman (dynamic programming), Ford and Fulkerson (network flows), Dantzig (linear programming) and many others. As a sports fan, this reminds me of the great baseball teams of the New York Yankees, basketball teams of the Boston Celtics and soccer teams of Manchester United. I was Ray Fulkerson’s colleague at Cornell in the 1970s. I have no memory of him giving an opinion of the significance of the arc-path formulation of the multicommodity flow problem. Even if he thought this was a fundamental contribution, his modesty would have prevented him from saying so. However I think that this early work influenced his later contributions on blocking and anti-blocking pairs of polyhedra [15], which studies polyhedra associated with combinatorial optimization problems that frequently have an exponential number of variables and provided a basic theory of integral polyhedra. Another way to derive Ford and Fulkerson’s path formulation is to begin with the arc formulation and note that the arc capacity constraints link all of the variables while the flow balance constraints can be separated by commodity. For each commodity the extreme points of the flow balance constraints are the origin-destination simple paths for that commodity. Feasible solutions to the whole problem are convex combinations of these extreme flows that satisfy the arc capacity constraints. So if we begin with a so-called master LP that just contains a few of these extreme flows for each commodity and solve it to optimality, we can use an optimal dual solution to price out the extreme flows not yet considered by solving a shortest path problem for each commodity. This is precisely what Ford and Fulkerson proposed simply beginning with the path formulation. This idea can be generalized to yield an algorithm for solving any LP by partitioning the constraints into a set of master constraints and a set of subproblem constraints. The resulting algorithm is what we call Dantzig–Wolfe decomposition [8]. I think it is rather odd that George Dantzig did not get his name attached to the simplex method but to this very important contribution still of surely lessor stature. Dantzig and Wolfe say: Credit is due to Ford and Fulkerson for their proposal for solving multicommodity network problems as it served to inspire the present development. However the contribution of Dantzig–Wolfe decomposition is very significant Documenta Mathematica

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in its own right since it does not depend on beginning with the exponential formulation. It could arise from an appropriate partition of the constraints into a small number that involved all of the variables and the rest that could be decomposed into individual subproblems involving only a relatively small subset of the variables. Think, for example, of a multiperiod problem with a few budget constraints involving variables from all of the periods and subproblems for each period, or a resource allocation problem involving a few constraints coupling all of the variables globally together with subproblems for each region. For these structures, and other similar ones, using Dantzig– Wolfe decomposition, a large LP can be decomposed into a master problem with a small number of constraints and an exponential number of variables corresponding to the extreme points of the subproblems, the solution of which represents convex combinations of these extreme points that satisfy the master constraints. Optimal dual solutions of the master problem provide prices to the subproblems, whose solutions yield new extreme point variables for the master. 2

Next steps – integer subproblems

The previous work relied only on LP. The multicommodity flow problem requires the generation of integer vectors that are incidence vectors of paths, but they can be found without the explicit imposition of integrality constraints. The first column generation work that involved integer variables appears to have been done by Gilmore and Gomory [17]. They studied the cutting stock problem: given a positive integer number d(i) of items of integer size a(i), determine the minimum number of stock rolls of integer size b needed to pack all of the items. Gilmore and Gomory proposed a model in which there is an integer variable corresponding to every possible way to cut a roll. Since a solution to the cutting of a single roll is a solution of an integer knapsack problem (a single constraint integer program (IP)), which can have an exponential number of solutions, this model contains an exponential number of variables. However, when the LP relaxation of the model is solved over a subset of variables, optimality can be proved or new columns can be added to improve the solution by solving an integer knapsack problem with objective function specified by the dual variables in an optimal LP solution and constraint specified by the item and role sizes. The knapsack problem can be solved reasonably efficiently by dynamic programming or branch-and-bound even though it is NP-hard. The application of this work described in [18] appears to be the first use of column generation in a practical problem. Gilmore and Gomory’s work on the cutting stock problem led to their work on the knapsack problem [19], and motivated Gomory’s work on the group problem [20], which has had a significant impact on the field of integer programming. Gilmore and Gomory only use the LP relaxation of their formulation of the cutting stock problem. They simply propose to round up the variables in an optimal LP solution to obtain a feasible solution to the IP. But this heuristic can be justified by the fact that, in general, the optimal LP solution value Documenta Mathematica

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provides a very tight bound on the optimal number of rolls. In fact, it has been shown empirically in [29] that for a very large number of randomly generated instances the difference is always less than one. Carefully contrived instances with a difference greater than one are known [25, 30], but it is not known whether a difference of two or larger can be obtained. Although rounding up a fractional solution can increase the objective function by the number of items (number of basic variables), it has been observed in [4] that the increase is no more than 4 % of the number of items. The whole point of this discussion is to emphasize that the Gilmore–Gomory formulation of the cutting stock problem provides a very tight relaxation. This is typically the case for such formulations leading to a tradeoff between a tight bound from an exponential formulation that can be challenging to solve and a compact (polynomial size) formulation with a much weaker bound. Although not stated by Gilmore and Gomory, and then lost in translation when the cutting stock problem is presented in basic operations research textbooks, there is a straightforward compact formulation of the cutting stock problem. Begin with an upper bound on the number of rolls required and a binary variable for each roll that is equal to one if the roll is used and zero otherwise. There are identical knapsack constraints for each potential roll with right-hand side b if its binary variable equals one, and zero otherwise and additional constraints requiring that the amount d(i) of the ith item must be cut. The LP relaxation of this formulation is terrible. It gives no information since it is easy to show that the bound is the total amount to be cut divided by b. Furthermore if this LP relaxation is used in a branch-and-bound algorithm, the performance is terrible not only because of the weak bound, but also because of the symmetry of the formulation since all rolls are the same. In fact, a compact formulation similar to the one above was given by Kantorovich [23] who introduced the cutting stock problem in 1939! The Gilmore–Gomory formulation applied to the bin packing specialization of the cutting stock problem in which d(i) = 1 for all i yields a set partitioning problem: given a ground set S and a set of subsets S(j), j = 1, . . ., n, find a minimum cardinality set of disjoint subsets whose union is S. In the bin packing problem S is the set of items and S(j) is a subset that fits into a bin. |S| = m is typically small, but n is exponential in m. This form of set partitioning and set covering (disjointness is not required) models arises in many combinatorial optimization problems. For example, in node coloring S is the set of nodes and S(j) is a subset of nodes that is a stable set (a set of nodes that can receive the same color since no pair of them is joined by an edge). Thus column generation for the LP relaxation of the node coloring set partitioning formulation involves solving a minimum weight stable set problem, where the node weights correspond to the dual variables in an optimal LP solution. Note that the column generation formulation eliminates the symmetry possessed by a compact formulation in which there is a variable for each nodecolor pair. The absence of symmetry is a very important property of the exponential formulation since symmetry is a major nemesis of branch-andDocumenta Mathematica

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bound algorithms. These models appear in many practical applications as well. Perhaps the one that has received the most attention in the literature is airline crew scheduling [6, 21], but there are many other applications to all kinds of transportation routing problems, scheduling problems, districting problems, coloring problems, etc. In the crew scheduling problem S is a set of flights that need to be flown over a given time horizon, say a day or a week, and S(j) is a subset of flights that can be flown by a single crew. The cost of using the subset S(j) is c(j). This cost function complicates the model introduced for bin packing and graph coloring since the objective function of total minimum cost is no longer a minimum cardinality objective function and a set of allowable flights is subject to complex rules concerning safety and other factors. Nevertheless, feasible subsets, which are called pairings, can be generated as constrained paths in a network and minimum cost constrained shortest paths for column generation can be generated as well. The first published paper that appears to discuss such a model in detail is [5]. It reports on crew scheduling methods used by airlines in the 1960s, several of whom were already using a set partitioning model. Some were trying to solve the IP by optimization algorithms using branch-and-bound or cutting planes. They recognized that the algorithms could only deal with a small number of pairings. So pairings were generated up front and then a subset was heuristically chosen to include in the IP model. A significant improvement to the approach of a single round of pairing generation followed by a single round of optimization was proposed in [27]. Given a feasible solution, a better solution might be found by a neighborhood search that selects a small subset of flights, generates all of the pairings that only cover these flights and then solves a set partitioning problem defined by these flights and pairings. If an improvement is found, this solution replaces the current pairings that cover these flights. The neighborhood search can be iterated until no improvements are found. This quasi-column generation process was used by many airlines throughout the 1980s and even later [1]. Nevertheless it could only achieve a local optimum, and although the solution quality might be good, optimality could not be claimed. Other approaches solved the full LP relaxation by some form of column generation, but only provided a subset of columns to the IP solver. Even without an exponential number of columns these IP can be difficult to solve. Standard branching on binary variables is not very effective since the branch with the binary variable at zero hardly restricts the problem. A branching rule proposed in [28], unrelated to column generation at the time, called follow-on branching, helped to alleviate this difficulty. In a simplified version of the rule, two adjacent arcs in the flight network associated with a fractional pairing are identified and then, on one branch, pairings that contain both of these flights are excluded, and on the other branch, pairings that contain one of them are excluded. It can be shown that such a pair of arcs exists in a fractional solution, and the fractional solution is excluded on both branches. This rule divides the solution space much more evenly than variable Documenta Mathematica

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branching. As we shall see, generalizations of this rule are very useful when column generation is incorporated in a branch-and-bound search. 3

Branch-and-price: solving integer programs by column generation

If a tree search (branch-and-bound) algorithm for an IP with an implicit exponential number of variables is designed to produce an optimal solution or even one with a prescribed optimality tolerance, it is necessary to do column generation throughout the tree. To the best of our knowledge, the first appearance in the literature of column generation within branch-and-bound is in [13]. There are interesting challenges in applying column generation to problems associated with nodes within the search tree. Foremost is that standard branching on variables, besides being inefficient, can complicate column generation. Consider a set partitioning problem where we branch on a single binary variable corresponding to some subset. The branch where the variable is fixed to one does not create a problem since we now have a smaller set partitioning problem. But in the branch where the variable is set to zero we need to impose on the column generation solver a constraint saying that this subset is not feasible. Such constraints will significantly hamper the efficiency of the column generator. However, a generalized version of the follow-on branching idea for crew scheduling makes it possible to preserve the efficiency of the column generation solver and also reasonably balances the solutions between the two newly created nodes. Consider a fractional column (subset) in an optimal solution of the LP relaxation. It can be shown that there are two elements in the column such that there is another fractional column containing only one of these elements. On one branch we exclude columns containing only one of these elements and on the other branch we exclude columns containing both. Not allowing only one of the elements to appear, i.e., both must appear together, amounts to combining the elements, while not allowing both to appear together involves adding a simple constraint. For example, in a node coloring problem where the elements are nodes and a feasible subset is a stable set, both appearing together is accomplished by replacing the two nodes by a super node with an edge from the super node to all other nodes that were connected to one or both or the original nodes, and not allowed to appear together is accomplished by adding an edge between the two nodes. We can think of this type of branching as branching on the variables from the original compact formulation instead of branching on the variables in the exponential set partitioning formulation. For example in the node coloring problem, the branching is on node variables. On one branch we require two nodes to have the same color and on the other the two nodes must get different colors. Early use of this branching rule are given in [10] for urban transit crew scheduling, [14] for vehicle routing, [2] for airline crew scheduling, [31] for bin packing, [11] for a survey of routing and scheduling applications and [26] for node coloring. Vanderbeck and Wolsey [34] Documenta Mathematica

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studies column generation branching with general integer variables. Barnhart et al. [7] unified this early literature by presenting a general methodology for column generation in IP and named the general technique branch-and-price. Vanderbeck [32] presents a general treatise on branching in column generation and gives some interesting new branching ideas in [33]. In the last decade there have been many successful applications of branch-andprice algorithms to practical problems and a completely different use in choosing neighborhoods for local search algorithms [22]. More information about column generation and branch-and-price algorithms can be found in Desrosiers and L¨ ubbecke [12], who present a primer on column generation, in a chapter of a collection of articles on column generation [9], and L¨ ubbecke and Desrosiers [24], who present a survey of techniques and applications of column generation in IP. References [1] R. Anbil, E. Gelman, B. Patty and R. Tanga (1991). Recent advances in crew pairing optimization at American Airlines. Interfaces 21, 62–74. [2] R. Anbil, R. Tanga and E.L. Johnson (1992). A global optimization approach to crew scheduling. IBM Systems Journal 31, 71–78. [3] L.E. Appelgren (1969). A column generation algorithm for a ship scheduling problem. Transportation Science 3, 53–68. [4] D.L. Applegate, L.S. Buriol, B.L. Dillard, D.S. Johnson and P.W. Shor (2003). The cutting-stock approach to bin packing: theory and experiments. In Proceedings of the Fifth Workshop on Algorithm Engineering and Experiments, R.E. Ladner ed. SIAM, 2–15. [5] J.P. Arabeyre, J. Fearnley, F.C. Steiger and W. Teather (1969). The airline crew scheduling problem: A survey. Transportation Science 3, 140–163. [6] C. Barnhart, A. Cohn, E.L. Johnson, D. Klabjan. G.L. Nemhauser, and P.Vance (2002). Airline crew scheduling. In Handbook in Transportation Science, R.W. Hall ed. Kluwer, 517–560. [7] C. Barnhart, E.L. Johnson, G.L. Nemhauser, M.W.P. Savelsbergh and P.H. Vance (1998). Branch-and-price: column generations for solving huge integer programs. Operations Research 46, 316–329. [8] G.B. Dantzig and P. Wolfe (1960). Decomposition principle for linear programs. Operations Research 8, 101–111. [9] G. Desaulniers, J. Desrosiers, and M. Solomon (2005). Column Generation, Springer. Documenta Mathematica

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[10] M. Desrochers and F. Soumis (1989). A column generation approach to the urban transportation problem. Transportation Science 23, 1–13. [11] J. Desrosiers, Y. Dumas, M.M. Solomon and F. Soumis (1995). Time constrained routing and scheduling. In Handbooks in Operations Research and Management Science 8, Network Routing, M.E. Ball, T.L. Magnanti, C. Monma and G.L. Nemhauser eds. Elsevier, 35–140. [12] J. Desrosiers and M.E. L¨ ubbecke (2005). A primer in column generation. In Column Generation, G. Desaulniers, J. Desrosiers, and M. Solomon eds. Springer, 1–32. [13] J. Desrosiers, F Soumis and M. Desrochers (1984). Routing with time windows by column generation. Networks 14, 545–565. [14] Y. Dumas, J. Desrosiers and F. Soumis (1991). The pickup and delivery problem with time windows. European Journal of Operations Research 54, 7–22. [15] L.R. Ford and D.R. Fulkerson (1958). A suggested computation for maximal multicommodity network flows. Management Science 5, 97–101. [16] D.R. Fulkerson (1971). Blocking and anti-blocking pairs of polyhedra. Mathematical Programming 1, 168–194. [17] P.C. Gilmore and R.E. Gomory (1961). A linear programming approach to the cutting-stock problem. Operations Research 9, 849–859. [18] P.C. Gilmore and R.E. Gomory (1963). A linear programming approach to the cutting stock problem—Part II. Operations Research 11, 863–888. [19] P.C. Gilmore and R.E. Gomory (1966). The theory and computation of knapsack functions. Operations Research 14, 1045–1074. [20] R.E. Gomory (1965). On the relation between integer and non-integer solutions to linear programs. Proceedings of the National Academy of Science 53, 260–265. [21] B. Gopalakrishnan and E.L. Johnson (2005). Airline crew scheduling: state-of-the-art. Annals of Operations Research 140, 305–337. [22] M. Hewitt, G.L. Nemhauser and M.W.P. Savelsbergh (2012). Branch-andprice guided search for integer programs with an application to the multicommodity fixed charge network flow problem. To appear in INFORMS Journal on Computing. [23] L.V. Kantorovich (1960). Mathematical methods of organizing and planning production. Management Science 6, 366–422. Translated from the Russian original 1939. Documenta Mathematica

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[24] M.E. L¨ ubbecke and J. Desrosiers (2005). Selected topics in column generation. Operations Research 53, 1007–1023. [25] O. Marcotte (1986). An instance of the cutting stock problem for which the rounding property does not hold. Operations Research Letters 4, 239–243. [26] A. Mehrotra and M.A. Trick (1996). A column generation approach for exact graph coloring. INFORMS Journal on Computing 8, 344–354. [27] J. Rubin (1973). A technique for the solution of massive set covering problems with application to airline crew scheduling. Transportation Science 7, 34–48. [28] D.M. Ryan and B. Foster (1981). An integer programming approach to scheduling. In Computer Scheduling of Public Transport Urban Passenger Vehicle and Crew Scheduling, A. Wren ed. North-Holland, 269–280. [29] G. Scheithauer and J. Terno (1995). The modified integer round-up property of the one-dimensional cutting stock problem. European Journal of Operational Research 84, 562–571. [30] G. Scheithauer and J. Terno (1997). Theoretical investigations on the modified integer round-up property for the one-dimensional cutting stock problem. Operations Research Letters 20, 93–100. [31] P.H. Vance, C. Barnhart, E.L. Johnson and G.L Nemhauser (1994). Solving binary cutting stock problems by column generation and branch-andbound. Computational Optimization and Applications 3, 111–130. [32] F. Vanderbeck (2000). On Dantzig–Wolfe decomposition in integer programming and ways to perform branching in a branch-and-price algorithm. Operations Research 48, 111–128. [33] F. Vanderbeck (2011). Branching in branch-and-price: a generic scheme. Mathematical Programming 130, 249–294. [34] F. Vanderbeck and L.A. Wolsey (1996). An exact algorithm for IP column generation. Operations Research Letters 19, 151–159.

George L. Nemhauser Georgia Institute of Technology Atlanta GA, USA [emailprotected]

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Who Solved the Hirsch Conjecture? ¨ nter M. Ziegler Gu

2010 Mathematics Subject Classification: 90C05 Keywords and Phrases: Linear programming

1

Warren M. Hirsch, who posed the Hirsch conjecture

In the section “The simplex interpretation of the simplex method” of his 1963 classic “Linear Programming and Extensions”, George Dantzig [5, p. 160] describes “informal empirical observations” that While the simplex method appears a natural one to try in the ndimensional space of the variables, it might be expected, a priori, to be inefficient as tehre could be considerable wandering on the outside edges of the convex [set] of solutions before an optimal extreme point is reached. This certainly appears to be true when n − m = k is small, (. . . ) However, empirical experience with thousands of practical problems indicates that the number of iterations is usually close to the number of basic variables in the final set which were not present in the initial set. For an m-equation problem with m different variables in the final basic set, the number of iterations may run anywhere from m as a minimum, to 2m and rarely to 3m. The number is usually less than 3m/2 when there are less than 50 equations and 200 variables (to judge from informal empirical observations). Some believe that on a randomly chosen problem with fixed m, the number of iterations grows in proportion to n. Thus Dantzig gives a lot of empirical evidence, and speculates about random linear programs, before quoting a conjecture about a worst case:

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Warren M. Hirsch (1918–2007) (http://thevillager.com/villager_223/ warrenhirsch.html) This is reiterated and also phrased geometrically in the problems for the same section [5, p. 168]: 13. (W. M. Hirsch, unsolved.) Does there exist a sequence of m or less pivot operations, each generating a new basic feasible solution (b.f.s.), which starts with some given b.f.s. and ends at some other given b.f.s., where m is the number of equations? Expressed geometrically: In a convex region in n − m dimensional space defined by n halfplanes, is m an upper bound for the minimum-length chain of adjacent vertices joining two given vertices? This is the “Hirsch conjecture” – a key problem in the modern theory of polyhedra, motivated by linear programming, backed up by a lot of experimental evidence. Dantzig thus gives credit to Warren M. Hirsch, who had gotten his Ph.D. at New York University’s Courant Institute in 1952, was on the faculty there from 1953 to his retirement 1988. We may note, however, that Hirsch has lasting fame also in other parts of science: Obituaries say that he is best known for his work in mathematical epidemiology. With hindsight, Dantzig’s two renditions of the problem point to many different facets of the later developments. In particular, random linear programs are mentioned – for which good diameter bounds were later proved in celebrated work by Karl Heinz Borgwardt [4]. As the present writer is a geometer at heart, let us translate Dantzig’s geometric version into current terminology (as in [21, Sect. 3.3]): The Hirsch conjecture: For n ≥ d ≥ 2, let ∆(d, n) denote the largest possible diameter of the graph of a d-dimensional polyhedron with n facets. Then ∆(d, n) ≤ n − d. Documenta Mathematica

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A first counterexample

We now know that the Hirsch conjecture – as stated by Dantzig – is false: The credit for this result goes to Victor Klee and David W. Walkup, who in Section 5 of their 1967 Acta paper [15] indeed gave an explicit example of a simple 4-dimensional polyhedron P4 with n = 8 facets and 15 vertices whose graph diameter is equal to δ(P4 ) = 5. Thus, indeed, ∆(4, 8) = 5, which disproved the Hirsch conjecture. Kim & Santos [12, Sect. 3.3] explain nicely how this polyhedron can be derived from a (bounded!) polytope Q4 of dimension 4 with 9 facets – found also by Klee & Walkup – that has two vertices x and y of distance 5, by moving the facet that does not meet x or y to infinity by a projective transformation. From much later enumerations by Altshuler, Bokowski & Steinberg [1] we now know that Q4 is unique with these properties among the 1142 different simple 4-dimensional polytopes with 9 facets. What a feat to find this object! However, instead of just celebrating their example and declaring victory, Klee and Walkup mounted a detailed study on a restricted version of the Hirsch conjecture, which considers (bounded) polytopes in place of (possibly unbounded) polyhedra: The bounded Hirsch conjecture: For n ≥ d ≥ 2, let ∆b (d, n) denote the largest possible diameter of the graph of a d-dimensional polytope with n facets. Then ∆b (d, n) ≤ n − d. As a consequence of the Klee–Walkup example, also using projective transformations, Mike Todd observed that the monotone version of the Hirsch conjecture is false even for polytopes: There is a simple 4-dimensional polytope with

Victor L. Klee (1925–2007) (Photo: L. Danzer, Bildarchiv des Mathematischen Forschungsinstituts Oberwolfach) Documenta Mathematica

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George Dantzig personal.htm)

(1914–2005)

(http://lyle.smu.edu/~jlk/personal/

8 facets, such that from a specified starting vertex and objective function every pivot sequence to the optimum needs at least 5 steps. 3

The Hirsch conjecture, Dantzig figures, and revisits

Published only one year after his classic book, Dantzig [6] presented the following as the first of his “Eight unsolved problems from mathematical programming”: a. Let Cn be an n-dimensional bounded polyhedral convex set defined by 2n distinct faces, n of which determine the extreme point p1 and the remaining n of which determine the extreme point p2 . Does there always exist a chain of edges joining p1 to p2 such that the number of edges in the chain is n? Dantzig did not mention Hirsch in this connection, but he also did not give any references, not even his own book which must just have been published when he compiled the problems. But clearly this is a special case of the Hirsch conjecture, with two restrictions, namely to the case of bounded polytopes with n = 2d facets, and with two antipodal vertices that do not share a facet. This is what Klee and Walkup call a “Dantzig figure.” Klee and Walkup clarified the situation, by proving that the following three fundamental conjectures on convex polytopes are equivalent: The Hirsch conjecture for polytopes: For all d-dimensional bounded polyhedra with n facets, n > d ≥ 2, ∆b (d, n) ≤ n − d. Documenta Mathematica

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Dantzig’s bounded d -step conjecture: For all d-dimensional simple polytopes with 2d facets, the distance between any two complementary vertices that don’t share a facet is d, for d ≥ 2. The nonrevisiting conjecture, by V. Klee and P. Wolfe: From any vertex of a simple convex polytope to any other vertex, there is a path that does not leave a facet and then later come back to it. Some of these implications are quite obvious: For example, a nonrevisiting path starts on a vertex that lies on (at least) d facets, and in every step it reaches a new facet, so its length clearly cannot be more than n − d. Other implications are harder, and in particular they were not established on a dimension-bydimension basis (but rather for fixed m = n − d). The restriction to simple polytopes in all these constructions (that is, ddimensional polytopes such that every vertex lies on exactly d facets) appears at the beginning of the fundamental Klee–Walkup paper. Indeed, right after introduction and preliminaries, Section 2 “Some reductions” starts with the observation 2.1. It is sufficient to consider simple polyhedra and simple polytopes when determining ∆(d, n) and ∆b (d, n). This is, as we will see, true, easy to eastablish, fundamental – and was quite misleading. 4

Francisco Santos solved the Hirsch conjecture

In May 2010, Francisco Santos from the University of Cantabria in Santander, submitted the following abstract to the upcoming Seattle conference “100 Years in Seattle: the mathematics of Klee and Gr¨ unbaum” dedicated to the outstanding geometers Victor Klee (who had passed away in 2007) and Branko Gr¨ unbaum (famous for his 1967 book on polytopes [9], which carried a chapter by V. Klee on diameters of polytopes): Title:

"A counter-example to the Hirsch conjecture"

Author:

Francisco Santos, Universidad de Cantabria

Abstract: I have been in Seattle only once, in November 2003, when I visited to give a seminar talk at U of W. Victor Klee was already retired (he was 78 at that time), but he came to the department. We had a nice conversation during which he asked "Why don’t you try to disprove the Hirsch Conjecture"? Although I have later found out that he asked the same to many Documenta Mathematica

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Francisco “Paco” Santos (*1968)

people, including all his students, the question and the way it was posed made me feel special at that time. This talk is the answer to that question. I will describe the construction of a 43-dimensional polytope with 86 facets and diameter bigger than 43. The proof is based on a generalization of the d-step theorem of Klee and Walkup. Francisco “Paco” Santos, *1968, was known in the polytopes community as an outstanding geometer, who had previously surprised the experts with constructions such as a 6-dimensional triangulation that does not admit a single “bistellar flip.” Thus, as a preprint of his paper was first circulating among a few experts, and then released on the arXiv [18], there was no doubt that this would be correct. Indeed, the announcement contained only one mistake, which was soon corrected: His visit to Seattle had not been in 2003, but in 2002. This is not the place to even sketch Santos’ magnificent construction. Let us just say that his starting point is a generalization of Dantzig’s d-step conjecture: Santos calls a spindle a polytope with two vertices x and y such that all facets contains one of them (but not both). If the polytope has dimension d, then it has n ≥ 2d facets. If such a spindle is simple, then n = 2d: This is the case of a Dantzig figure. So the key for Santos’ approach is to not do the reduction to simple polytopes, but to consider spindles that are not simple. The d-step conjecture for spindles asks for a path of length d between the vertices x and y in any spindle. This happens to exist for d = 3 (exercise for you), and also for d = 4 (not so easy – see Santos et al. [20]). But for d = 5 there is a counterexample, which Santos devised using intuition from a careful Documenta Mathematica

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A Santos spindle, from [19] analysis of the Klee–Walkup example P4 , and which he cleverly explained and visualized in 2- and 3-dimensional images. This example can then be lifted, using Klee–Walkup type “wedging” techniques, to yield a counterexample to the d-step conjecture (and hence the Hirsch conjecture), for d = 43: ∆(43, 86) > 43. Later “tweaking” and “optimization” yielded counterexamples in lower dimensions, arriving at an explicit example of a 20-dimensional Dantzig figure with 40 facets and 36,425 vertices and graph diameter 21 – proving that ∆(20, 40) > 21. See Matschke, Santos & Weibel [16]. 5

If there is a short path, there must be a way to find it

If you want to prove the Hirsch conjecture, or at least prove good upper bounds for the diameter of polytopes, one natural approach is to ask for numerical or combinatorial strategies to find short paths. Indeed, the interest from linear programming certainly is not to only establish the existence of short paths, but to specify pivot rules that find one. Certainly the expectation of Hirsch, Dantzig, and others was that the usual pivot rules used for linear programming (at the time) would not need more than a linear number of steps, which, a fortiori, would establish the existence of “reasonably” short paths. That hope was seriously damaged by a seminal paper by Victor Klee and George Minty from 1972, with the innocuous title “How good is the simplex algorithm?” [14]. The answer was “It is bad”: Klee and Minty constructed linear programs, certain d-dimensional “deformed cubes,” soon known as the “Klee–Minty cubes”, on which the usual largest coefficient pivot rule would take 2d steps. Documenta Mathematica

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c Zadeh’s letter to Victor Klee ( G. M. Ziegler [22], http://www.scilogs.de/ wblogs/blog/mathematik-im-alltag/)

But would a different pivot rule be better? Linear? Establish the Hirsch conjecture? The Klee–Minty breakthrough started a sequence of papers that constructed variants of the “deformed cube” construction, on which the classical pivot rules for lineare programming, one by one, were shown to be exponential in a worst case – an industry that Manfred Padberg criticised as worstcasitis in [17, p. 70]. (The geometric background was formalized as “deformed products” in Amenta & Ziegler [2].) Two pivot rules remained, and defied all attacks, namely • random pivots, and • minimizing revisits. The latter idea, perhaps inspired by Robert Frost’s famous “road less travelled by,” was proposed by the mathematician (and now controversial businessman) Norman Zadeh, who had once offered $ 1000 for a proof or disproof that his “least entered rule” was polynomial: This prize was finally in January 2011 collected, at IPAM, by a doctoral student, Oliver Friedman from Munich, who had used game-theoretic methods to construct linear programs on which Zadeh’s rule is exponential [7]. At the same time, Friedmann, Hansen & Zwick also showed that the “random pivot” rule is exponential [8], thus for the time being destroying all hopes for any “reasonable” pivot rule for the simplex algorithm with polynomial worstcase behaviour. Documenta Mathematica

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Oliver Friedmann (Photo: E. Kim) 6

The Hirsch conjecture is not solved

Clearly, Hirsch and Dantzig were interested in an upper bound on the maximal number of pivots for the simplex algorithm. Santos’ example shows that the upper bound ∆b (d, n) ≤ n−d does not hold in general, but all the lower bounds we have right now are quite weak: From glueing techniques applied to Santos’ examples we get linear lower bounds of the type ∆b (d, n) ≥

21 20 (n

− d)

for very large n and d, while the best available upper bounds by Kalai & Kleitman [11] resp. by Barnette and Larman [3] ∆(d, n) ≤ nlog2 2d

and

∆(d, n) ≤

1 d 12 2 n

are very mildly sub-exponential, resp. linear in n but exponential in d (and hence, for example, exponential for the case n = 2d of the d-step conjecture). The huge gap between these is striking. And if we interpret Hirsch’s question as asking for a good (linear?) upper bound for the worst-case behaviour of the Hirsch conjecture, then all we can say as of now is: We honestly don’t know. Much more could be said – but we refer the readers to Santos’ paper [18], to the surveys by Klee & Kleinschmidt [13] and Kim & Santos [12], and to Gil Kalai’s blog [10] instead. References [1] Amos Altshuler, J¨ urgen Bokowski, and Leon Steinberg. The classification of simplicial 3-spheres with nine vertices into polytopes and nonpolytopes. Discrete Math., 31:115–124, 1980. [2] Nina Amenta and G¨ unter M. Ziegler. Deformed products and maximal shadows. In B. Chazelle, J. E. Goodman, and R. Pollack, editors, Advances Documenta Mathematica

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¨ nter M. Ziegler Gu in Discrete and Computational Geometry (South Hadley, MA, 1996), volume 223 of Contemporary Mathematics, pages 57–90, Providence RI, 1998. Amer. Math. Soc.

[3] David W. Barnette. An upper bound for the diameter of a polytope. Discrete Math., 10:9–13, 1974. [4] Karl Heinz Borgwardt. The Simplex Method. A Probabilistic Analysis, volume 1 of Algorithms and Combinatorics. Springer-Verlag, Berlin Heidelberg, 1987. [5] George B. Dantzig. Linear Programming and Extensions. Princeton University Press, Princeton, 1963. Reprint 1998. [6] George B. Dantzig. Eight unsolved problems from mathematical programming. Bulletin Amer. Math. Soc., 70:499–500, 1964. [7] Oliver Friedmann. A subexponential lower bound for Zadeh’s pivoting rule for solving linear programs and games. In In Proceedings of the 15th Conference on Integer Programming and Combinatorial Optimization, IPCO’11, New York, NY, USA, 2011. [8] Oliver Friedmann, Thomas Hansen, and Uri Zwick. Subexponential lower bounds for randomized pivoting rules for the simplex algorithm. In In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC’11, San Jose, CA, USA, 2011. [9] Branko Gr¨ unbaum. Convex Polytopes, volume 221 of Graduate Texts in Math. Springer-Verlag, New York, 2003. Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler (original edition: Interscience, London 1967). [10] Gil Kalai. Combinatorics and more. Blog, http://gilkalai.wordpress. com/. [11] Gil Kalai and Daniel J. Kleitman. A quasi-polynomial bound for the diameter of graphs of polyhedra. Bulletin Amer. Math. Soc., 26:315–316, 1992. [12] Edward D. Kim and Francisco Santos. An update on the Hirsch conjecture. Jahresbericht der DMV, 112:73–98, 2010. [13] Victor Klee and Peter Kleinschmidt. The d-step conjecture and its relatives. Math. Operations Research, 12:718–755, 1987. [14] Victor Klee and George J. Minty. How good is the simplex algorithm? In O. Shisha, editor, Inequalitites, III, pages 159–175. Academic Press, New York, 1972. Documenta Mathematica

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[15] Victor Klee and David W. Walkup. The d-step conjecture for polyhedra of dimension d < 6. Acta Math., 117:53–78, 1967. [16] Benjamin Matschke, Francisco Santos, and Christophe Weibel. The width of 5-dimensional prismatoids. Preprint, February 2012, 28 pages, http:// arxiv.org/abs/1202.4701. [17] Manfred Padberg. Linear Optimization and Extensions, volume 12 of Algorithms and Combinatorics. Springer-Verlag, Heidelberg, second edition, 1999. [18] Francisco Santos. A counterexample to the Hirsch conjecture. Preprint http://arxiv.org/abs/1006.2814, 27 pages, June 2010; Annals of Math. 176 (2012), to appear (published online Nov. 2011). ¨ [19] Francisco Santos. Uber ein Gegenbeispiel zur Hirsch-Vermutung. Mitteilungen der DMV, 18:214–221, 2010. Translated by J. Pfeifle. [20] Francisco Santos, Tamon Stephen, and Hugh Thomas. Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids. Discrete Comput. Geometry, 47:569–576, 2012. [21] G¨ unter M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. Revised edition, 1998; seventh updated printing 2007. [22] G¨ unter M. Ziegler. Typical and extremal linear programs. In M. Gr¨otschel, editor, The Sharpest Cut: The Impact of Manfred Padberg and His Work, volume 4 of MPS-SIAM Series on Optimization, chapter 14, pages 217– 230. SIAM, Philadelphia, PA, 2004.

G¨ unter M. Ziegler Inst. Mathematics Freie Universit¨at Berlin Arnimallee 2 14195 Berlin Germany [emailprotected]

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Pope Gregory, the Calendar, and Continued Fractions Friedrich Eisenbrand

Abstract. The success of many activities of modern civilization crucially depends on careful planning. Some activities should be carried out during a certain period of the year. For example: When is the right time of the year to sow, when is the right time to plow? It is thus no surprise that calendars are found in literally every ancient civilization. The earth revolves around the sun in about 365.2422 days. An accurate calendar can thus not provision the same number of days every year if the calendar should be synchronous with the seasons. This article is about the problem of approximating a given number by a rational number with small denominator, continued fractions and their relationship to the Gregorian calendar with its leap-year rule that is still in use today and keeps the calendar synchronized for a very long time. 2010 Mathematics Subject Classification: 11J70, 11Y16, 11A55 Keywords and Phrases: Calendar, Diophantine approximation, continued fractions

The Julian calendar and Gregory’s reform The number 365.2422 is close to 365 + 1/4. If this was precisely the duration of one year in days, then the following rule would result in an exact calendar. Each year that is divisible by 4 consists of 366 days and each other year consists of 365 days. The mean duration of a calendar year is thus 365 + 1/4. In other words, each year that is divisible by 4 will be a leap year. This leap year rule was imposed by Julius Cesar in 45 B.C. Already at this time, astronomers calculated the duration of a year in days fairly accurately and it was clear that the calendar would be behind by one day in roughly 130 years. Documenta Mathematica

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Friedrich Eisenbrand

In 1582, when the Julian calendar was evidently out of sync by a large extent, pope Gregory the XIII imposed the following calendar reform. As before, every year that is divisible by 4 is a leap-year, except for those divisible by 100 but not by 400. The mean duration of a year of the Gregorian calendar is thus 365 + 97/400. Best approximations What is the mathematical challenge behind the design of an accurate leap-year rule? The task is to approximate the number 0.2422 by a rational number p/q with p, q ∈ N+ such that q as well as the error E = |.2422 − p/q| is small. The mean duration of a calendar year is then 365 + p/q if the calendar provisions p leap years every q years. The smaller the q, the simpler should be the leap-year rule. In the Julian calendar, p/q = 1/4. The rule “Each year divisible by four is a leap year” is easy to remember. In 1/E years, the calendar will then be ahead by one day or behind by one day depending on whether p/q is smaller or larger than 0.2422. Finding a convenient and sufficiently accurate leap-year rule is related to approximating a real number α ∈ R≥0 by a rational number p/q in a good way. In the following we always assume that p is a natural number or 0 and that q is a positive natural number when we speak about the representation p/q of a rational number. The rational number p/q is a best approximation of α if for any other rational number p′ /q ′ 6= p/q one has |α − p/q| < |α − p′ /q ′ | if q ′ ≤ q. Going back to the calendar problem, this makes sense. If there exists an approximation p′ /q ′ of 0.2422 with q ′ ≤ q that results in a smaller error, then we could hope that we can find a leap year rule that accommodates for p′ leap years in q ′ years instead of the one that accommodates for p leap years in q years that is just as easy to remember. Furthermore, the calendar would be more accurate. Continued fractions Continued fractions have been used to approximate numbers for a very long time and it seems impossible to attribute their first use to a particular researcher or even to a particular ancient civilization. Keeping the best approximation problem in mind however, the application of continued fractions seems natural. Suppose our task is to approximate α ∈ R≥0 by a rational number with small denominator. If α is not a natural number then we can re-write α = ⌊α⌋ + (α − ⌊α⌋) 1 . = ⌊α⌋ + 1/(α − ⌊α⌋) Documenta Mathematica

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The number β = 1/(α − ⌊α⌋) is larger than one. If β is not a natural number, one continues to expand the number β and obtains α = ⌊α⌋ +

1 1 ⌊β⌋ + 1/(β − ⌊β⌋)

.

The continued fraction expansion of α is inductively defined as the sequence α if α ∈ N and ⌊α⌋, a1 , a2 , . . . otherwise, where a1 , a2 , . . . is the continued fraction expansion of 1/(α − ⌊α⌋). On the other hand, a finite sequence of integers b0 , . . . , bn , all positive, except perhaps b0 gives rise to the continued fraction 1

hb0 , . . . , bn i = b0 + b1 +

.

1 ..

.+

1 bn

If the sequence a0 , a1 , . . . is the continued fraction expansion of α ∈ R≥0 and if its length is at least k + 1, then the k-th convergent of α is the continued fraction 1 ha0 , . . . , ak i = a0 + 1 a1 + 1 .. .+ ak Let us compute the first convergents of the number α = 365.2422. Clearly, a0 is 365. To continue, it is convenient to represent α as a rational number α = 1826211/5000. Clearly α − ⌊α⌋ is the remainder of the division of 1826211 by 5000 divided by 5000. One has 1826211 = 5000 · 365 + 1211. Thus we continue to expand 5000/1211 and obtain a1 = 4. The remainder of the division of 5000 by 1211 is 156 which means that we next expand 1211/156 which results in a2 = 7. The remainder of this division is 119 and we next expand 156/119 resulting in a3 = 1, then 119/37 yielding a4 = 3 and 37/8 yields a5 = 4. At this point we can record an important observation. If α = p/q is a rational number, then its continued fraction expansion is precisely the sequence of quotients of the division-with-remainder steps that are carried out by the Euclidean algorithm on input p and q. Also, for arbitrary real α ∈ R≥0 , the function fk (x) = ha0 , . . . , ak−1 , xi defined for x > 0 is strictly increasing in x if k is even and decreasing if k is odd. Furthermore, if k is even, then ak is the largest integer with ha0 , . . . , ak i ≤ α and if k is odd then ak is the largest integer such that ha0 , . . . , ak i ≥ α. Documenta Mathematica

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The quality of the Gregorian calendar The third convergent of 365.2422 is 1

365 +

= 365 + 8/33.

1

4+

7+

1 1

According to Rickey [6], the Persian mathematician, philosopher and poet Omar Khayyam (1048 - 1131) suggested a 33-year cycle where the years 4, 8, 12, 16, 20, 24, 28 and 33 should be leap years. Thus the mean-duration of a year according to his suggestion would be exactly the value of the third convergent. How does this compare to the mean duration of a year of the Gregorian calendar. We calculate both error terms E1 = |365.2422 − 365 + 8/33| = 0.000224242424242432 E2 = |365.2422 − 365 + 97/400| = 0.000299999999999995 and surprisingly, one finds that Omar Khayyam’s leap-year rule is more accurate. Using the third convergent, his calendar will be imprecise by one day in roughly 4459.45 years, whereas Gregory’s calendar will be off by one day in “only” 3333.33 years. Still the leap-year rule of the Gregorian calendar is convenient, as it relates nicely with our decimal number system and is simple to remember. However, why is it a good idea to approximate a number by its convergent? What is the relation of the convergents of a number with its best approximations? Best approximations and convergents We now explain the relationship of convergents of α ∈ R≥0 and best approximations. The subject is nicely treated in [2]. Let a0 , a1 , . . . be a sequence of natural numbers where again all are positive except perhaps a0 and consider the two sequences gk and hk that are inductively defined as g−1 g−2 1 0 gk gk−1 gk−1 gk−2 ak 1 = , = , k ≥ 0. h−1 h−2 0 1 hk hk−1 hk−1 hk−2 1 0 (1) It follows from a simple inductive argument that, if βk is the number βk = gk /hk , then one has ha0 , . . . , ak i = βk for k ≥ 0. Now the process of forming convergents admits a nice geometric interpretation. Notice that, since the ai are integers and since the determinant of gk gk−1 (2) hk hk−1 Documenta Mathematica

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g h=α·g

vk−1

vk 0

h

hk−1

Figure 1: An illustration of the geometric interpretation of convergents is 1, such a matrix (2) is a basis of the standard lattice Z2 . This means that each vector in Z2 can be obtained by multiplying the matrix (2) with an integral 2-dimensional vector and conversely, the result of such a multiplication g is always an integral 2-dimensional vector. If vk = hkk then the line with slope α through 0 is sandwiched between the vectors vk and vk−1 in the positive orthant, see Figure 1. In Figure 1, the rational number gk−1 /hk−1 is larger than α. Since there is no integer point in the shaded region, any other rational number p/q ≥ α with p/q − α ≤ gk−1 /hk−1 − α must have a denominator that is larger than hk−1 . One says that gk−1 /hk−1 is a best approximation from above. Similarly, gk /hk is a best approximation from below. At this point it is already clear that one of the convergents is a best approximation. Next we show that the following best approximation problem can be solved in polynomial time. Given a rational number α ∈ Q>0 and a positive integer M , compute the best approximation of α with denominator bounded by M , i.e., compute a rational number p/q with p ≤ M such that |α − p/q| is minimum. The algorithm is described in [2], see also [1], and is as follows. One computes the convergents α as long as the denominator (h-component) of the latest convergent is bounded by M . Since the denominators double every second round, the number of steps is bounded by the encoding length of M . Suppose that this is the k-th convergent and we denote the columns of the matrix (2) again by vk and vk−1 . In the next round, the new first column would be vk−1 + ak+1 · vk but the h-component of this vector exceeds M . Instead, one computes now the largest µ ∈ N0 such that the h-component of vk−1 + µ · vk does not exceed M . If we denote the resulting vector by u then still u, vk is a basis of Z2 but the second component of u + vk exceeds M . The situation is depicted in Figure 2. Any rational number p/q that approximates α better Documenta Mathematica

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u

vk h

h≤M Figure 2: An illustration of the algorithm solving the best approximation problem than u and vk is in the cone C spanned by u and vk C = {λ1 u + λ2 vk : λ1 , λ2 ≥ 0}. But if this rational number is different from the one represented by u and v, then λ1 and λ2 must be strictly positive. However, since u and vk form a lattice-basis, λ1 and λ2 are positive integers and thus the h-component q of the corresponding vector exceeds M . Thus u or vk is a solution to the bestapproximation problem. Further historical remarks Continued fractions are a true classic in mathematics and it is impossible to give a thorough historical account. In this final section I content myself with a very brief discussion of computational issues related to best approximations and continued fractions and some recent results. The simultaneous best approximation problem is the high-dimensional counterpart to the best approximation problem that we discussed. Here, one is given a rational vector and a denominator bound and the task is to find another rational vector where each component has the same denominator that is bounded by the prescribed denominator bound. The objective is to minimize the error in the ℓ∞ -norm. Lagarias [3] has shown that this problem is NP-hard and applied the LLL-algorithm [4] to approximate this optimization problem. Variants of this simultaneous best approximation problem are also shown to be hard to approximate [7]. Sch¨onhage [8] showed how to compute convergents in a quasilinear amount of bit-operations. Recently Novocin, Stehl´e and Villard [5] have shown that a variant of LLL-reduction depends on the bit-size of the largest input coefficient in a similar way. Documenta Mathematica

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References [1] M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 1988. [2] A. Ya. Khinchin. Continued fractions. Dover Publications Inc., Mineola, NY, russian edition, 1997. Reprint of the 1964 translation. The first Russion edition was published in 1935. [3] J. C. Lagarias. The computational complexity of simultaneous diophantine approximation problems. SIAM J. Computing, 14(1):196–209, 1985. [4] A. K. Lenstra, H. W. Lenstra, and L. Lov´asz. Factoring polynomials with rational coefficients. Math. Annalen, 261:515–534, 1982. [5] Andrew Novocin, Damien Stehl´e, and Gilles Villard. An lll-reduction algorithm with quasi-linear time complexity: extended abstract. In Lance Fortnow and Salil P. Vadhan, editors, STOC, pages 403–412. ACM, 2011. [6] V. Frederick Rickey. Mathematics of the Gregorian calendar. The Mathematical Intelligencer, 7(1):53–56, 1985. [7] Carsten R¨ ossner and Jean-Pierre Seifert. Approximating good simultaneous Diophantine approximations is almost NP-hard. In Mathematical foundations of computer science 1996 (Cracow), volume 1113 of Lecture Notes in Comput. Sci., pages 494–505. Springer, Berlin, 1996. [8] A. Sch¨ onhage. Schnelle Berechnung von Kettenbruchentwicklungen. Acta Informatica, 1:139–144, 1971.

Friedrich Eisenbrand EPFL SB IMA MA C1 573 1015 Lausanne Switzerland [emailprotected]

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¨ wner–John Ellipsoids Lo Martin Henk

2010 Mathematics Subject Classiﬁcation: 52XX, 90CXX Keywords and Phrases: L¨owner-John ellipsoids, volume, ellipsoid method, (reverse) isoperimetric inequality, Kalai’s 3n -conjecture, norm approximation, non-negative hom*ogeneous polynomials

1

The men behind the ellipsoids

Before giving the mathematical description of the L¨owner–John ellipsoids and pointing out some of their far-ranging applications, I brieﬂy illuminate the adventurous life of the two eminent mathematicians, by whom the ellipsoids are named: Charles Loewner (Karel L¨owner) and Fritz John. Karel L¨ owner (see Figure 1) was born into a Jewish family in L´any, a small town about 30 km west of Prague, in 1893. Due to his father’s liking for German

Figure 1: Charles Loewner in 1963 (Source: Wikimedia Commons) Documenta Mathematica

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style education, Karel attended a German Gymnasium in Prague and in 1912 he began his studies at German Charles-Ferdinand University in Prague, where he not only studied mathematics, but also physics, astronomy, chemistry and meteorology. He made his Ph.D. in 1917 under supervision of Georg Pick on a distortion theorem for a class of holomorphic functions. In 1922 he moved to the University of Berlin, where he made his Habilitation in 1923 on the solution of a special case of the famous Bieberbach conjecture. In 1928 he was appointed as non-permanent extraordinary professor at Cologne, and in 1930 he moved back to Prague where he became ﬁrst an extraordinary professor and then a full professor at the German University in Prague in 1934. After the complete occupation of Czech lands in 1939 by Nazi Germany, L¨ owner was forced to leave his homeland with his family and emigrated to the United States. From this point on he changed his name to Charles Loewner. He worked for a couple of years at Louisville, Brown and Syracuse University, and in 1951 he moved to Stanford University. He died in Stanford in 1968 at the age of 75. Among the main research interests of Loewner were geometric function theory, ﬂuid dynamics, partial diﬀerential equations and semigroups. Robert Finn (Stanford) wrote about Loewner’s scientiﬁc work: “Loewners Ver¨ oﬀentlichungen sind nach heutigen Maßst¨aben zwar nicht zahlreich, aber jede f¨ ur sich richtungsweisend.”1 Fritz John2 was born in Berlin in 1910 and studied mathematics in G¨ottingen where he was most inﬂuenced by Courant, Herglotz and Lewy. Shortly after Hitler had come to power in January 1933, he – as a Non-Aryan – lost his scholarship which gave him, in addition to the general discrimination of NonAryans, a very hard ﬁnancial time. In July 1933, under supervision of Courant he ﬁnished his Ph.D. on a reconstructing problem of functions, which was suggested to him by Lewy. With the help of Courant he left Germany in the beginning of 1934 and stayed for one year in Cambridge. Fortunately, in 1935 he got an assistant professorship in Lexington, Kentucky, where he was promoted to associate professor in 1942. Four years later, 1946, he moved to New York University where he joined Courant, Friedrichs and Stoker in building the institute which later became the Courant Institute of Mathematical Sciences. In 1951 he was appointed full professor at NYU and remained there until his retirement 1981. He died in New Rochelle, NY, in 1994 at the age of 84. For his deep and pioneering contributions to diﬀerent areas of mathematics which include partial diﬀerential equations, Radon transformations, convex geometry, numerical analysis, ill-posed problems etc., he received many awards and distinctions. For detailed information on life and impact of Karel L¨owner and Fritz John we refer to [16, 25, 27, 35, 36, 37, 39, 40].

1 “Compared

to today’s standards, Loewner’s publications are not many, yet each of them is far reaching.” 2 For a picture see the article of Richard W. Cottle [13] in this volume. Documenta Mathematica

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The ellipsoids

Before presenting the L¨ owner–John ellipsoids let me brieﬂy ﬁx some notations. An ellipsoid E in the n-dimensional Euclidean space Rn is the image of the unit ball Bn , i.e., the ball of radius 1 centered at the origin, under a regular aﬃne transformation. So there exist a t ∈ Rn , the center of the ellipsoid, and a regular matrix T ∈ Rn×n such that E = t + T Bn = {t + T y : y ∈ Bn } = {x ∈ Rn : ||T −1 (x − t)|| ≤ 1},

(1)

where || · || denotes the Euclidean norm. By standard compactness arguments it can be easily seen that every convex body K ⊂ Rn , i.e., convex compact set with interior points, has an inscribed and circ*mscribed ellipsoid of maximal and minimal volume, respectively.

Figure 2: Maximal inscribed ellipse of a ﬂat diamond, and minimal circ*mscribed ellipse (circle) of a regular triangle To prove, however, that these extremal volume ellipsoids are uniquely determined requires some work. In the planar case n = 2, this was shown by F. Behrend3 in 1937/38 [7, 8]. O.B. Ader, a student of Fritz John in Kentucky, treated a special 3-dimensional case [1], and the ﬁrst proof of uniqueness of these ellipsoids in general seems to have been given by Danzer, Laugwitz and Lenz in 1957 [14] and independently by Zaguskin [45]. In his seminal paper Extremum problems with inequalities as subsidiary conditions [26], Fritz John extends the Lagrange multiplier rule to the case of (possibly inﬁnitely many) inequalities as side constraints. As an application of his optimality criterion he shows that for the minimal volume ellipsoid t+T Bn , say, containing K it holds t+

1 T B n ⊂ K ⊆ t + T Bn . n

(2)

In other words, K can be sandwiched between two concentric ellipsoids of ratio n. According to Harold W. Kuhn [30], the geometric problem (2) and related questions from convex geometry were John’s main motivation for his paper [26]. John also pointed out that for convex bodies having a center of symmetry, i.e., 3 Felix Adalbert Behrend was awarded a Doctor of Science at German University in Prague in 1938 and most likely, he discussed and collaborated with Karel L¨ owner on the ellipsoids.

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Figure 3: Minimal volume ellipses together with their concentric copies scaled by 21 for the triangle and by √12 for the square there exists a c ∈ Rn √ such that K = c − K = {c − y : y ∈ K}, the factor 1/n can be replaced by 1/ n and that both bounds are best possible as a simplex and a cube show (see Figure 3). Actually, his optimality criterion gives more information about the geometry of minimal (or maximal) volume ellipsoids and together with a reﬁnement/supplement by Keith Ball from 1992 [3] (see also Pelczy´ nski [38] and [4, 21, 29]) we have the following beautiful characterization: Theorem 2.1 (John). Let K ⊂ Rn be a convex body and let K ⊆ Bn . Then the following statements are equivalent: i) Bn is the unique minimal volume ellipsoid containing K. ii) There exist contact points u1 , . . . , um ∈ bdK ∩ bdBn , i.e., lying in the boundary of K and Bn , and positive numbers λ1 , . . . , λm , m ≥ n, such that m m X X λi ui = 0 and In = λi (ui u⊺i ), i=1

i=1

where In is the (n × n)- identity matrix. For instance, let Cn = [−1, 1]n be the cube of √ edge length 2 centered at the origin. C is contained in the ball of radius n centered at the origin, i.e., n √ nBn , which is the minimal volume ellipsoid containing Cn . To see this, we observe that the statement above is invariant with respect √ to scalings of Bn . Thus it suﬃces to look for contact points in bdCn ∩ bd nBn satisfying ii). P n Obviously, all the 2 vertices u of C are contact points and since u = 0 i n i P and (ui u⊺i ) = 2n In we are done. But do we need all of them? Or, in general, are there upper bounds on the number of contact points needed for the decomposition of the identity matrix in Theorem 2.1 ii)? There are! In the general case the upper bound is n(n + 3)/2 as it was pointed out by John. For symmetric bodies we can replace it by n(n + 1)/2. Hence we can ﬁnd at most n(n + 1)/2 vertices of the cube such that the unit ball is also the minimal volume ellipsoid of the convex hull of these vertices. For the number of contact points for “typical” convex bodies we refer to Gruber [22, 23]. Documenta Mathematica

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For maximal volume inscribed ellipsoids we have the same characterization as in the theorem above. Hence we also see that Bn is the maximal volume ellipsoid contained in Cn . Here we take as contact points the unit vectors (see Figure 3). According to Busemann [11], L¨owner discovered the uniqueness of the minimal volume ellipsoid but “did not publish his result” (see also [12, p. 90]), and in honor of Karel L¨ owner and Fritz John these extremal volume ellipsoids are called L¨ owner–John ellipsoids. Sometimes they are also called John-L¨owner ellipsoids (see, e.g., [9]), just John-ellipsoids, when the emphasis is more on the decomposition property ii) in Theorem 2.1 (see, e.g., [19, 4]), or it also happens that the maximal inscribed ellipsoids are called John-ellipsoids and the L¨owner-ellipsoids are the circ*mscribed ones (see, e.g., [24]). 3

Ellipsoids in action

From my point of view the applications can be roughly divided into two classes, either the L¨ owner–John ellipsoids are used in order to bring the body into a “good position” by an aﬃne transformation or they serve as a “good&easy” approximation of a given convex body. I start with some instances of the ﬁrst class, since problems from this class were the main motivation to investigate these ellipsoids. To simplify the language, we call a convex body K in L¨ owner–John-position, if the unit ball Bn is the minimal volume ellipsoid containing K. Reverse geometric inequalities. For a convex body K ⊂ Rn let r(K) be the radius of a largest ball contained in K, and let R(K) be the radius of the smallest ball containing K. Then we obviously have R(K)/r(K) ≥ 1 and, in general, we cannot bound that ratio from above, as, e.g., ﬂat or needle-like bodies show (see Figure 2). If we allow, however, to apply aﬃne transformations to K, the situation changes. Assuming that K is in its L¨owner–John-position, by (2) we get R(K)/r(K) ≤ n and so (cf. [33]) 1≤

max

K convex body

R(α(K)) ≤ n. α regular affine transf. r(α(K)) min

The lower bound is attained for ellipsoids and the upper bound for simplices. The study of this type of reverse inequalities or “aﬃne invariant inequalities” goes back to the already mentioned work of Behrend [7] (see also the paper of John [26, Section 3]) and is of great importance in convex geometry. Another, and more involved, example of this type is a reverse isoperimetric inequality. Here the ratio of the surface area F(K) to the volume V(K) of a convex body K is studied. The classical isoperimetric inequality states that among all bodies of a given ﬁxed volume, the ball has minimal surface area, and, again, ﬂat bodies show that there is no upper bound. Based on John’s Documenta Mathematica

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Theorem 2.1, however, Ball [2] proved that simplices give an upper bound, provided we allow aﬃne transformations. More precisely, we have 1

1

F(Bn ) n−1 1

V(Bn ) n

≤

max

K convex body

min

α regular affine transf.

F(α(K)) n−1 1

V(α(K)) n

1

≤

F(Sn ) n−1 1

,

V(Sn ) n

where Sn is a regular n-simplex. For more applications of this type we refer to the survey [17]. Faces of symmetric polytopes. One of my favorite and most surprising applications is a result on the number of vertices f0 (P ) and facets fn−1 (P ), i.e., (n − 1)-dimensional faces, of a polytope P ⊂ Rn which is symmetric with respect to the origin. For this class of polytopes, it is conjectured by Kalai that the total number of all faces (vertices, edges, . . . , facets) is at least 3n − 1, as for instance in the case of the cube Cn = [−1, 1]n . So far this has been veriﬁed in dimensions n ≤ 4 [41], and not much is known about the number of faces of symmetric polytopes in arbitrary dimensions. One of the very few exceptions is a result by Figiel, Lindenstrauss and Milman [15], where they show ln(f0 (P )) ln(fn−1 (P )) ≥

1 n. 16 √

In particular, either f0 (P ) or fn−1 (P ) has to be of size ∼ e n . For the proof it is essential that √ in the case of symmetric polytopes the factor n in (2) can be replaced by n. For more details we refer to [5, pp. 274]. Preprocessing in algorithms. Also in various algorithmic related problems in optimization, computational geometry, etc., it is of advantage to bring ﬁrst the convex body in question close to its L¨owner–John-position, in order to avoid almost degenerate, i.e., needle-like, ﬂat bodies. A famous example in this context is the celebrated algorithm of Lenstra [34] for solving integer programming problems in polynomial time in ﬁxed dimension. Given a rational polytope P ⊂ Rn , in a preprocessing step an aﬃne transformation α is constructed such that α(P ) has a “spherical appearance”, which means that R(α(P ))/r(α(P )) is bounded from above by a constant depending only on n. Of course, this could be easily done, if we could determine a L¨owner–John ellipsoid (either inscribed or circ*mscribed) in polynomial time. In general this seems to be a hard task, but there are polynomial time algorithms which compute a (1 + ǫ)-approximation of a L¨owner–John ellipsoid for ﬁxed ǫ. For more references and for an overview of the current state of the art of computing L¨ owner–John ellipsoids we refer to [44] and the references therein. In some special cases, however, we can give an explicit formula for the minimal volume ellipsoid containing a body K, and so we obtain a “good&easy” approximation of K. This brings me to my second class of applications of L¨ owner–John ellipsoids. Documenta Mathematica

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H P

Figure 4: The L¨ owner–John ellipse of a half-ellipse Khachiyan’s ellipsoid allgorithm. The famous polynomial time algorithm of Khachiyan for solving linear programming problems is based on the idea to construct a sequence of ellipsoids of strictly decreasing volume containing the given polytope until either the center of an ellipsoid lies inside our given polytope or the volume of the ellipsoids is so small that we can conclude that the polytope must be empty (roughly speaking). This “ellipsoid method” goes back to works of N. Z. Shor [43] and Judin and Nemirovski˘ı [28] (see also the articles of Robert E. Bixby [10] and David Shanno [42] in this volume). Assuming that our polytope P is contained in an ellipsoid t + T Bn , say, we are faced with the question what to do if t ∈ / P . But then we know that one of the inequalities describing our polytope P induces a hyperplane H passing through the center t, such that P is entirely contained in one of the halfspaces H + , say, associated to H. Hence we know P ⊂ (t + T Bn ) ∩ H + , and in order to iterate this process we have to ﬁnd a “small” ellipsoid containing the half-ellipsoid (t + T Bn ) ∩ H + . Here it turns out that the L¨owner–John ellipsoid of minimal volume containing (t + T Bn ) ∩ H + (see Figure 4) can be explicitly calculated by a formula (see, e.g., [20, p. 70]) and the ratio of the volumes of two consecutive ellipsoids in the sequence is less than e−1/(2n) . To turn this theoretic idea into a polynomial time algorithm, however, needs more work. In this context, we refer to [20, Chapter 3], where also variants of this basic ellipsoid method are discussed. Extremal geometric problems. In geometric inequalities, where one is interested in maximizing or minimizing a certain functional among all convex bodies, the approximation of the convex body by (one of) its L¨owner–John ellipsoids gives a reasonable ﬁrst (and sometimes optimal) bound. As an example we consider the Banach–Mazur distance d(K, M ) between two convex bodies K, M ⊂ Rn . Here, d(K, M ) is the smallest factor δ such that there exist an aﬃne transformation α and a point x ∈ Rn with K ⊆ α(M ) ⊆ δ K + x. This distance is symmetric and multiplicative, i.e., d(K, M ) = d(M, K) ≤ d(M, L) d(L, K). Of course, this distance perfectly ﬁts to L¨owner–John ellipsoids and by (2) we have d(Bn , K) ≤ n for every convex body K. So we immediately get that the Documenta Mathematica

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Banach-Mazur distance between any pair of convex bodies is bounded, namely d(K, M ) ≤ d(Bn , K) d(Bn , M ) ≤ n2 . But how good is this bound? This is still an open problem and for the current best lower and upper bounds as well as related questions on the Banach-Mazur distance we refer to [19, Sec. 7.2]. 4

Beyond Ellipsoids

Looking at (2) and Theorem 2.1, it is quite natural to ask, what happens if we replace the class of ellipsoids, i.e., the aﬃne images of Bn , by parallelepipeds, i.e., the aﬃne images of the cube Cn , or, in general, by the aﬃne images of a given convex body L. This question was studied by Giannopoulos, Perissinaki and Tsolomitis in their paper John’s theorem for an arbitrary pair of convex bodies [18]. They give necessary and suﬃcient conditions when a convex body L has minimal volume among all its aﬃne images containing a given body K which nicely generalize Theorem 2.1. One consequence is that for every convex body K, there exists a parallelepiped t + T Cn such that (cf. (2) and see also Lassak [31]) 1 T C n ⊂ K ⊂ t + T Cn . t+ 2n − 1 Observe, that in this more general setting we lose the uniqueness of an optimal solution. Another obvious question is: what can be said about minimal circ*mscribed and maximal inscribed ellipsoids when we replace the volume functional by the surface area, or, in general, by so the called intrinsic volumes? For answers in this context we refer to Gruber [23]. In view of (2), ellipsoids E = T Bn with center 0 may be described by an inequality of the form E = {x ∈ Rn : p2 (x) ≤ 1}, where p2 (x) = x⊺ T −⊺ T −1 x ∈ R[x] is a hom*ogeneous non-negative polynomial of degree 2. Given a convex body K symmetric with respect to the origin, the center t in (2) of the minimal volume ellipsoid is the origin and so we can restate (2) as follows: for any 0symmetric convex body K there exists a non-negative hom*ogeneous polynomial p2 (x) of degree 2 such that

1 p2 (x) n

21

1

≤ |x|K ≤ p2 (x) 2 for all x ∈ Rn ,

(3)

where |x|K = min{λ ≥ 0 : x ∈ λ K} is the gauge or Minkowski function of K. In fact, this formulation can also be found at the end of John’s paper [26]. Since | · |K deﬁnes a norm on Rn and any norm can be described in this way, (3) tells us, how well a given arbitrary norm can be approximated by a hom*ogeneous polynomial of degree 2, i.e., by the Euclidean norm. So what can we gain if we allow higher degree non-negative hom*ogeneous polynomials? In [6], Barvinok studied this question and proved that for any norm | · | on Rn and Documenta Mathematica

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any odd integer d there exists a non-negative hom*ogeneous polynomial p2d (x) of degree 2d such that 1

p2d (x)

d+n−1 d

1 ! 2d

1

≤ |x| ≤ p2d (x) 2d for all x ∈ Rn .

Observe, for d = 1 we get (3) and thus (2) for symmetric bodies, but in general it is not known whether the factor d+n−1 is best possible. Barvinok’s proof d is to some extent also an application of John’s theorem as in one step it uses (2) in a certain d+n−1 -dimensional vector space. In [6] there is also a variant d for non-symmetric gauge functions (non-symmetric convex bodies) which, in particular, implies (2) in the case d = 1. In a recent paper Jean B. Lasserre [32] studied the following even more general problem: Given a compact set U ⊂ Rn and d ∈ N, ﬁnd a hom*ogeneous polynomial g of degree 2d such that its sublevel set G = {x ∈ Rn : g(x) ≤ 1} contains U and has minimum volume among all such sublevel sets containing U . It turns out that this is a ﬁnite-dimensional convex optimization problem and in [32, Theorem 3.2] a characterization of the optimal solutions is given which “perfectly” generalizes Theorem 2.1. In particular, the optimal solutions are also determined by ﬁnitely many “contact points”. Acknowledgements. I would like to thank very much Peter M. Gruber, Jaroslav Nesetril and Ivan Netuka for all their help and information regarding the history of the L¨ owner–John ellipsoids. For many helpful comments on earlier drafts I want to thank Matthias Henze, Mar´ıa Hern´andez Cifre, Eva Linke and Carsten Thiel. References [1] O.B. Ader. An aﬃne invariant of convex bodies. Duke Math. J., 4(2):291– 299, 1938. [2] K. Ball. Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. (2), 44(2):351–359, 1991. [3] K. Ball. Ellipsoids of maximal volume in convex bodies. Geom. Dedicata, 41:241–250, 1992. [4] K. Ball. An Elementary Introduction to Modern Convex Geometry. Cambridge University Press. Math. Sci. Res. Inst. Publ., 31:1–58, 1997. [5] A. Barvinok. A course in convexity, volume 54 of Graduate Studies in Mathematics. AMS, 2002. [6] A. Barvinok. Approximating a norm by a polynomial. GAFA seminar notes (Springer Lect. Notes Math.), 1807:20–26, 2003. Documenta Mathematica

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¨ [7] F. Behrend. Uber einige Aﬃninvarianten konvexer Bereiche. Math. Annalen, 113:713–747, 1937. ¨ [8] F. Behrend. Uber die kleinste umbeschriebene und die gr¨oßte einbeschriebene Ellipse eines konvexen Bereichs. Math. Annalen, 115(1):379– 411, 1938. [9] M. Berger. Geometry Revealed. Springer book, pages 1–840, 2010. [10] R. E. Bixby A Brief History of Linear and Mixed-Integer Programming Computation, this volume. [11] H. Busemann. The Foundations of Minkowskian Geometry. Comment. Math. Helv., 24:156–187, 1950. [12] H. Busemann. The geometry of geodesics. Academic Press Inc., New York, N. Y., 1955. [13] R.W. Cottle William Karush and the KKT theorem, this volume. ¨ [14] L. Danzer, D. Laugwitz, and H. Lenz. Uber das L¨ownersche Ellipsoid und sein Analogon unter den einem Eik¨orper einbeschriebenen Ellipsoiden. Arch. Math., 8:214–219, 1957. [15] T. Figiel, J. Lindenstrauss, and V.D. Milman. The dimension of almost spherical sections of convex bodies. Acta Mathematica, 139:53–94, 1977. [16] R. Finn. Nachlass von Charles Loewner. DMV Mitteilungen, 17(1):58, 2009. [17] R.J. Gardner. The Brunn-Minkowski inequality. Bull. Am. Math. Soc., New Ser., 39(3):355–405, 2002. [18] A. Giannopoulos, I. Perissinaki, and A. Tsolomitis. John’s theorem for an arbitrary pair of convex bodies. Geom. Dedicata, 84(1-3):63–79, 2001. [19] A. Giannopoulos and V.D. Milman. Euclidean structure in ﬁnite dimensional normed spaces. In Handbook of the Geometry of Banach Spaces, pages 709–777. North-Holland, 2001. [20] M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver. Geometric algorithms and combinatorial optimization. Algorithms and Combinatorics. Springer, 2nd edition, 1993. [21] P.M. Gruber and F.E. Schuster. An arithmetic proof of John’s ellipsoid theorem. Arch. Math., 85:82–88, 2005. [22] P.M. Gruber. Minimal ellipsoids and their duals Rend. Circ. Mat. Palermo (2), 37(1):35–64, 1988. Documenta Mathematica

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[23] P.M. Gruber. Application of an idea of Voronoi to John type poblems. Adv. Math., 218(2):309–351, 2008. [24] P.M. Gruber. John and Loewner Ellipsoids. 46(4):776–788, 2011.

Discrete Comp. Geom.,

[25] S. Hildebrandt. Remarks on the life and work of Fritz John. Commun. Pure Appl. Math., 51(9-10):971–989, 1998. [26] F. John. Extremum problems with inequalities as subsidiary conditions. In Studies and Essays presented to R. Courant on his 60th Birthday, pages 187–204. Interscience Publishers, 1948. [27] F. John. Collected Papers. Volumes 1,2. Birkh¨auser, 1985. [28] D.B. Judin and A.S. Nemirovski˘ı. Estimation of the informational com` plexity of mathematical programming problems. Ekonom. i Mat. Metody, 12(1):128–142, 1976. [29] F. Juhnke. Embedded maximal ellipsoids and semi-inﬁnite optimization. Beitr. Algebra Geom., 35:163–171, 1994. [30] H. W. Kuhn. Nonlinear Programming: A Historical Note. In History of mathematical programming, pages 82–96. North-Holland, 1991. [31] M. Lassak. Approximation of Convex Bodies by Centrally Symmetric Bodies. Geom. Dedicata, 72:1–6, 1998. [32] J. B. Lasserre. Level sets and non Gaussian integrals of positively hom*ogeneous functions. arXiv:1110.6632v3, 2011. ¨ [33] K. Leichtweiss. Uber die aﬃne Exzentrizit¨at konvexer K¨orper. Arch. Math., 10:187–198, 1958. [34] H.W. Lenstra, jr. Integer programming with a ﬁxed number of variables. Math. Oper. Res., 8(4):538–548, 1983. [35] Ch.Loewner. Collected papers. Birkh¨auser, 1988. [36] J. Moser. Obituaries – Fritz John. Notices of the AMS, 42(2):256–257, 1995. [37] I. Netuka. Charles Loewner and the L¨owner ellipsoid. Pokroky Mat. Fyz. Astronom. (Czech), 38(4):212–218, 1993. [38] A. Pelczy´ nski. Remarks on John’s theorem on the ellipsoid of maximal volume inscribed into a convex symmetric body in Rn . Note di Matematica, 10(suppl. 2):395–410, 1990. [39] M. Pinl. Kollegen in einer dunklen Zeit. Jber. Deutsch. Math.-Verein, 72:176, 1970. Documenta Mathematica

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[40] M. Pinl. Kollegen in einer dunklen Zeit. Schluss. Jber. Deutsch. Math.Verein, 75:166–208, 1973. [41] R. Sanyal, A. Werner, and G.M. Ziegler. On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes. Discrete Comp. Geom., 41(2):183– 198, 2008. [42] D. Shanno Who invented the interior-point method?, this volume. ˇ [43] N.Z. Shor. Use of the space expansion operation in problems of convex function minimalization, Kibernetika,1:6–12, 1970. [44] M.J. Todd and E.A. Yildirim. On Khachiyan’s algorithm for the computation of minimum-volume enclosing ellipsoids. Discrete Appl. Math., 155(13):1731–1744, 2007. [45] V.L. Zaguskin. Circ*mscribed and inscribed ellipsoids of extremal volume. Usp. Mat. Nauk, 13(6 (84)):89–93, 1958.

Martin Henk Fakult¨at f¨ ur Mathematik Otto-von-GuerickeUniversit¨at Magdeburg Universit¨atsplatz 2 39106 Magdeburg Germany [emailprotected]

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A Brief History of Linear and Mixed-Integer Programming Computation Robert E. Bixby

2010 Mathematics Subject Classification: 90C05, 90C10 Keywords and Phrases: Linear programming, mixed-integer programming, simplex algorithm, branch-and-bound, implementation, computer

The early years For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computation. In contrast to the economists of his time, Dantzig viewed LP not just as a qualitative tool in the analysis of economic phenomena, but as a method that could be used to compute actual answers to specific real-world problems. Consistent with that view, he proposed an algorithm for solving LPs, the simplex algorithm [12]. To this day the simplex algorithm remains a primary computational tool in linear and mixed-integer programming (MIP). In [11] it is reported that the first application of Dantzig’s simplex algorithm to the solution of a non-trivial LP was Laderman’s solution of a 21 constraint, 77 variable instance of the classical Stigler Diet Problem [41]. It is reported that the total computation time was 120 man-days! The first computer implementation of an at-least modestly general version of the simplex algorithm is reported to have been on the SEAC computer at the then National Bureau of Standards [25]. (There were apparently some slightly earlier implementations for dealing with models that were “triangular”, that is, where all the linear systems could be solved by simple addition and subtraction.) Orchard-Hays [35] reports that several small instances having as many as 10 constraints and 20 variables were solved with this implementation. The first systematic development of computer codes for the simplex algorithm began very shortly thereafter at the RAND Corporation in Santa Monica, California. Dantzig’s initial LP work occurred at the Air Force following Documenta Mathematica

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the end of World War II, influenced in part by military logistics problems that arose during the war. In 1952 Dantzig moved from the Air Force to the RAND Corporation, apparently with the specific purpose of focusing on the further development of his fundamental new ideas. Part of the effort was to build computer implementations of the simplex algorithm, and Orchard-Hays was assigned the task of working with Dantzig. The result was a four-year collaboration at RAND that laid the foundation for the computational development of the subject. The start did not go smoothly. The simplex algorithm was at that point far from a well-defined computational procedure, and the computers of the day were nothing like what we think of as a computer today. Their first implementation used a device known as a Card Programmable Calculator (CPC). As the name suggests, it wasn’t really a computer, but as Orchard-Hays [35] described it “an ancient conglomeration of tabulating equipment, electro-mechanical storage devices, and an electronic calculator (with tubes and relays), long since forgotten. One did not program in a modern sense, but wired three patch-boards which became like masses of spaghetti”. The first implementation computed an explicit inverse at each iteration, and Dantzig was appalled when he saw the result [35]; the future of the simplex algorithm didn’t look promising. He then recalled an idea proposed to him by Alex Orden, the product-form of the inverse. This method, which remained a staple of simplex implementations for over twenty years, was the starting point for a second and more successful CPC implementation. It was reportedly capable of handling LPs with up to 45 constraints and 70 variables and was used to solve a 26 constraint, 71 variable instance of the Stigler model. Total computation time was reported to be about 8 hours, a good portion of that time being spent manually feeding cards into the CPC. That was 1953. In 1954–55 the algorithms were improved and re-implemented on an IBM 701, IBM’s first real “scientific computer”. This implementation could handle LPs with 101 constraints, and was used in extensive computations on a model devised by the economist Alan Manne. This appears to have been the first real application of the simplex algorithm. The 701 implementation was followed in 1955–56 by an implementation for the IBM 704. This code was capable of handling LPs with up to 255 constraints, including explicit constraints for any upper bounds. It became known as RSLP1, and seems to have been the first code to be distributed for use by a wider audience. It was later improved to handle 512 constraints, and released for use by CEIR, Inc. around 1958-59 under the name of SCROL. LP was coming of age and beginning to enjoy significant use in the oil industry. Orchard-Hays moved from RAND to CEIR in Arlington, Va., in 1956 and began the development of the LP/90 code for the IBM 7090. It was capable of handling up to 1024 constraints. LP/90 was released in 1961-62, with improvements continuing into 1963. This code was followed by LP/90/94 for the IBM 7094, released in 1963/64. This code was then taken over by CEIR, Ltd. in the UK. The LP/90/94 code can fairly be characterized as the culmination of the Documenta Mathematica

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first-generation of LP codes, and this seems to be the last really successful code over which Orchard-Hays had significant influence. With it a new generation of developers emerged to continue the computational development of LP and, in the not-to-distant future, MIP. A key figure in motivating these developments was E. M. L. (Martin) Beale in the UK. Among those who worked with Beale and were influenced by his vision of mathematical programming were R. E. Small and Max Shaw, followed by John Tomlin and John Forrest, both of whom continue to influence the field to this day. LP/90/94 was also a milestone because it became, by all accounts, the first commercially used MIP code based upon branch-and-bound [9]. The conversion of this code to handle mixed-integer problems seems to have been initiated around 1964–65 by Beale and Small [4]. They used an approach suggested by Land and Doig [29] with dichotomous branching as proposed by Dakin [14]. This code was then taken over by Max Shaw in 1965 [39]: Back in the 60s the IBM 7094 was a 36 bit word machine with 32K words of storage. It was nevertheless a super computer of its time. A team in the USA at CEIR INC. lead by William OrchardHays wrote a standalone LP system (LP 90/94) that mixed linear programming with brilliant system design that could solve LP problems up to 1000 rows or so. This code was written exclusively in 7094 machine code and used all manner of advanced techniques to maximise computing efficiency. I never met Bill Orchard-Hays and his team but when I studied their code I was most impressed. The revised simplex method of George Dantzig was implemented such that the transformation vectors (we called them etas) were held on tape and were read forward to update vectors being expressed in terms of the basis, added to etas for vectors brought into the basis, and read backward to compute the price or feasibility advantage of vectors to be brought into the solution. Shaw reports that this code was used in the first successful applications of MIP, which included: • Re-location of factories in Europe by Philips Petroleum • The selection of ships and transport aircraft to support deployment of UK military assets • Refinery infrastructure investments by British Petroleum • Selecting coal mines for closure by the UK National Coal Board In his own words: There was some excitement for customers using the LP 90/94 system in 1967-8 as they had never been able earlier to get optimal results to their mixed-integer models. Documenta Mathematica

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This really demonstrated for the first time, contrary to common belief, that a search procedure based on branch-and-bound could be used to solve real-world MIPs to optimality. That was true in spite of the fact that the algorithmic opportunities on the machines of the day were severely limited. Again, quoting Shaw: The version of the 7094 used by CEIR only had tape storage. This caused us to search to the bottom of each branch of the tree of bounded solutions until we got an integer value; and then track back up the tree using the bound obtained from the best integer solution so far. The 70s and 80s: The next generation This brings us to the 1970s. The computational aspects of the subject were now close to twenty years old and both LP simplex codes and branch-and-bound codes for MIP, though primitive, were available. It was in a very real sense the end of the Orchard-Hays era, one strongly influenced by his pioneering implementations of the simplex algorithm. It also marked the introduction of the IBM 360 class of computers. The expanded capabilities of these machines meant not only that problems could be solved more quickly, but perhaps more importantly that new ideas and methods could be tried that would have been unworkable on the previous generation of computers. It was also the beginning of a period of great promise for linear and mixed-integer programming. For LP, important ideas such as the implicit treatment of bounds within the simplex algorithm, which reduced the number of explicit constraints in the model, the use of LU-factorizations, the use of sophisticated LU-updates, based upon the Forrest-Tomlin [18] variant of the Bartels-Golub [3] update, and improved variable-selection paradigms such as devex pricing, as proposed by Paula Harris at British Petroleum [24]. The dual simplex algorithm, proposed by Lemke in 1954 [30] also became a fairly standard part of LP codes, though its use was restricted almost exclusively to re-optimization within MIP branch-and-bound trees (amazingly, the ability to explicitly deal with dual infeasibilities does not seem to have emerged until the mid-1990s). The basic form of these algorithms, developed in the early 70s, seems to have remained more-or-less constant into the mid-1980s. The implementations were almost exclusively written in assembler code and highly tuned to exploit the specific characteristics of the target machine. On the integer programming side there was also major progress. A number of completely new codes were introduced. These new codes offered a tight integration between the underlying LP solver and MIP code. And the MIP codes themselves became much more sophisticated. Tree search moved beyond the very inefficient LIFO search dictated by earlier computer architectures. Sophisticated node and variable selection procedures were developed, including the important notion of pseudo-costs, still heavily in use today. Many of these Documenta Mathematica

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developments are nicely documented in [19] and [28]. The net result was that MIP was developing into a more powerful tool beginning to see more extensive applications in practice. However, while these codes did continue to be refined and improved, at a certain fundamental level they also remained in a largely unchanged form. Indeed, they remained largely unchanged until the late-1990s! This is a remarkable testimony to their effectiveness. However, it was also a form of roadblock to further developments in the subject: they made MIP a viable practical tool for the first time, but they also helped create totally unrealistic expectations for what remained fundamentally a primitive technology. The first generation of these new codes, developed and/or released around 1970, included FMPS [40], UMPIRE [17], MPSX [5], MPS III, and APEX. These were followed by the introduction of MPSX/370 (for the IBM 370) around 1974 [6], an improved version of MPSX, SCICONIC around 1976, an improved version of UMPIRE, and finally APEX III, the final version of the APEX codes, released around 1982. (See [19] and [28] for further details on these systems.) And in 1980 the Whizard extension of MPS III was developed at Ketron, which had earlier purchased MPS III from Management Science. Whizard was developed jointly by Eli Hellerman and Dennis Rarick, but also worked on extensively by John Tomlin and Jim Welch among others at Ketron [43]. It was a remarkable LP code for its time, including very efficient LUfactorization and LU-update capabilities, and among the first really successful presolve and postsolve capabilities for LP, based to some extent on ideas from the apparently quite advanced FMPS presolve implementation [43]. During this period, two additional important developments occurred. In 1977, the MINOS code, developed at Stanford primarily by Michael Saunders, was released. This was primarily a non-linear programming code, but included a very good, stable implementation of the primal simplex algorithm. Around the same time, in 1979, the XMP code developed by Roy Marsten, using the Harwell LA05 linear-algebra routines, was also released [32]. Both codes were written in portable FORTRAN, and were among the first portable codes in general use. (Some earlier versions of FMPS and UMPIRE were also written in FORTRAN.) Moreover, XMP had an additional, important property: it was written with the idea that it could be embedded in other codes, and thus used as a LP-solving-subroutine in “larger” LP-based algorithmic procedures. The most powerful solvers of the day, written largely as closed systems, were not easily used in this way and represented a serious hindrance most particularly to research in integer programming. This situation is well described by remarks of Gr¨ otschel and Holland [21], commenting on their use of MPSX/370 in work on the traveling salesman problem. They note that if the LP-package they were using had been “better suited for a row generation process than MPSX is, the total speed-up obtained by faster (cut) recognition procedures might be worth the higher programming effort”. Another key development during this period was the introduction around 1980 of the IBM personal computer (PC). Personal computers were not new Documenta Mathematica

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at that time, but the release of the IBM PC marked the beginnings of the business applications of PCs, and it was the event that led to the realization that PCs could be used as platforms for the development of practical LP and MIP codes. It was several years before widely-available MIP codes for PCs were developed, but LP codes began to emerge rather quickly, probably as early as 1983. Sharda and Somarajan [38] report on several such codes, including early versions for the still commonly used LINDO code. The first versions of the XpressMP [15] code were also finding industry use [2] in 1983. Of course the PCs available in those days were a mere shadow of the powerful desktop computers now available. In [38] computational results were reported for a number of PC codes, including LINDO, comparing these codes to MPSX/370 on a small set of LP test problems. The PC codes were run on an IBM PC with an 8087 math co-processor and 640K of RAM. MPSX was run on an IBM 3081D mainframe. LINDO was written in FORTRAN, as presumably were most of the PC codes of that time. Based upon the LINPACK benchmarks for those machines (http://www.netlib.org/benchmark/performance.pdf), one could estimate that the 3081D was roughly 15 times faster than the PC being used. The largest instances used in [38] had roughly 1000 constraints and 1000 variables. LINDO solved 14 of the 16 instances, the best of any of the PC codes tested, taking 5100 seconds in one case, while MPSX was never slower than 13 seconds on any of the models, and solved all 16. Based upon the geometric means of the ratios of the solution times for LINDO versus MPSX/370, LINDO was slightly more than 166 times slower! A fair conclusion from these numbers was that PC codes did, in some cases, provide a useful alternative to the powerful mainframe codes of the day, but were still far behind in overall performance, even taking into account the differences in machine speed. These results seem to confirm the general feeling at the time that LP codes had reached a final level of maturity. Machines would no doubt get faster, but after nearly 40 years of development, the simplex algorithm was viewed as not likely to see further significant improvements. Events were to prove this belief to be totally wrong. Two additional developments occurred during this period that would have fundamental effects on the future of LP (and hence MIP). In 1979, L. Khachiyan [27] showed for the first time that LPs could be solved in polynomial time. This was not an unexpected result, given the fact that LP was known to be in NP and co-NP; nevertheless, it was a fundamental advance, not least of which because of its important theoretical implications in the theory of combinatorial optimization [22]. The applicability to LP computation was however limited and this use of Khachiyan’s algorithm was quickly abandoned. Modern LP codes The work of Khachiyan was followed in 1984 by the paper of N. Karmarkar [26]. Karmarkar used projective transformations to demonstrate a polynomial-time bound for LP that was not only far better than the bounds for Khachiyan’s method, it also corresponded to a computational approach that was applicable Documenta Mathematica

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in practice. Karmarkar’s paper led to a remarkable flurry of theoretical work in linear programming and related areas that, in many ways, continues to this day in convex programming and related subjects [37]. On the computational side, AT&T developed the KORBX system [8], in what turned out to be a largely unsuccessful attempt to commercially exploit Karmarkar’s breakthrough. However, at the same time, researchers were quick to recognize the connections between Karmarkar’s theoretical contribution and earlier work of Fiacco and McCormick on log-barrier methods. This realization eventually led to the development of a class of algorithms known as primal-dual log-barrier algorithms. These results are well documented on the computational side in the work of Lustig, Marsten, and Shanno [31], who developed the OB1 FORTAN code implementing early versions of this log-barrier algorithm. This code was generally available around 1991 and together with the improvements happening during that same period with simplex algorithms – in codes such as CPLEX and OSL – this spelled the end for the KORBX code. While OB1 itself also failed to be commercially successful, it nevertheless was the leading barrier code of its day and generated an enormous amount of interest and activity. The period around 1990 was a remarkably active period in LP. The work of Karmarkar had stimulated a rebirth of interest in LP, both on the theoretical and computation sides. Not only did this lead to a better understanding and improved implementations of barrier algorithms, it also led to a rebirth of interest in simplex algorithms and is responsible to a degree for some of the early developments in the CPLEX LP code, first released in 1988. At about the same time, IBM also released its OSL code, the designated replacement for MPSX/370, developed primarily by John Forrest and John Tomlin. These two codes – CPLEX and OSL – were the dominant LP codes in the early 1990s, and included implementations of both primal and dual simplex algorithms as well as, eventually, barrier algorithms. For the CPLEX code, many of these developments are documented in [7]. Among the most important advances that occurred during this time were the following: • The emergence of the dual simplex algorithm as a general purpose solver (not just restricted to use in branch-and-bound algorithms) • The development of dual steepest-edge algorithms (using a variant proposed in [16]) • Improved Cholesky factorization methodology for barrier algorithms and the introduction of parallelism in these algorithms • Vastly improved linear algebra in the application of simplex algorithms for large, sparse models [20]. In [7] I reported in detail on the overall improvements in the CPLEX LP code from 1988 through 2002, and subsequently updated these results in 2004. The following is a summary of these results: Documenta Mathematica

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Algorithmic improvement (machine independent) Best of barrier, primal simplex, and dual simplex: Machine improvement: Total improvement (3300 · 2000):

3300× 1600× 5,280,000×

These results show that in a period of sixteen years, from 1988 to 2004, by at least some measure, the average speed of at least one LP code – independent of any machine effects – improved by a factor of roughly 3300, far in excess of the improvements in the speed of computing machines over that same period; moreover, combining the effects of the algorithms and the machines gives an improvement factor exceeding six orders of magnitude, nothing short of remarkable. Note that we have used here as our algorithm the best of barrier, primal, and dual. One can argue whether this is a legitimate approach, but it is the one that I have used. It means that, for each model in the test set, each of the three algorithms was run, and the solution time of the fastest of the three was taken as the solution time for the model. It should also be noted that crossover to a basis was used in all cases when the barrier algorithm was applied. This was done in large part because, in all of the major commercial implementations of barrier algorithms, crossover is considered an integral part of the algorithm. It serves to compensate for the numerical difficulties often encountered by barrier algorithms. In addition, the vast majority of LPs that are solved from scratch in practice are the root solves of MIPs, and a basis is then essential to exploit the advanced-start capabilities of simplex algorithms in the branch-and-bound (or now more correctly, branch-and-cut) search tree. Using barrier algorithms within the tree is generally impractical. The above results represent a fundamental change in a subject that twentyfive years ago was considered fully mature. It is interesting to also examine in more detail what is behind these numbers. One finds that of the three listed algorithms, primal simplex is now rarely the winner. Dual and barrier dominate; moreover, because of current trends in computing machinery, with individual processors making relatively little progress, and most increased power coming from increasing the number of cores per CPU chip, the ability to exploit parallelism is becoming more and more important. Barrier algorithms can and have been very effectively parallelized, while there has been essentially no success in parallelizing simplex algorithms. The result is that barrier algorithms are increasingly the winning algorithm when solving large linear programs from scratch. However, since crossover to a basis is an essential part of barrier algorithms, and this step is fundamentally a simplex computation and hence sequential, the fraction of time taken by crossover is ever increasing. The improvements we have seen in LP computation are clearly good news for the application of these technologies. Indeed, this has led to the common view among practitioners that LP is a “solved problem”: it is now common Documenta Mathematica

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that LPs with several hundred thousand constraints and variables are solved without difficulty. However, there remains considerable room for improvement. The numerical difficulties that are often encountered with barrier algorithms and particularly in the subsequent crossover step represent a major hurdle; moreover, for integer programming (a subject we will return to shortly) computational tests show that, in current practice, roughly 2 % of real-world MIPs are blocked in their solution by the difficulty of the underlying LPs. This combined with the fact that since 2004 there have been essentially no improvements in the standard LP algorithms, means that LP is threatening in the future to again become a significant bottleneck in our ability to solve real-world problems of interest. Modern MIP codes Let me now return to the topic of computation in MIP. While LP is a fundamental technique in the modern application of quantitative techniques to the solution of real-world problems, in the context of optimization, it is MIP that dominates. As previously noted in this paper, MIP codes passed an important milestone in the early 1970’s with the introduction of several powerful new codes – notably SCICONIC, MPSX/370 and MPS III with Whizard – using what were then state-of-the art implementations of simplex algorithms tightly integrated with LP based branch-and-bound, and combined with a wide variety of generally simple, but very effective heuristic techniques to improve the overall search. That was an important step forward in the field. However, the dominance of these codes also led to stagnation in the field. In the years between mid-60s and the late 90s, there was a steady stream of fundamental theoretical work in integer programming and related areas of combinatorial optimization. Important parts of this work were motived by the seminal paper of Dantzig, Fulkerson and Johnson in 1954 [13]. Other fundamental contributions in this period included the work of Gomory on pure integer programs, the work on Edmonds on matching and polyhedral combinatorics, subsequent work by Padberg, Gr¨otschel, Wolsey and others developing and applying cutting-plane techniques (with roots in the paper of Dantzig, Fulkerson and Johnson [13] as well as the work of Edmonds), and a substantial body of theory of disjunctive programming developed primarily by Balas. In addition, there were very important papers by Crowder, Johnson, and Padberg for 0/1 pure integer programs [10] and Van Roy and Wolsey for general MIP [42] that demonstrated the practical effectiveness of cutting-plane techniques and MIP presolve reductions in solving collections of real-world MIPs, MIPs that appeared intractable using traditional branch-and-bound. Indeed, in both of these cases, existing commercial codes (in the first case MPSX/370 and in the second SCICONIC) were directly modified to demonstrate the efficacy of these ideas. In spite of that fact, there was no real change in the generally available commercial codes. They got faster, but only because maDocumenta Mathematica

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chines got faster and LP algorithms improved. The basic MIP algorithms in use remained largely those developed in the 70s. To complete our story of the evolution of MIP software to the present, let me now return to some of the important codes that were developed subsequent to the developments in the 70s. There is a long list, but foremost among these have been XpressMP, the first MIP version being released in 1989, CPLEX MIP, with the first release in 1991, and much more recently, the Gurobi [23] mixed-integer solver, first released in 2009. It should also be mentioned that during this period there was also another solver that was influential to the development of the subject, the MINTO code developed at Georgia Tech [34] and first released around 1991. This code was not intended to be a competitor to commercial solvers, and it was never really used widely in applications. However, it was a milestone in the sense that it was the first general purpose MIP code to make systematic use of cutting-plane techniques, a set of methods that have subsequently proved to be fundamental in the development of MIP. Moreover, though this code was a research code, it was clearly well implemented and provided an important test of the efficacy of these methods. Another key software development in this period was the introduction of the concept of a callable library as first realized in the initial versions of CPLEX. The idea behind this structure, which involved an early example of what was effectively an object-oriented design, was to treat LP as a kind of black-box tool that could be used as an embedded solver in the development of other algorithmic techniques, most importantly in algorithms for solving integer programs. This callable library approach was enormously successful, and became the model for essentially all future codes in this domain. Let me now turn to a discussion of the computational progress that has occurred since the early 90s. In late 2007, I undertook a massive computational test using the CPLEX codes that had been released over the years. This test made use of an extensive library of real-world problems that had been collected from academic and industry sources over a period of almost twenty years. From this extensive library, a test set of 1892 representative models was selected. Using these models, and using a bank of identical computing machines, I recompiled each of the corresponding twelve CPLEX released versions – from Version 1.2 (the first version having MIP) through CPLEX 11 – to run on the target machine. I then ran each of the 1892 models with each different CPLEX version, using a time limit of 30,000 seconds, roughly 1/3 of a day. I then compared consecutive versions by taking each model that was solved to optimality by at least one of the two versions, computing the ratios of the solve times (using the time limit for models that did not solve to optimality), and then computing the geometric means of these ratios. The results of these tests are summarized in the chart below: This chart can be read as follows. The scale on the left refers to the bars in the chart and the scale on the right to the piecewise-linear line through the middle. First looking at the bars, we see, for example that in this test CPLEX 2.1 was approximately 3.1 times faster than CPLEX 1.2, and that each subsequent Documenta Mathematica

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version, with the arguable exception of CPLEX 6.0, represented a significant improvement over the previous version. Two particular bars in this chart stand out, the one comparing CPLEX 3.0 to 2.1 and the one comparing CPLEX 6.5 to 6.0. The first of these, representing an improvement factor of nearly 5.5, corresponds to the maturity of the dual simplex algorithm. The second and by far the biggest improvement occurred in 1998, a speedup exceeding a factor of 10.0. How and why did this happen? The way I like to describe it is as follows. As noted above, the late 90s were preceded by a period of some thirty years of important theoretical and computational developments, many clearly relevant to MIP computation, but virtually none of which had been implemented in commercial codes. The conclusion was clear. It was time to change that. With CPLEX version 6.5 a systematic program was undertaken to include as many of these ideas as possible. You see the result in the chart. The net effect was that in 1998 there was a fundamental change in our ability to solve real-world MIPs. With these developments it was possible, arguably for the first time, to use an out-of-the box solver together with default settings to solve a significant fraction of non-trivial, real-world MIP instances. I would venture to say that if you had asked any of the top MIP researches in the field prior to that time if that would have been possible, they would have said no. The subject had changed, and changed fundamentally. The piecewise-linear line through the graph is an attempt to capture the overall magnitude of that change. It was computed by multiplying the effects of the individual improvements, producing a projected, machine-independent improvement of a factor of over 29,000. And, this trend has continued. Tests carried out in 2009 using public benchmarks maintained by Hans Mittelmann at the University of Arizona [33] indicated that Gurobi 1.0, the first release of the Gurobi solver, had performance Documenta Mathematica

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that was roughly equivalent to that of CPLEX 11.0. Since the release of Gurobi 1.0, we have measured the improvements for subsequent releases, up through the current 5.0 release. Using the standard approach of taking ratios of solve times and computing geometric means, the total improvement was a factor of 16.2, and this on top of the factor of 29,000 in the period prior to 2009, yielding a combined machine-independent factor far exceeding that for LP; moreover, this phenomenon is not restricted to CPLEX and Gurobi. The recent Mittelmann benchmarks demonstrate equally impressive performance by other codes, notably XpressMP and the open-source solver SCIP [1]. It’s a great story for the future of our subject, and it shows no signs of stopping. Acknowledgment. The author would like to thank Robert Ashford, John Gregory, Ed Rothberg, Max Shaw and John Tomlin for several useful e-mail exchanges that contributed to this article. References [1] Achterberg, T. 2009. SCIP: solving constraint integer programs. Math. Programming Computation, 1 (1) 1–41. [2] Ashford, R. 2012. Private communication. [3] Bartels, R. H., G. H. Golub. 1969. The simplex method of linear programming using LU decomposition. Communications of the Association for Computing Machinery 12 266–268. [4] Beale, E. M. L., R. E. Small. 1965. Mixed integer programming by a branch and bound technique, Proc. IFIP Congress, Vol. 2 (W. Kalench, Ed.), Macmillan, London (1965) 450-451. [5] Benichou, M., J. M. Gauthier, P. Girodet, G. Hentges, G. Ribi`ere, O. Vincent. 1971. Experiments in mixed-integer linear programming. Math. Programming 1 76–94. [6] Benichou, M., J. M. Gauthier, G. Hentges, G. Ribi`ere. 1977. The efficient solution of large scale linear programming problems. Some algorithmic techniques and computational results. Math. Programming 13 280–322. [7] Bixby, R. E. 2002. Solving real-world linear programs: a decade and more of progress. Operations Research 50 (1) 1–13. [8] Carolan, W. J., J. E. Hill, J. L. Kennington, S. Niemi, S. J. Wichmann. 1990. An empirical evaluation of the KORBX algorithms for military airlift applications. Operations Research. 38 (2) 240–248. [9] Cook, W. 2012. Markowitz and Manne + Eastman + Land and Doig = Branch and Bound, this volume. Documenta Mathematica

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[10] Crowder, H., E. L. Johnson, M. Padberg. 1983. Solving large-scale zero-one linear programming problems. Operations Research 31 (5) 803–834. [11] Dantzig, G. 1963. Linear Programming and Extensions. Princeton University Press, Princeton. [12] Dantzig, G. 1948. Programming in a linear structure, U.S. Air Force Comptroller, USAF, Washington, D.C. [13] Dantzig, G., D. R. Fulkerson, S. Johnson. 1954. Solution of a large scale traveling salesman problem. Operations Research 2 393–410. [14] Dakin, R. J. 1965. A tree search algorithm for mixed integer programming problems, Computer Journal 8 250–255. [15] Fair Isaac Corporation. 2012. Xpress-Optimizer reference manual. (http://www.fico.com/en/Products/DMTools/xpress-overview/ Pages/Xpress-Optimizer.aspx) [16] Forrest, J. J., D. Goldfarb. 1992. Steepest-edge simplex algorithms for linear programming. Math. Programming 57 341–374. [17] Forrest, J. J. H., J. P. H. Hirst, J. A. Tomlin. 1974. Practical solution of large mixed integer programming problems with UMPIRE. Management Science 20 (5) 736–773. [18] Forrest, J. J. H., J. A. Tomlin. 1972. Updated triangular factors of the basis to maintain sparsity in the product form simplex method. Math. Programming 2 263–278. [19] Geoffrion, A. M., R. E. Marsten. 1972. Integer programming algorithms: a framework and state-of-the-art survey. Management Science 18 465–491. [20] Gilbert, J. R., T. Peierls. 1988. Sparse partial pivoting in time proportional to arithmetic operations. SJSSC 9 862–874. [21] Gr¨ otschel, M., O. Holland. 1991. Solution of large-scale symmetric travelling salesman problems. Math. Programming 51 141–202. [22] Gr¨ otschel, M., L. Lov´ asz, A. Schrijver. 1981. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1 169–197. [23] Gurobi Optimization, Inc. 2012. Gurobi optimizer reference manual. (http://www.gurobi.com) [24] Harris, P. J. J. 1974. Pivot selection methods of the devex LP code. Math. Programming 5 1–28. [25] Hoffman, A., A. Mannos, D. Sokolowsky, D. Wiegmann. 1953. Computational experience in solving linear programs. SIAM J. 1 1–33. Documenta Mathematica

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[26] Karmarkar, N. 1984. A new polynomial-time algorithm for linear programming, Combinatorica 4 373–395. [27] Khachiyan, L. G. 1979. A polynomial algorithm in linear programming (in Russian). Doklady Akademii Nauk SSSR 244 1094–1096. [28] Land, A., S. Powell. 1979. Computer codes for problem of integer programming. Annals of Discrete Mathematics 5 221–269. [29] Land, A., A. G. Doig. 1960. An automatic method of solving discrete programming problems. Econometrica 28 (3) 597–520. [30] Lemke, C. E. 1954. The dual method of solving the linear programming problem. Naval Res. Logist. Quart. 1 36–47. [31] Lustig, I. J., R. Marsten, D. F. Shanno. 1994. Interior point methods for linear programming: Computational state of the art. ORSA J. Comput. 6(1) 1–14. [32] Marsten, R. E. 1981. XMP: A structured library of subroutines for experimental mathematical programming. ACM Trans. Math. Software 7 481–497. [33] Mittelmann, H. 2012. Benchmarks for Optimization Software (http:// plato.asu.edu/bench.html). [34] Nemhauser, G. L., M. W. P. Savelsbergh, G. C. Sigismondi. 1994. MINTO, A Mixed INTeger Optimizer. Operations Research Letters 15 47–58. [35] Orchard-Hays, W. 1990. History of the development of LP solvers. Interfaces 20 (4) 61–73. [36] Schrijver, A. 2012. This volume. [37] Shanno, D. F. 2012. This volume. [38] Sharda, R, C. Somarajan. 1986. Comparative performance of advanced microcomputer systems. Comput. & Ops. Res. 13 (2/3) 131–147. [39] Shaw, M. 2012. Private communication. [40] Sperry-Univac. 1975. Sperry-Univac 1100 Series Functional Mathematical Programming System (FMPS) Programming Reference UP-8198. [41] Stigler, G. J. 1945. The cost of subsistence, J. Farm Econom. 27 (2) 303–314. [42] Van Roy, T. J., L. A., Wolsey. 1987. Solving mixed integer programming problems with automatic reformulation. Operations Research 35 (1) 45– 57. [43] Tomlin, J. 2012. Private communication. Documenta Mathematica

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A Brief History Robert E. Bixby 8 Briarwood Ct. Houston, Texas, 77019 USA [emailprotected]

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Discrete Optimization Stories

There are a number of very good surveys of the history of combinatorial optimization (briefly CO). I want to recommend to the reader two outstanding articles: [5] covers the area until 1960 and [2] the history of integer programming in the last ∼ 50 years. And there is the encyclopedic 3-volume book [6] which is an unsurpassable source book for the historical development of CO. Nevertheless, the articles in this section shed some new light on certain historical aspects of CO. The original plan of this book included further remarkable CO-stories. They had to be abandoned for space reasons. But I want to mention two of them in this introduction because there are good sources available where the details can be found. Let me begin with a most astonishing discovery. One of the first algorithms of CO I heard about was the Hungarian method which has been viewed by many as a prototype of algorithm design and efficiency. Harold Kuhn presented it in 1955 in [3]. Having used ideas and results of J. Egerv´ary and D. K˝onig, he gave his algorithm (generously) the name Hungarian method. In 2004 the journal Naval Research Logistics Quarterly (briefly NRL) established a new “best paper award” to recognize outstanding research published in NRL. [3] was selected as the best paper published since 1954 in NRL, and A. Frank [2] wrote a moving paper about “Kuhn’s Hungarian Method” in NRL. In 2005 A. Frank organized a conference in Budapest entitled “Celebration Day of the 50th Anniversary of the Hungarian Method” at which I highlighted the role the Hungarian algorithm has played in practical applications such as vehicle scheduling. Soon thereafter, on March 9, 2006 I received an e-mail from Harold Kuhn that started as follows: Dear Friends: As participants in the 50th Birthday celebration of the Hungarian Method, you should be among the first to know that Jacobi discovered an algorithm that includes both Koenig’s Theorem and the Egervary step. I was told about Jacobi’s paper by Francois Ollivier who has a website with the original papers and French and English translations. They were published in Latin after his death and so the work was done prior to 1851!!! Documenta Mathematica

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Figure 1: Carl G. J. Jacobi c BBAW) (

Figure 2: Jacobi’s grave c Iris Gr¨otschel) (

What a surprise! The Hungarian method had appeared for the first time in a paper, written in Latin, attempting to establish a bound on the degree of a system of partial differential equations and which was only published posthumously in Jacobi’s collected works. The original manuscript can be found in the “Jacobi Nachlass” of the BBAW archive in Berlin. I will not go into the details of the story since Harold Kuhn has written up all the circ*mstances in his recent article [4], where one can find all the relevant references. I just want to remark that the Jacobi mentioned is Carl Gustav Jacob Jacobi, see Fig. 1, after whom the Jacobi matrix is named. Jacobi was born in Potsdam in 1804, became Professor in K¨onigsberg in 1826, moved to Berlin in 1843, and died in 1851. Jacobi has an “honorary grave” (Ehrengrab) on the “Friedhof der Berliner Dreifaltigkeitsgemeinde” in Berlin, see Fig. 2. The second story is of completely different nature. It is about mathematics done under extreme circ*mstances. I just want to quote pieces of a paper [7] written by Paul Tur´ an, one of the great Hungarian figures of combinatorics, about some of his experiences in World War II. In 1940 Tur´ an had to work on railway building in a labor camp in Transylvania and proved what we call Tur`ans theorem today. In his words: . . . I immediately felt that here was the problem appropriate to the circ*mstances. I cannot properly describe my feelings during the next few days. The pleasure of dealing with a quite unusual type of problem, the beauty of it, the gradual nearing of the solution, and finally the complete solution made these days really ecstatic. The feeling of some intellectual freedom and being, to a certain extent, spiritually free of oppression only added to this ecstasy. The second experience I want to mention is about Tur´an’s discovery of the crossing number. He writes: Documenta Mathematica

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In July 1944 the danger of deportation was real in Budapest, and a reality outside Budapest. We worked near Budapest, in a brick factory. There were some kilns where the bricks were made and some open storage yards where the bricks were stored. All the kilns were connected by rail with all the storage yards. The bricks were carried on small wheeled trucks to the storage yards. All we had to do was to put the bricks on the trucks at the kilns, push the trucks to the storage yards, and unload them there. We had a reasonable piece rate for the trucks, and the work itself was not difficult; the trouble was only at the crossings. The trucks generally jumped the rails there, and the bricks fell out of them; in short this caused a lot of trouble and loss of time which was rather precious to all of us (for reasons not to be discussed here). We were all sweating and cursing at such occasions, I too; but nolens-volens the idea occurred to me that this loss of time could have been minimized if the number of crossings of the rails had been minimized. But what is the minimum number of crossings? Let us all hope that mathematics discoveries will never again have to be made under such circ*mstances. Martin Gr¨otschel References [1] W. Cook, Fifty-plus years of combinatorial integer programming, in M. J¨ unger (ed.) et al., 50 years of integer programming 1958–2008. From the early years to the state-of-the-art, Springer, Berlin, 2010, pp. 387–430. [2] A. Frank, On Kuhn’s Hungarian Method – a tribute from Hungary, Naval Research Logistics 52 (2005), 2–5. [3] H. Kuhn, The Hungarian Method for the Assignment Problem, Naval Research Logistics Quart. 2 (1955), 83–97. [4] H. Kuhn, A tale of three eras: The discovery and rediscovery of the Hungarian Method, European Journal of Operational Research, 219 (2012), 641– 651. [5] A. Schrijver, Combinatorial Optimization: Springer, Berlin, 2002.

Polyhedra and Efficiency,

[6] A. Schrijver, On the history of combinatorial optimization. in: K. Aardal (ed.) et al., Discrete optimization. Elsevier, Amsterdam, Handbooks in Operations Research and Management Science 12, 2005, pp. 1–68. [7] P. Tur´ an, A Note of Welcome, Journal of Graph Theory, 1 (1977), 7–9.

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The Origins of Minimal Spanning Tree Algorithms – Bor˚ uvka and Jarn´ık ˇil and Helena Neˇ ˇilova ´ Jaroslav Neˇ setr setr

2010 Mathematics Subject Classification: 01-00, 05-03, 90-03, 01A60, 01A70, 05C85, 05C90, 68R15 Keywords and Phrases: Minimal spanning tree, Bor˚ uvka, Jarn´ık, history of computing

1

Introduction

In this paper we discuss the early history of Minimum Spanning Tree problem and its solution. The MST problem is a corner stone of combinatorial optimization and its history is rich. It has been described in detail in several places, for example, one can mention [22] which gives a general overview of the history of combinatorial optimization; historically exhaustive paper [9]; another historical paper which contains the first commented translation of the original papers of Bor˚ uvka into English [19]; the paper [13] which deals with early papers by Jarn´ık; and papers [18] and particularly [16], which cover the later rich development from contemporary perspective. Here we complement this by concentraiting on the very early beginning of this development before 1930. It is accepted by now that two papers [1], [2] by Bor˚ uvka in 1926 and Jarn´ık [11] in 1930 are the first papers providing a solution to Minimum Spanning Tree problem. We document this together with remarks illustrating the milieu of this discovery and personalities of both authors (and Bor˚ uvka in particular). 2

Paper No. 1

Otakar Bor˚ uvka published three papers in 1926, two of which are our optimization papers: the paper [2] appeared in a local mathematical journal in Brno and the other in an engineering magazine Elektrotechnick´ y obzor [1] (Electrotechnical Overview). The paper [2] has 22 pages and it was repeatedly described as unnecessary complicated. Paper [1] has a single page and it is little known (for example, it is not listed among his scientific works neither in [20] nor [4]). Documenta Mathematica

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However we believe that this is the key paper. It demonstrates how clearly Bor˚ uvka understood the problem and its algorithmic solution. The paper is very short and thus we can include the English translation in full (the original paper was written in Czech). 2.1

ˇ´ıspe ˇvek k r ˇeˇ ´ zky ekonomicke ´ stavby Translation of “Pr sen´ı ota ´ ch s´ıt´ı” elektrovodny Dr. Otakar Bor˚ uvka A CONTRIBUTION TO THE SOLUTION OF A PROBLEM OF ECONOMIC CONSTRUCTION OF ELECTRIC POWER-LINE NETWORKS In my paper “On a certain minimal problem”(to appear in Pr´ ace moravsk´e pˇr´ırodovˇedeck´e spoleˇcnosti) I proved a general theorem, which, as a special case, solves the following problem: There are n points given in the plane (in the space) whose mutual distances are all different. We wish to join them by a net such that 1. Any two points are joined either directly or by means of some points, 2. The total length of the net would be the shortest possible. It is evident that a solution of this problem could have some importace in electricity power-line network design; hence I present the solution briefly using an example. The reader with a deeper interest in the subject is referred to the above quoted paper. I shall give a solution of the problem in the case of 40 points given in Fig. 1. I shall join each of the given points with the nearest neighbor. Thus, for example, point 1 with point 2, point 2 with point 3, point 3 with point 4 (point 4 with point 3), point 5 with point 2, point 6 with point 5, point 7 with point 6, point 8 with point 9, (point 9 with point 8), etc. I shall obtain a sequence of polygonal strokes 1, 2, . . . , 13 (Fig. 2). I shall join each of these strokes with the nearest stroke in the shortest possible way. Thus, for example, stroke 1 with stroke 2, (stroke 2 with stroke 1), stroke 3 with stroke 4, (stroke 4 with stroke 3), etc. I shall obtain a sequence of polygonal strokes 1, 2, . . . , 4 (Fig. 3) I shall join each of these strokes in the shortest way with the nearest stroke. Thus stroke 1 with stroke 3, stroke 2 with stroke 3 (stroke 3 with stroke 1), stroke 4 with stroke 1. I shall finally obtain a single polygonal stroke (Fig. 4), which solves the given problem.

2.2

ˇ´ıspe ˇ ve ˇk k r ˇeˇ ´mu ekonomicke ´ konRemarks on “Pr sen´ı proble ´ ch s´ıt´ı” strukce elektrovodny

The numbering of Figures is clear from a copy of the original article which we include below. Documenta Mathematica

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Figure 1: Bor˚ uvka’s short paper [1]

This paper is written in a nearly contemporary style. An example given (40 cities) is derived from the original motivation of Bor˚ uvka’s research which was a problem related to the electrification of south-west Moravia. (See Section 6 about further details of Bor˚ uvka’s motivation.) Paper [2] contains yet another example with 74 cities. The electrification of South-Moravia was an actual topic in the early 20th century and it was very close to the editors of the Elektrotechnick´ y obzor. (Note also that South-Moravia is one of the developed and cultured parts of Europe. It is and has been for centuries fully industrialized and yet a wine growing, rich and beautiful country. The core part of it is now protected by UNESCO.) As a good analyst Bor˚ uvka viewed the assumption on distinct distances as unimportant. Once he told us: “if we measure distances, we can assume that Documenta Mathematica

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they are all different. Whether distance from Brno to Bˇreclav is 50 km or 50 km and 1 cm is a matter of conjecture” [5]. We tried to keep the view of the original article. A careful reader can observe that the last figure (Fig. 4) in Bor˚ uvka’s paper [1] is reversed. This was noted already by Bor˚ uvka in 1926 as seen from our depicted copy which he mailed to Prof. Bydˇzovsk´ y). Of course, the Elektrotechnick´ y obzor is not a mathematical journal. Yet, this was a proper place to publish the result. The magazine was founded in 1910 (and it has been published by that name until 1991 when it merged with other journals under the name Elektro). It was the first Czech journal focussed on electricity. It was founded by Vladim´ır List, engineer and professor in Brno (who served as president of the Czech Technical University in Brno and, among other things, was Chairman of the International standards organization ISA). He advocated the systematic electrification of Moravia and convinced authorities to build public high voltage trasmission lines. Bor˚ uvka began his studies at the the Technical University in Brno. 3

Contemporary setting

Before discussing the paper [2] let us include, for comparision, the well known contemporary formulations of the Minimum Spanning Tree problem, Bor˚ uvka’s algorithm and the proof, see, e.g., [23]. Problem (MST). Let G = (V, E) be an undirected connected graph with n vertices and m edges. For each edge e let w(e) be a real weight of the edge e and let us assume that w(e) 6= w(e′ ) for e 6= e′ . Find a spanning tree T = (V, E ′ ) of the graph G such that the total weight w(T ) is minimum. Bor˚ uvka’s algorithm 1. Initially all edges of G are uncolored and let each vertex of G be a trivial blue tree. 2. Repeat the following coloring step until there is only one blue tree. 3. Coloring step: For every blue tree T , select the minimum-weight uncolored edge incident to T . Color all selected edges blue. Proof (Correctness of Bor˚ uvka’s algorithm). It is easy to see that at the end of Bor˚ uvka’s algorithm the blue colored edges form a spanning tree (in each step the distinct edge-weights guarantee to get a blue forest containing all vertices). Now we show that the blue spanning tree obtained by Bor˚ uvka’s algorithm is the minimum spanning tree and that it is the only minimum spanning tree of the given graph G. Indeed, let T be a minimum spanning tree of G and let T ∗ be the blue spanning tree obtained by the algorithm. We show that T = T ∗ . Assume to the contrary T 6= T ∗ . Let e∗ be the first blue colored edge of T ∗ which does not belong to T . Let P be the path in T joining the vertices of e∗ . It is clear that at the time when the edge e∗ gets blue color at least one of the edges, say e, of P is uncolored. By the algorithm w(e) > w(e∗ ). However, Documenta Mathematica

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Figure 2: Last pages of paper [2] then T − e + e∗ is a spanning tree with smaller weight, a contradiction. Thus T = T ∗. This algorithm is called parallel merging or forest growing. It needs only log | V | iterations while each iteration needs | E | steps. The speed up of this (and other MST) algorithm was intensively studied, see, e.g., [16] for a survey. 4

Bor˚ uvka’s paper [2]

In the present terminology [1] is an outline of [2], and [2] is the full version of [1]. [2] is written in Czech with an extensive (6 pages) German summary. This also contributed to the fact that [2] is better known than [1]. The following is the translation of the beginning of the paper. Dr. Otakar Bor˚ uvka ON A CERTAIN MINIMUM PROBLEM In this article I am presenting a solution of the following problem: Let a matrix M of numbers rαβ (α, β = 1, 2, . . . , n; n ≥ 2), all positive and pairwise different, with the exception of rαβ = 0 and Documenta Mathematica

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ˇil and Helena Neˇ ˇilova ´ Jaroslav Neˇ setr setr rαβ = rβα be given. From this matrix a set of nonzero and pairwise different numbers should be chosen such that (1) For any p1 , p2 mutually different natural numbers ≤ n, it would be possible to choose a subset of the form rp1 c2 , rc2 c3 , rc3 c4 , . . . , rcq−2 cq−1 , rcq−1 p2 . (2) The sum of its elements would be smaller than the sum of elements of any other subset of nonzero and pairwise different numbers, satisfying the condition (1).

Paper [2] then proceeds by constructing the solution. What was written in [1] in an easy way, takes in this paper a very complicated form and Bor˚ uvka needs four full pages (pages 37–40) to elaborately explain the first iteration of his algorithm. Why does it take so long? In a private conversation Bor˚ uvka explained this in a contextual way: “I have been young, this was a very new and non-standard topic and thus I have been afraid that it will not be published. So I made it a little more mathematical”, [5]. That, of course, may be a part of the truth. Another reason is certainly the absence of good notation and mainly special notions (such as chain, path, or connectivity). Bor˚ uvka elaborately constructs each component of the first iteration by describing the corresponding forest by means of (sort of) a pointer machine: first he finds a maximum path P containing a given point then he starts with a new vertex and finds a maximum path P ′ which either is disjoint with P or terminates in a vertex of P and so on. Then he combines these paths to tree-components. In the iterative step he already proceeds more easily (page 41). The final set is denoted by J. The author then verifies all the properties of the set J. This is (on page 41) divided into 5 theorems (numbered I, II, III, IV, V) which are proved in the rest of the paper on p. 43–52. The proofs, of course, follow the elaborate construction of the set J. The paper ends (p. 51) with a remark on a geometric interpretation (in kdimensions) of the result and an example of the solution for a particular planar set with 74 points is given. The German summary covers the construction of the set J and states Theorems I, II, III, IV, V. It is interesting to note that at three places of the article (in the proof of Theorem III) he arrives on p. 46 to the exchange axiom in the following rudimental form K ′′ ≡ K ′ − [mq], [mn]. He does not, of course, mention cycles (as in Whitney) or more general algebraic setting (as in Van der Waerden). That had to wait another decade (and this is covered in another article of this book, see [7]). Bor˚ uvka’s approach is a brute force approach par excellence. Not knowing any related literature (and there was almost none, graph theory and even alDocumenta Mathematica

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Figure 3: Proof of Theorem III, paper [2]

gorithms were not yet born1 ) and feeling that the problem is very new, he produced a solution. On the way he arrived at the key exchange axiom which is in the heart of all greedy-type algorithms for MST. He was just solving a concrete engineering problem and in a stroke of a genius he isolated the key statement of contemporary combinatorial optimization. But he certainly was not a Moravian engineer (as it is sometimes erroneously claimed). He was rather an important and well connected mathematician (see Section 6). 5

ˇch Jarn´ık [11] Vojte

Bor˚ uvka was lucky. His contribution was recognised and his article [2] has been quoted by both Kruskal [14] and Prim [19] – papers which became the standard references in the renewed interest in the MST in sixties. [2] became the most quoted paper of Bor˚ uvka. The first reaction to Bor˚ uvka came however almost immediately from Vojtˇech Jarn´ık [11]. Paper [11] published in the same journal, has the same title as [2] which is explained by its subtitle “from a letter 1 For

comparison, K¨ onig’s book appeared in 1936. It is interesting to note that K¨ onig describes his book as “absolute graph theory” and neither optimization (i.e., MST) nor enumeration is covered by this book. Documenta Mathematica

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to O. Bor˚ uvka‘”2 . This paper has only five pages with two pages of German summary. The paper begins as follows: In your article “About a minimum problem” (Pr´ ace moravsk´e pˇr´ırodovˇedeck´e spoleˇcnosti, svazek III, spis 3) you solved an interesting problem. It seems to me that there is yet another, and I believe, simpler solution. Allow me to describe to you my solution. Let n elements be given, I denote them as numbers 1, 2, . . . , n. From these elements I form 21 n(n − 1) pairs [i, k], where i 6= k; i, k = 1, 2, . . . , n. I consider the pair [k, i] identical with pair [i, k]. To every pair [i, k] let there be associated a positive number ri,k (ri,k = rk,i ). Let these numbers be pairwise different. We denote by M the set of all pairs [i, k]. For two distinct natural numbers p, q ≤ n, I call a chain (p, q) any set of pairs from M of the following form: [p, c1 ], [c1 , c2 ], [c2 , c3 ], . . . , [cs−1 , cs ], [cs , q]

(1)

Also a single pair [p, q] I call a chain (p, q). A subset H of M I call a complete subset (kˇ c for short) if for any pair of distinct natural numbers p, q ≤ n, there exists a chain (p, q) in H (i.e., a chain of form (1) all of whose pairs belong to H). There are kˇ c; as M itself is kˇ c. If [i1 , k1 ], [i2 , k2 ], . . . , [it , kt ]

(2)

is a subset K of set M , we put t X

rij ,kj = R(K).

j=1

If for a complete set K the value R(K) is smaller than or equal to the values for all other complete sets , then I call K a minimal complete set in M (symbolically mkˇ c). As there exists at least one kˇ c and there are only finitely many kˇ c, there exists at least one mkˇ c. The problem, which you solved in your paper, can be formulated as follows: Problem: Prove that there exists a unique mkˇ c and give a formula for its construction. Remark: Sets satisfying (1) are, of course now, called path, trail, walk; Jarn´ık considers (1) as a family – repetitions are allowed). Of course kˇ c corresponds to spanning connected subgraphs and mkˇ c corresponds to minimum 2 This

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Figure 4: Jarn´ık’s formula for M ST spanning tree. There is no mention of trees in this paper. However, in the proof Jarn´ık defines “connected set of entries‘”. These definitions are key to his simplification of Bor˚ uvka. On p. 60 Jarn´ık begins to decribe his solution: Let us now introduce a certain subset J of M as follows: Definition of set J. J = [a1 , a2 ], [a3 , a4 ], . . . , [a2n−3 , a2n−2 ] where a1 , a2 , . . . are defined as follows: First step. Choose as a1 any of elements 1, 2, . . . , n. Let a2 be defined by the relation ra1 ,a2 = min ra1 ,l

(l = 1, 2, . . . , n; l 6= a1 ).

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ˇil and Helena Neˇ ˇilova ´ Jaroslav Neˇ setr setr we define a2k−1 , a2k by ra2k−l ,a2k = min ri,j where i ranges over all numbers a1 , a2 , . . . , a2k−2 and j ranges over all the remaining numbers from 1, 2, . . . , n. Moreover, let a2k−1 be one of the numbers in (5) such that a2k is not among the numbers in (5). It is evident that in this procedure exactly k of the numbers in (5) are different, so that for k < n the k-th step can be performed. The solution of our problem is then provided by the following: Proposition: 1. J is mkˇ c. 1. There is no other mkˇ c. 1. J consists of exactly n − 1 pairs.

This construction is today called the tree growing procedure. It is usually called Prim’s algorithm [20]; to establish justice we call this in [17] (and elsewhere) the Jarn´ık-Prim algorithm. Jarn´ık (1897–1970) was less lucky than Bor˚ uvka in the credits to his work in combinatorial optimization. His solution was almost entirely neglected until very recently, [6] being perhaps the earliest exception. Even more so: the same negligence (see, e.g., [8]) relates to his joint paper with K¨ossler [12] which is probably the earliest paper dealing with the Steiner Tree Problem (see [13] for history and additional information on this part of Jarn´ık’s work). This is surprising because Jarn´ık was (and still is) a famous mathematician. Already in 1930 (after two years in G¨ ottingen with E. Landau) he was well known (and better known than Bor˚ uvka). It is interesting to note how quickly Jarn´ık reacted to the “exotic” Bor˚ uvka paper. One can only speculate that this probably motivated him to continue (with K¨ossler) with the “Steiner tree problem” [12]. Like Bor˚ uvka, he never returned to these problems again. 6

Bor˚ uvka’s century

At the end of the last millenium more authors (e.g., G. Grass, I. Kl´ıma, B.-H. L´evy) attempted to summarize the passing century as “my” century. But in a way, this was Bor˚ uvka’s century: born in 1899 he died in 1995. He was born to a middle class Czech family. His father Jan Bor˚ uvka was a respected school principal at his birthplace in Uhersk´ y Ostroh. He was elected a honorable citizien of the town. The school garden, which he founded, was a safe haven for young Otakar. He attended the school of his father and later the gymnasium in Uhersk´e Hradiˇstˇe. He excelled in all subjects. This was already during the First World War (1914–1918) and on the advice of his parents, Bor˚ uvka switched to the military gymnasium in Hranice and then to military academy in M¨ odling (near Vienna). As he recollects, the sole reason of this was to escape the military draft during the war. While he respected good teachers at both institutions, he did not like this period very much (riding a horse being an Documenta Mathematica

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Figure 5: Otakar Bor˚ uvka (archive of the authors)

exception). So immediatelly after the end of the war he resigned and returned home to independent Czechoslovakia. He continued his studies at the Technical University in Brno and then at the Masaryk University in Brno. It is there where he met professor Maty´ aˇs Lerch. Lerch (1860–1922) was perhaps the first modern Czech mathematician who obtained the prestigeous Grand Prix de Academie de Paris in 1900, published over 230 papers and was in contact with leading mathematicians of his time (he also attended the old gymnasium in Rakovn´ık, a dear place to the authors of this article). Lerch chose Bor˚ uvka as his assistant in 1921 and had a profound influence on him. Bor˚ uvka writes that possibly thanks to Lerch he became a mathematician. He considered himself as the heir to Lerch’s legacy and initiated in 1960 the installment of Lerch’s memorial plaque in Brno. Unfortunately, Lerch died early in 1922. However, at that time Bor˚ uvka was fortunate to meet another strong mathematician, ˇ ˇ Eduard Cech (1893–1960), and he became his assistant in 1923. Cech, a few years Bor˚ uvka’s senior and very active person in every respect, suggested to him ˇ to start working in differential geometry. Cech asked Bor˚ uvka to complete some computations in his ongoing work and to become acquainted with what was then a very new method of rap`ere mobile of Elie Cartan. Bor˚ uvka succeeded ˇ and was rewarded by Cech who arranged his stay in Paris during the academic year 1926/27. Before this, in winter 1925/26, Bor˚ uvka met Jindˇrich Saxel, an employee of Z´ apadomoravsk´e elektr´ arny (West-Moravian Powerplants), who was not acaDocumenta Mathematica

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demically educated and yet suggested to Bor˚ uvka a problem related to electrification of South-West Moravia. Bor˚ uvka remembers ([4], p. 52) that in the solution he was inspired by Lerch’s attitude towards applications and that he worked intensively on the problem. We already know the outcome of this. In spring 1927 Bor˚ uvka lectured in Paris about [2] at a seminar (of Cambridge mathematician J. L. Coolidge). He writes: “despite (and perhaps because of ) this very unconventional topic, the lecture was received very well with an active discussion” ([4], p. 59). In Paris he worked intensively with E. Cartan and became a lifelong friend of Cartan’s family (particularly of his son Henri, future president of IMU, whom Bor˚ uvka invited to Brno in 1969). Back in Brno, in winter 1927/28, Bor˚ uvka passed a habilitation (with a thesis ˇ on the Γ-function and, again on a suggestion of E. Cech, obtained a Rockefeller scholarship to Paris for the academic year 1929/30. In Paris he continued his research motivated by intensive contacts with E. Cartan and met other leading ´ Picard, M. Fr´echet, mathematicians of his time (J. Hamadard, B. Segre, E. ´ E. Goursat, H. Lebesgue). After one year in Paris he received (thanks to involvement of E. Cartan “in whose interest it was to expand his methods to Germany” [4], p. 67) the Rockefeller scholarship to Hamburg. In Hamburg he visited W. Blaschke but Bor˚ uvka mentions also E. Artin, H. Zassenhaus, E. K¨ ahler and E. Sperner. It is interesting to note that S. S. Chern followed Bor˚ uvka’s path a few years later (from Hamburg 1934, to Paris 1936). Chern quoted Bor˚ uvka and “even called some statements by my name” ([4], p. 67). This is also the case with, e.g., the Frenet-Bor˚ uvka theorem, see [10]. In 1931 Bor˚ uvka returned to Brno and stayed there basically for the rest of his life. He was then 32, had spent at least four years abroad meeting many of the eminent mathematicians of his time. He was an individualist (typically not ˇ writing joint papers). This is illustrated by the fact that although Cech invited him to take part in his newly founded (and later internationally famous) topological seminar in Brno, he declined. But Bor˚ uvka was an influential teacher. He progressed steadily at the university and in the society. However, the war which broke out in 1939 brought many changes to Bor˚ uvka’s life. All Czech universities were closed by the Nazis. Bor˚ uvka and his circle of friends were arrested by the Gestapo at Christmas 1941. In his memoirs [4], he recalls this at length in the chapter called “On the threshold of death”. Among others, his friend Jindˇrich Saxel was executed in 1941. It is interesting to note, that the West-Moravian Powerplants recollected Bor˚ uvka’s work on MST and made him a generous job offer (which he declined). During his life, Bor˚ uvka changed his research topic several times. He was fully aware of his position in Brno and took responsibility for the future development there. He wrote basic books on group theory and groupoids (during the World War II). After the war he started his seminar on differential equations. [4] contains contributions of his students in all areas of his activities. Due to the space limitations and the scope of this article we end the historical overview of Bor˚ uvka’s century here. Bor˚ uvka was deeply rooted in the Moravian soil. For Brno mathematics he was the founding father. Not in the Documenta Mathematica

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Figure 6: Bor˚ uvka’s grave at the Central Cemetery in Brno

sense of politics (which he luckily avoided most of his life) but in the sense of scientific activity which by far transcended the provincial focus of Brno of his time. In this respect he can be compared, e.g., to Leoˇs Jan´aˇcek. This is not a mere speculation: Bor˚ uvka played several instruments and the conductor Zdenˇek Chalabala was a close friend to both Jan´aˇcek and Bor˚ uvka. The authors of this text knew Bor˚ uvka in his last years. He was a grand old man, yet modest, and still interested in the new developments. He was aware of his MST fame. He would be certainly pleased to know that the late J. B. Kruskal immediately replied to an invitation to write a memorial article on Bor˚ uvka [15]. The quiet strength of Bor˚ uvka is felt even posthumously. Fig. 6 depicts Bor˚ uvka’s remarkable grave at the Central Cemetery in Brno. Acknowledgement.

Supported by the grant ERC-CZ 1201 CORES.

References [1] O. Bor˚ uvka, Pˇr´ıspˇevˇek k ˇreˇsen´ı ot´azky ekonomick´e stavby elektrovodn´ ych s´ıt´ı (Contribution to the solution of a problem of economical construction of electrical networks), Elektronick´y obzor 15 (1926), 153–154. Documenta Mathematica

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[2] O. Bor˚ uvka, O jist´em problem minim´aln´ım (About a certain minimum problem), Pr´ ace morav. pˇr´ırodovˇed. spol. v Brnˇe III (3) (1926), 37–58 (in Czech, German summary). [3] O. Bor˚ uvka, Nˇekolik vzpom´ınek na matematick´ y ˇzivot v Brnˇe (Some recollections of the mathematical life in Brno), Pokroky matematiky, fyziky a astronomie 22 (1977), 91–99. [4] O. Bor˚ uvka (ed. J. Malina), Universitas Masarykiana, Granos Plus, Brno 1996, ISBN 80-902004-0-0. [5] O. Bor˚ uvka, Personal communications and discussions with J. Neˇsetˇril (around 1980). ˇ ık and V. Doleˇzal and M. Fiedler, Kombinatorick´ [6] K. Cul´ a anal´yza v praxi, SNTL, Prague, 1967. [7] W. H. Cunningham, The Coming of the Matroids, this volume. [8] R. L. Graham and M. Gr¨otschel and L. Lov´asz (eds.), Handbook of Combinatorics, North-Holland, Amsterdam, 1995. [9] R. L. Graham and P. Hell, On the history of the minimum spanning tree problem, Ann. History Comput. 7 (1985), 43–57. [10] P. A. Griffits, Some Reflexions on the Mathematical Contributions of S. S. Chern, in: S. S. Chern, S. – Y. Cheng, G. Tian, P. Li (eds.), A mathematician and his mathematical work: selected papers of S.S. Chern, Springer 1978, 76 – 82. [11] V. Jarn´ık, O jist´em problem minim´aln´ım, Pr´ ace morav. pˇr´ırodovˇed. spol. v Brnˇe VI (4) (1930), 57–63. [12] V. Jarn´ık and M. K¨ ossler, O minim´aln´ıch grafech obsahuj´ıc´ıch n dan´ ych ˇ bod˚ u, Cas. pro pˇestov´ an´ı matematiky 63 (1934), 223–235. [13] B. Korte and J. Neˇsetˇril, Vojtˇech Jarn´ık’s work in combinatorial optimization, Discrete Mathematics 235 (2001), 1–17. [14] J. B. Kruskal, On the shortest spanning subtree of a graph and the travelling salesman problem, Proc. Amer. Math. Soc. 7 (1956), 48–50. [15] J. B. Kruskal, A reminiscence about shortest spanning subtrees, Arch. Math. 33 (1–2) (1997), 13–14. [16] M. Mareˇs, The saga of minimum spanning trees, Computer Sci. Review 2 (2008), 165–221. [17] J. Matouˇsek and J. Neˇsetˇril, Invitation to discrete mathematics, Oxford Univ. Press, Oxford 1998, 2008 Documenta Mathematica

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[18] J. Neˇsetˇril, Some remarks on the history of MST-problem, Arch. Math. 33 (1997), 15–22. [19] J. Neˇsetˇril and E. Milkov´a and H. Neˇsetˇrilov´a, Otakar Bor˚ uvka on minimum spanning tree problem. Translation of both the 1926 papers, comments, history, (2001), 3–36. [20] R. C. Prim, The shortest connecting network and some generalization, Bell Systems Tech. J. 36 (1957), 1389–1401. [21] Arch. Math. (a special Bor˚ uvka issue) 33, 1–2, (1997). [22] A. Schrijver, On the history of combinatorial optimization (till 1960), in: K. Aardal, G.L. Nemhauser, R. Weismantel (eds.), Handbook of Discrete Optimization Elsevier, Amsterdam, 2005, 1–68. [23] R. E. Tarjan, Data Structures and Network algorithms, SIAM, 1987.

Jaroslav Neˇsetˇril IUUK, Faculty of Mathematics and Physics Charles University Malostransk´e n´ am. 25 118 00 Prague 1 Czech Republic

Helena Neˇsetˇrilov´a PEF, Czech University for Life Sciences Kam´ yck´a 129 165 21 Prague 6 Czech Republic [emailprotected]

[emailprotected]

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The Coming of the Matroids William H. Cunningham

2010 Mathematics Subject Classification: 90C27, 05B35 Keywords and Phrases: Matroid, submodular function, polymatroid

Introduction In 1964, thirty years after their introduction, and having lived a quiet life until then, matroids began to get the attention of optimizers. Just a few years later, as a result of exciting research achievements as well as enthusiastic promotion, the theory of matroids and submodular functions had become an integral part of discrete optimization. Whitney Matroid theory starts with the paper [22] of Hassler Whitney in 1935. A matroid may be defined to be a family of “independent” subsets of a finite ground set S, satisfying • Every subset of an independent set is independent • For any A ⊆ S all maximal independent subsets of A (called bases of A) have the same cardinality (called the rank r(A) of A). Of course, if we take S to be the set of columns of a matrix, and the independent sets to be the ones that are linearly independent, we get a first example, called a linear matroid. Another important class consists of the graphic ones – here S is the set of edges of a graph G and a subset is independent if it forms a forest. Whitney established some equivalent versions of the axioms, highlighted the above two examples, and proved several basic results. In particular, he showed that, given a matroid M , one gets a second dual matroid M ∗ by declaring independent all the sets whose deletion from S do not lower its rank. This generalizes the notion of duality in planar graphs. In addition, he observed that the rank function r satisfies what we now call the submodular property: For all subsets A, B of S r(A) + r(B) ≥ r(A ∪ B) + r(A ∩ B). Documenta Mathematica

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There were other researchers who suggested ideas similar to Whitney’s. None of these early papers appears to contain any suggestion of a connection with optimization. In retrospect, one might observe that the definition implies that a certain trivial algorithm solves the optimization problem of finding a largest independent set. Rado In the next twenty years, there was little in the way of followup work to Whitney’s paper. One exception, not widely appreciated at the time, was a paper [14] of Richard Rado in 1942. Rado gave a matroid generalization of Hall’s theorem on matching. This famous theorem says that if G is a bipartite graph with parts S, T , then T can be matched into S if and only if for every subset A of T , |N (A)| ≥ |A|. (Here N (A) denotes the neighbourset of A.) Rado’s “Independent Transversal Theorem” is perhaps the first significant result in matroid theory. Theorem 1. Let G be a bipartite graph with parts S, T , and let M be a matroid on S. Then T can be matched to an independent set of M , if and and only if, for every subset A of T , r(N (A)) ≥ |A|. Tutte In the late fifties Bill Tutte published several deep results on matroid theory [18], [19]. Tutte’s background is interesting. A chemistry student at the beginning of the war, he was recruited to the Bletchley Park codebreaking project. His brilliant contributions to that effort were kept secret for more than fifty years. See Copeland [1] for details. At the end of the war Tutte returned to Cambridge as a mathematician, and a Fellow of Trinity College; the fellowship was a partial reward for his war work. In his thesis he studied “nets”, a generalizations of graphs, which he has described [21] as being “half-way to matroids”. He eventually published much of this work in the setting of matroid theory. Tutte solved several of the fundamental problems suggested by the work of Whitney. These included characterizing the matroids that are graphic, those that arise from matrices over the binary field, and those that are regular (that is, arise from matrices over every field). These basic results are already of importance to optimizers. Understanding the graphic matroids, is the key to understanding which linear programming problems are reducible, by row operations and variable-scaling, to network flow problems. Moreover, as Tutte showed, the regular matroids are precisely the ones realizable by totally unimodular matrices, which Tutte characterized. However, Tutte’s matroid papers were difficult and their connections with optimization were not immediately recognized. Documenta Mathematica

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The Sixties It was in the 1960’s that matroids and submodularity became an important subject in optimization. The dominant figure of the period was Jack Edmonds. Not only did he discover beautiful theorems and algorithms. He also championed his subject tirelessly, defining a vocabulary that is still in use, and an agenda – efficient algorithms and polyhedral descriptions – that is still being followed. By 1969 Edmonds and his work had been featured at a major international conference, and he had written for its proceedings the milestone paper [2]. Edmonds, Lehman, and Matroid Partition Like Tutte, Jack Edmonds had an interesting background; see his own lively account in [3]. After his undergraduate years, which included study at two universities and a year out of school, he enrolled in the graduate program in mathematics at the University of Maryland. He completed a master’s thesis, in which he proved a fundamental result in topological graph theory, but left Maryland before completing the doctoral program. He was fortunate to obtain a position in the Applied Mathematics Division of the National Bureau of Standards in Washington. Here, in an operations research group headed by Alan Goldman, he was exposed to problems in combinatorial optimization. Edmonds has written “That is where graduate school started for me, with Alan Goldman”. In 1961, while participating in a workshop at the Rand Corporation, he discovered the key idea that led to his solution of the matching problem. Over the next couple of years, he worked out algorithms and polyhedral descriptions for matching and degree-constrained subgraphs (for more on this, see Pulleyblank [13]). Since Tutte had proved the basic existence theorem in matching theory, Edmonds was certainly aware of his work. However, he credits Alfred Lehman for inspiring him to consider matroids as a natural setting for posing and attacking algorithmic problems. The two met in spring 1964, shortly after Lehman solved the Shannon switching game, a game played on a graph. In fact, Lehman [10] had invented and solved a more general game, played on a matroid. His solution did not however, provide efficient algorithms to decide which player had the winning strategy. For one variant of Lehman’s game, the condition for a certain player to have a winning strategy is that the ground set have two disjoint bases. Edmonds characterized this property, and more generally solved the problem of finding in a matroid M a largest set that is the union of k independent sets, at the same time providing an algorithm. The algorithm is efficient, assuming that there is an efficient algorithm to recognize independence in M . This and related results completed the solution of Lehman’s game. Then with Ray Fulkerson, Edmonds solved a yet more general problem, as follows. Suppose that we are given matroids M1 , . . . , Mk on S. Call a set I partitionable if it can be expressed Documenta Mathematica

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as the union of k sets Ii , where Ii is independent in Mi for each i. Theorem 2 (Matroid Partition Theorem). The maximum size of a set I partitionable with respect to M1 , . . . , Mk is equal to the minimum, over subsets A of S, of k X |S\A| + ri (A). i=1

Here ri denotes the rank function of Mi . Their proof is an efficient algorithm to find the optimal I and A. It is easy to obtain from the Matroid Partition Theorem a formula for the maximum number of disjoint bases of a given matroid, and for the minimum number of independent sets that cover S. In fact, the technique provides many applications to packing and covering. The First Conference Jack Edmonds organized the first conference on matroids. It was called a “Seminar on Matroids” and was held at NBS August 31 to September 11, 1964. He has written [4] that, when organizing the meeting, he “could not find more than six people who had heard the term” matroid. But there, according to Tutte [21], “the theory of matroids was proclaimed to the world”. Edmonds arranged for Tutte to give a series of lectures on his work, and to write for publication a new exposition [20] of his main structural results. Edmonds presented his own work related to partitioning and Lehman’s game. Participants included Ray Fulkerson and Gian-Carlo Rota; the latter campaigned to change the term “matroid” to “combinatorial geometry”. Tutte and Edmonds were not convinced, and the movement was ultimately not successful, but there was a period in the seventies when it seemed the new term might be winning out. One paper [9] suggested that was the case, and tut-tutted that the term “matroid” was “still

Figure 1: The Seminar on Matroids, NBS, 1964. First row, second from left, Ray Fulkerson, third from left, Bill Tutte. (Photo courtesy of William Pulleyblank) Documenta Mathematica

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Figure 2: The Seminar on Matroids, NBS, 1964. First row, right, Jack Edmonds, third from right, Gian-Carlo Rota. (Photo courtesy of William Pulleyblank)

used in pockets of the tradition-bound British Commonwealth”. (At that time both Tutte and Edmonds were in Waterloo.) Matroid Intersection There are several theorems essentially equivalent to the Matroid Partition Theorem, and they are important in their own right. These equivalent statements serve to emphasize the power of the theorem and algorithm. However, almost inevitably there have been independent discovery and rediscovery of results. In fact Rado’s Theorem 1 is one of these. Another of the equivalent theorems is known as Tutte’s Linking Theorem; see [12]. Tutte called it Menger’s Theorem for Matroids. But for optimizers, undoubtedly the most important of these versions is Edmonds’ Matroid Intersection Theorem, which he discovered by applying the Matroid Partition Theorem to M1 and the dual of M2 . Theorem 3 (Matroid Intersection Theorem). Let M1 , M2 be matroids on S. The maximum size of a common independent set is equal to the minimum over subsets A of S of r1 (A) + r2 (S\A). This theorem generalizes the famous K˝onig min-max theorem for the maximum size of a matching in a bipartite graph. Since the more general weighted version of that problem (essentially, the optimal assignment problem) was well known to be solvable, Theorem 3 cries out for a weighted generalization. So, given two matroids on S and a weight vector c ∈ RS , can we find a common independent set of maximum weight? Or, can we describe the convex hull of common independent sets? First, let’s back up and deal with a single matroid. Documenta Mathematica

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The Matroid Polytope By 1964 Jack Edmonds had already solved the weighted matching problem, in the process, proving the matching polyhedron theorem. The fact that a greedy algorithm finds an optimal spanning tree of a graph was well known. Its proof did not require polyhedral methods, but Alan Goldman asked a natural question – can we describe the convex hull of spanning trees? By this time Edmonds was well into matroids, and realized (this was also known to Rado [15]) that the greedy algorithm finds a maximum weight basis of a matroid. So getting the polytope of independent sets was a breeze. Theorem 4 (Matroid Polytope Theorem). Let M be a matroid on S with rank function r. The convex hull of characteristic vectors of independent sets is P (M ) = {x ∈ RS : x ≥ 0, x(A) ≤ r(A) for all A ⊆ S}. Edmonds proved the theorem by proving that, for any weight vector c ∈ RS , the LP problem maximize cT x subject to x ∈ P (M ) is solved by the greedy algorithm. We will see his method in more detail shortly. Edmonds’ Amazing Theorem Now suppose we have two matroids M1 , M2 on S and we want to describe the convex hull of common independent sets, which we write, with abuse of notation, as P (M1 ∩ M2 ). Clearly, every common extreme point of any two polyhedra is an extreme point of their intersection. In general, there will be other extreme points as well. It would be a rare situation indeed for the two polyhedra to fit together so neatly, that the only extreme points of the intersection were the common extreme points. But this is the case if the two polyhedra are matroid polyhedra! In lectures, Edmonds sometimes referred to his result – indeed, deservedly – as “my amazing theorem”. Theorem 5 (Matroid Intersection Polytope Theorem). Let M1 , M2 be matroids on S. Then P (M1 ∩ M2 ) = P (M1 ) ∩ P (M2 ). Now, having generalized from one matroid to two, and from maximum cardinality to maximum weight, Edmonds went further, generalizing the matroid concept. The polyhedron P (M ) has the property that for every weight vector c, the greedy algorithm optimizes cT x over P (M ). Edmonds discovered a more general class of polyhedra having this property. And, one that permits generalization of the Amazing Theorem, too. Polymatroids Edmonds considered nonempty polyhedra of the form P (f ) = {x ∈ RS : x ≥ 0, x(A) ≤ f (A) for all A ⊆ S}, where f is submodular. He called such a polyhedron a polymatroid. It turns out that any such P (f ) can be Documenta Mathematica

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described by an f which is also increasing and satisfies f (∅) = 0. Such functions are now called polymatroid functions. Of course, matroid rank functions are polymatroid functions, and matroid polyhedra are polymatroids. Generalizing his method for matroids, he considered the dual LP problems max cT x : x ≥ 0, x(A) ≤ f (A) for all A ⊆ S

min P

P

(f (A)yA : A ⊆ S) subject to

(1)

(2)

(yA : A ⊆ S, e ∈ A) ≥ ce , for all e ∈ S yA ≥ 0, for all A ⊆ S.

Now order S as e1 . . . , en such that ce1 ≥ · · · ≥ cem ≥ 0 ≥ cem+1 ≥ · · · ≥ cen , and define Ti to be {e1 , . . . , ei } for 0 ≤ i ≤ n. The greedy algorithm is: Put xei = f (Ti ) − f (Ti−1 ) for 1 ≤ i ≤ m and xj = 0 otherwise. The dual greedy algorithm is: Put yTi = cei − cei+1 for 1 ≤ i ≤ m − 1, put yTm = cem and put all other yA = 0. The resulting solutions satisfy the LP optimality conditions for (1) and (2). It is also clear that if f is integral, then so is x, and if c is integral, then so is y. In particular, this proves a significant generalization of Theorem 4. As we shall see, it proves much more. Polymatroid Intersection Now here is the topper – Edmonds puts all three directions of generalization together. Theorem 6 (Weighted Polymatroid Intersection). Let f1 , f2 be polymatroid functions on S, and let c ∈ RS . Consider the LP problem max cT x x(A) ≤ f1 (A), for all A ⊆ S

(3)

x(A) ≤ f2 (A), for all A ⊆ S xe ≥ 0, for all e ∈ S. and its dual problem min P

P

1 2 (r1 (A)yA + r2 (A)yA : A ⊆ S) subject to

1 2 (yA + yA : A ⊆ S, e ∈ A) ≥ ce , for all e ∈ S 1 2 yA , yA ≥ 0, for all A ⊆ S.

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(a) If f1 , f2 are integer-valued, then (3) has an integral optimal solution. (b) If c is integral, then (4) has an integral optimal solution. We will sketch Edmonds’ ingenious proof. Consider an optimal solution yˆ1 , yˆ2 of (4). The problem of optimizing over y 1 while keeping y 2 fixed at yˆ2 is an LP problem of the form (2), which can be optimized by the dual greedy algorithm. Therefore, we can replace yˆ1 by the output of that algorithm. Now we can fix y 1 and similarly replace yˆ2 . We conclude that (4) has an optimal solution that is an optimal solution to a problem in which the constraint matrix has a very special structure. Namely, its columns split into two sets, each of which consists of the characteristic vectors of a telescoping family of subsets of S. Edmonds proved – it is a nice exercise – that such a matrix is totally unimodular. It follows that (4) has an optimal solution that is integral, assuming that c is integral, proving (b). Now with the benefits of hindsight, we can invoke the theory of total dual integrality, and (a) is proved. In fact, Edmonds did not have that tool. He used another argument, again a clever indirect use of total unimodularity, to prove (a). There are several important consequences of the above theorem. For example, taking f1 , f2 to be matroid rank functions, we get the Amazing Theorem. Taking each cj = 1, we get the following important result. Theorem 7 (Polymatroid Intersection Theorem). Let f1 , f2 be polymatroid functions on S. Then max(x(S) : x ∈ P (f1 ) ∩ P (f2 )) = min(f1 (A) + f2 (S\A) : A ⊆ S). Moreover, if f1 , f2 are integer-valued, then x can be chosen integral. Postlude In the years since the sixties, much progress has been made, far too much to summarize here. I mention a few highlights, relating them to the work of the sixties. The books of Frank [6] and Schrijver [17] can be consulted for more detail. Submodularity and Convexity Let us call a function f supermodular if −f is submodular, and call it modular if it is both submodular and supermodular. It is easy to see that a function f is modular if and only if it satisfies f (A) = m(A) + k for some m ∈ RS and k ∈ R. Then we have the beautiful Discrete Separation Theorem of Frank [5]. Theorem 8. Let f, g be functions defined on subsets of S such that f is submodular, g is supermodular, and f ≤ g. Then there exists a modular function h such that f ≤ h ≤ g. Moreover, if f and g are integer-valued, then h may be chosen integer-valued. Documenta Mathematica

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In fact, this theorem can be proved from the Polymatroid Intersection Theorem 7, and conversely. Its first part resembles a well-known result about the separation of convex and concave functions by an affine function. Actually, there is a connection. Lov´ asz [11] defined the extension fˆ to RS+ of a set function f , using ideas suggested by the dual greedy algorithm. He then proved that fˆ is convex if and only if f is submodular. Using this, one can derive the first part of Frank’s theorem from the convexity result. Submodular Function Minimization The problem of minimizing a submodular function (given by an evaluation oracle) is fundamental. Its special cases include finding a minimum capacity s, t-cut in a directed graph, and (in view of the Matroid Intersection Theorem) finding the maximum size of a common independent set of two given matroids. A good characterization of the minimum follows from the work of Edmonds [2]. One way to describe it is this. One can reduce the problem of minimizing a submodular function g to the problem of minimizing f (A) + u(S\A), where u ≥ 0 and f is a polymatroid function. But max(x(S) : x ∈ P (f ), x ≤ u) = min(f (A) + u(S\A) : A ⊆ S). This is a special case of the Polymatroid Intersection Theorem 7, but it can easily be proved directly. Now suppose we have A and x giving equality above. Then x ∈ P (f ) can be certified by expressing it as the convex combination of a small number of extreme points of P (f ), and each extreme point can be certified by the polymatroid greedy algorithm. So much for characterizing the minimum. What about an algorithm to find the minimum? The first efficient algorithm was found by Gr¨otschel, Lov´asz and Schrijver [7], based essentially on the equivalence, via the ellipsoid method, of separation and optimization. More recently, Iwata, Fleischer, and Fujishige [8] and Schrijver [16] gave combinatorial algorithms. Both use explicitly the method of certifying membership in P (f ) described above. Weighted Polymatroid Intersection The problem of finding an efficient algorithm for weighted polymatroid intersection, and other closely related models such as optimal submodular flows, was left open by Edmonds. (He, and also Lawler, did solve the special case of weighted matroid intersection.) Efficient combinatorial algorithms now exist. One may summarize their development as follows. Lawler and Martel and also Sch¨ onsleben gave efficient algorithms for the maximum component-sum problem. Cunningham and Frank combined this with a primal-dual approach to handle general weights. These algorithms need as a subroutine one of the above algorithms for submodular function minimization. Documenta Mathematica

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Matroid Intersection and Matching Weighted versions of matroid intersection and matching have a common special case, optimal bipartite matching. In addition they share similar attractive results – polyhedral descriptions, and efficient solution algorithms. It is natural, therefore, to ask whether there exists a common generalization to which these results extend. Several candidates have been proposed. The most important one, proposed independently by Edmonds and Lawler, has several equivalent versions, one of which goes as follows. Given a graph G and a matroid M on its vertex-set, a matroid matching is a matching of G whose covered vertices form an independent set in M . It turned out that finding a maximum-weight matroid matching, even when the weights are all 1, is a hard problem. However, in the late seventies Lov´ asz found an efficient algorithm and a min-max formula for the case where M arises from a given linear representation. Recently, Iwata and Pap independently have found efficient algorithms for the weighted version, answering a question that was open for more than thirty years. References [1] J. Copeland, Colossus: The Secrets of Bletchley Park’s Codebreaking Computers, Oxford University Press, 2006. [2] J. Edmonds, Submodular functions, matroids, and certain polyhedra in: R. Guy et al. (eds) Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, 69–87. [3] Edmonds, Jack, A glimpse of heaven, in: J.K. Lenstra et al. (eds), History of Mathematical Programming North-Holland, Amsterdam, 1991. 32–54. [4] J. Edmonds, Matroid partition, in: M. Juenger et al. (eds.) Fifty Years of Integer Programming Springer Verlag, Heidelberg, 2010, 199–217. [5] A. Frank, An algorithm for submodular functions on graphs, Ann. Discrete Math 16 (1982), 97–210. [6] A. Frank, Connections in Combinatorial Optimization, Oxford, U.K., 2011. [7] M. Gr¨ otschel, L. Lov´ asz and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), 169– 197. [8] S. Iwata, L. Fleischer, and S. Fujishige, A Combinatorial strongly polynomial algorithm for minimizing submodular functions, J. ACM 48 (2001), 761–777. [9] D.G. Kelly and G.-C. Rota, Some problems in combinatorial theory, in: A Survey of Combinatorial Theory North-Holland, Amsterdam, 1973, pp. 309–312. Documenta Mathematica

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[10] A. Lehman, A Solution of the Shannon switching game, J. SIAM 12 (1964). 687–725. [11] L. Lov´ asz, Submodular functions and convexity in: Bachem et al. (eds.) Mathematical Programming: The State of the Art, Springer Verlag 1982. [12] J. Oxley, Matroid Theory, Oxford University Press, Oxford, 2011. [13] W. R. Pulleyblank, Edmonds, matching, and the birth of polyhedral combinatorics, this volume. [14] R. Rado, A theorem on independence relations, Quarterly J. Math. Oxford (2) 13(1942), 83–89. [15] R. Rado, Note on independence functions, Proc. London Math Soc (3) 7(1957), 300–320. [16] A. Schrijver, A combinatorial algorithm minimizing submodular functions in polynomial time, J. Comb. Theory B 80(2000), 346–355. [17] A. Schrijver, Combinatorial Optimization: Springer-Verlag Berlin, 2003.

Polyhedra and Efficiency.

[18] W.T. Tutte, A hom*otopy theorem for matroids, I and II, Trans. AMS 88(1958), 153–184. [19] W.T. Tutte, Matroids and graphs, Tran. AMS 89(1959), 527–552. [20] W.T. Tutte, Lectures on matroids, J. Res. NBS 69B (1965), 1–47. [21] W.T. Tutte, The coming of the matroids, Surveys in Combinatorics, LMS Lecture Note Series 267 (1999), 3–14. [22] H. Whitney, The abstract properties of linear dependence, Am. J.Math. 57(1935), 509–533.

William H. Cunningham Department of Combinatorics & Optimization University of Waterloo Waterloo, ON Canada, N2L 3G1 [emailprotected]

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On the History of the Shortest Path Problem Alexander Schrijver

2010 Mathematics Subject Classification: 01A60, 05-03, 05C38, 05C85, 90C27 Keywords and Phrases: Shortest path, algorithm, history It is difficult to trace back the history of the shortest path problem. One can imagine that even in very primitive (even animal) societies, finding short paths (for instance, to food) is essential. Compared with other combinatorial optimization problems, like shortest spanning tree, assignment and transportation, the mathematical research in the shortest path problem started relatively late. This might be due to the fact that the problem is elementary and relatively easy, which is also illustrated by the fact that at the moment that the problem came into the focus of interest, several researchers independently developed similar methods. Yet, the problem has offered some substantial difficulties. For some considerable period heuristical, nonoptimal approaches have been investigated (cf. for instance Rosenfeld [1956], who gave a heuristic approach for determining an optimal trucking route through a given traffic congestion pattern). Path finding, in particular searching in a maze, belongs to the classical graph problems, and the classical references are Wiener [1873], Lucas [1882] (describing a method due to C.P. Tr´emaux), and Tarry [1895] – see Biggs, Lloyd, and Wilson [1976]. They form the basis for depth-first search techniques. Path problems were also studied at the beginning of the 1950’s in the context of ‘alternate routing’, that is, finding a second shortest route if the shortest route is blocked. This applies to freeway usage (Trueblood [1952]), but also to telephone call routing. At that time making long-distance calls in the U.S.A. was automatized, and alternate routes for telephone calls over the U.S. telephone network nation-wide should be found automatically. Quoting Jacobitti [1955]: When a telephone customer makes a long-distance call, the major problem facing the operator is how to get the call to its destination. In some cases, each toll operator has two main routes by which the call can be started towards this destination. The first-choice route, of course, is the most direct route. If this is busy, the second choice is made, followed by other available choices at the operator’s Documenta Mathematica

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Alexander Schrijver discretion. When telephone operators are concerned with such a call, they can exercise choice between alternate routes. But when operator or customer toll dialing is considered, the choice of routes has to be left to a machine. Since the “intelligence” of a machine is limited to previously “programmed” operations, the choice of routes has to be decided upon, and incorporated in, an automatic alternate routing arrangement.

Matrix methods for unit-length shortest path 1946–1953 Matrix methods were developed to study relations in networks, like finding the transitive closure of a relation; that is, identifying in a directed graph the pairs of points s, t such that t is reachable from s. Such methods were studied because of their application to communication nets (including neural nets) and to animal sociology (e.g. peck rights). The matrix methods consist of representing the directed graph by a matrix, and then taking iterative matrix products to calculate the transitive closure. This was studied by Landahl and Runge [1946], Landahl [1947], Luce and Perry [1949], Luce [1950], Lunts [1950, 1952], and by A. Shimbel. Shimbel’s interest in matrix methods was motivated by their applications to neural networks. He analyzed with matrices which sites in a network can communicate to each other, and how much time it takes. To this end, let S be the 0, 1 matrix indicating that if Si,j = 1 then there is direct communication from i to j (including i = j). Shimbel [1951] observed that the positive entries in S t correspond to pairs between which there exists communication in t steps. An adequate communication system is one for which the matrix S t is positive for some t. One of the other observations of Shimbel [1951] is that in an adequate communication system, the time it takes that all sites have all information, is equal to the minimum value of t for which S t is positive. (A related phenomenon was observed by Luce [1950].) Shimbel [1953] mentioned that the distance from i to j is equal to the number of zeros in the i, j position in the matrices S 0 , S 1 , S 2 , . . . , S t . So essentially he gave an O(n4 ) algorithm to find all distances in a directed graph with unit lengths. Shortest-length paths If a directed graph D = (V, A) and a length function l : A → R are given, one may ask for the distances and shortest-length paths from a given vertex s. For this, there are two well-known methods: the ‘Bellman-Ford method’ and ‘Dijkstra’s method’. The latter one is faster but is restricted to nonnegative length functions. The former method only requires that there is no directed circuit of negative length. The general framework for both methods is the following scheme, described in this general form by Ford [1956]. Keep a provisional distance function d. Documenta Mathematica

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Initially, set d(s) := 0 and d(v) := ∞ for each v 6= s. Next, iteratively, choose an arc (u, v) with d(v) > d(u) + l(u, v) and reset d(v) := d(u) + l(u, v). (1) If no such arc exists, d is the distance function. The difference in the methods is the rule by which the arc (u, v) with d(v) > d(u) + l(u, v) is chosen. The Bellman-Ford method consists of considering all arcs consecutively and applying (1) where possible, and repeating this (at most |V | rounds suffice). This is the method described by Shimbel [1955], Bellman [1958], and Moore [1959]. Dijkstra’s method prescribes to choose an arc (u, v) with d(u) smallest (then each arc is chosen at most once, if the lengths are nonnegative). This was described by Leyzorek, Gray, Johnson, Ladew, Meaker, Petry, and Seitz [1957] and Dijkstra [1959]. A related method, but slightly slower than Dijkstra’s method when implemented, was given by Dantzig [1958], and chooses an arc (u, v) with d(u) + l(u, v) smallest. Parallel to this, a number of further results were obtained on the shortest path problem, including a linear programming approach and ‘good characterizations’. We review the articles in a more or less chronological order. Shimbel 1955 The paper of Shimbel [1955] was presented in April 1954 at the Symposium on Information Networks in New York. Extending his matrix methods for unit-length shortest paths, he introduced the following ‘min-sum algebra’: Arithmetic For any arbitrary real or infinite numbers x and y x + y ≡ min(x, y) and xy ≡ the algebraic sum of x and y. He transferred this arithmetic to the matrix product. Calling the distance matrix associated with a given length matrix S the ‘dispersion’, he stated: It follows trivially that S k k ≥ 1 is a matrix giving the shortest paths from site to site in S given that k − 1 other sites may be traversed in the process. It also follows that for any S there exists an integer k such that S k = S k+1 . Clearly, the dispersion of S (let us label it D(S)) will be the matrix S k such that S k = S k+1 . This is equivalent to the Bellman-Ford method. Although Shimbel did not mention it, one trivially can take k ≤ |V |, and hence the method yields an O(n4 ) algorithm to find the distances between all pairs of points. Documenta Mathematica

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Alexander Schrijver Shortest path as linear programming problem 1955–1957

Orden [1955] observed that the shortest path problem is a special case of a transshipment problem (= uncapacitated minimum-cost flow problem), and hence can be solved by linear programming. Dantzig [1957] described the following graphical procedure for the simplex method applied to this problem. Let T be a rooted spanning tree on {1, . . . , n}, with root 1. For each i = 1, . . . , n, let ui be equal to the length of the path from 1 to i in T . Now if uj ≤ ui + di,j for all i, j, then for each i, the 1 − i path in T is a shortest path. If uj > ui + di,j , replace the arc of T entering j by the arc (i, j), and iterate with the new tree. Pn Trivially, this process terminates (as j=1 uj decreases at each iteration, and as there are only finitely many rooted trees). Dantzig illustrated his method by an example of sending a package from Los Angeles to Boston. (Edmonds [1970] showed that this method may take exponential time.) In a reaction to the paper of Dantzig [1957], Minty [1957] proposed an ‘analog computer’ for the shortest path problem: Build a string model of the travel network, where knots represent cities and string lengths represent distances (or costs). Seize the knot ‘Los Angeles’ in your left hand and the knot ‘Boston’ in your right and pull them apart. If the model becomes entangled, have an assistant untie and re-tie knots until the entanglement is resolved. Eventually one or more paths will stretch tight – they then are alternative shortest routes. Dantzig’s ‘shortest-route tree’ can be found in this model by weighting the knots and picking up the model by the knot ‘Los Angeles’. It is well to label the knots since after one or two uses of the model their identities are easily confused. A similar method was proposed by Bock and Cameron [1958]. Ford 1956 In a RAND report dated 14 August 1956, Ford [1956] described a method to find a shortest path from P0 to PN , in a network with vertices P0 , . . . , PN , where lij denotes the length of an arc from i to j. We quote: Assign initially x0 = 0 and xi = ∞ for i 6= 0. Scan the network for a pair Pi and Pj with the property that xi − xj > lji . For this pair replace xi by xj + lji . Continue this process. Eventually no such pairs can be found, and xN is now minimal and represents the minimal distance from P0 to PN . So this is the general scheme described above (1). No selection rule for the arc (u, v) in (1) is prescribed by Ford. Documenta Mathematica

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Ford showed that the method terminates. It was shown however by Johnson [1973a, 1973b, 1977] that Ford’s liberal rule can take exponential time. The correctness of Ford’s method also follows from a result given in the book Studies in the Economics of Transportation by Beckmann, McGuire, and Winsten [1956]: given a length matrix (li,j ), the distance matrix is the unique matrix (di,j ) satisfying di,i = 0 for all i,

(2)

di,k = min(li,j + dj,k ) for all i, k with i 6= k. j

Good characterizations for shortest path 1956-1958 It was noticed by Robacker [1956] that shortest paths allow a theorem dual to Menger’s theorem: the minimum length of an P0 − Pn path in a graph N is equal to the maximum number of pairwise disjoint P0 − Pn cuts. In Robacker’s words: the maximum number of mutually disjunct cuts of N is equal to the length of the shortest chain of N from P0 to Pn . A related ‘good characterization’ was found by Gallai [1958]: A length function l : A → Z on the arcs of a directed graph (V, A) does not give negative-length directed circuits, if and only if there is a function (‘potential’) p : V → Z such that l(u, v) ≥ p(v) − p(u) for each arc (u, v). Case Institute of Technology 1957 The shortest path problem was also investigated by a group of researchers at the Case Institute of Technology in Cleveland, Ohio, in the project Investigation of Model Techniques, performed for the Combat Development Department of the Army Electronic Proving Ground. In their First Annual Report, Leyzorek, Gray, Johnson, Ladew, Meaker, Petry, and Seitz [1957] presented their results. First, they noted that Shimbel’s method can be speeded up by calculating S k by iteratively raising the current matrix to the square (in the min-sum matrix algebra). This solves the all-pairs shortest path problem in time O(n3 log n). Next, they gave a rudimentary description of a method equivalent to Dijkstra’s method. We quote: (1) All the links joined to the origin, a, may be given an outward orientation. [. . . ] (2) Pick out the link or links radiating from a, aaα , with the smallest delay. [. . . ] Then it is impossible to pass from the origin to any other node in the network by any “shorter” path than aaα . Consequently, the minimal path to the general node α is aaα . Documenta Mathematica

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Alexander Schrijver (3) All of the other links joining α may now be directed outward. Since aaα must necessarily be the minimal path to α, there is no advantage to be gained by directing any other links toward α. [. . . ] (4) Once α has been evaluated, it is possible to evaluate immediately all other nodes in the network whose minimal values do not exceed the value of the second-smallest link radiating from the origin. Since the minimal values of these nodes are less than the values of the second-smallest, third-smallest, and all other links radiating directly from the origin, only the smallest link, aaα , can form a part of the minimal path to these nodes. Once a minimal value has been assigned to these nodes, it is possible to orient all other links except the incoming link in an outward direction. (5) Suppose that all those nodes whose minimal values do not exceed the value of the second-smallest link radiating from the origin have been evaluated. Now it is possible to evaluate the node on which the second-smallest link terminates. At this point, it can be observed that if conflicting directions are assigned to a link, in accordance with the rules which have been given for direction assignment, that link may be ignored. It will not be a part of the minimal path to either of the two nodes it joins. [. . . ] Following these rules, it is now possible to expand from the secondsmallest link as well as the smallest link so long as the value of the third-smallest link radiating from the origin is not exceeded. It is possible to proceed in this way until the entire network has been solved.

(In this quotation we have deleted sentences referring to figures.) Bellman 1958 After having published several papers on dynamic programming (which is, in some sense, a generalization of shortest path methods), Bellman [1958] eventually focused on the shortest path problem by itself, in a paper in the Quarterly of Applied Mathematics. He described the following ‘functional equation approach’ for the shortest path problem, which is the same as that of Shimbel [1955]. There are N cities, numbered 1, . . . , N , every two of which are linked by a direct road. A matrix T = (ti,j ) is given, where ti,j is time required to travel from i to j (not necessarily symmetric). Find a path between 1 and N which consumes minimum time. Bellman remarked: Documenta Mathematica

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Since there are only a finite number of paths available, the problem reduces to choosing the smallest from a finite set of numbers. This direct, or enumerative, approach is impossible to execute, however, for values of N of the order of magnitude of 20. He gave a ‘functional equation approach” The basic method is that of successive approximations. We choose (0) an initial sequence {fi }, and then proceed iteratively, setting (k+1)

fi

(k)

= Min(tij + fj ], j6=i

(k+1)

fN

i = 1, 2, · · · , N − 1, = 0,

for k = 0, 1, 2 · · · , . (0)

As initial function fi Bellman proposed (upon a suggestion of F. Haight) (0) to take fi = ti,N for all i. Bellman noticed that, for each fixed i, starting (0) (k) with this choice of fi gives that fi is monotonically nonincreasing in k, and stated: It is clear from the physical interpretation of this iterative scheme that at most (N − 1) iterations are required for the sequence to converge to the solution. Since each iteration can be done in time O(N 2 ), the algorithm takes time O(N 3 ). As for the complexity, Bellman said: It is easily seen that the iterative scheme discussed above is a feasible method for either hand or machine computation for values of N of the order of magnitude of 50 or 100. In a footnote, Bellman mentioned: Added in proof (December 1957): After this paper was written, the author was informed by Max Woodbury and George Dantzig that the particular iterative scheme discussed in Sec. 5 had been obtained by them from first principles. Dantzig 1958 The paper of Dantzig [1958] gives an O(n2 log n) algorithm for the shortest path problem with nonnegative length function. It consists of choosing in (1) an arc with d(u) + l(u, v) as small as possible. Dantzig assumed (a) that one can write down without effort for each node the arcs leading to other nodes in increasing order of length and (b) that it is no effort to ignore an arc of the list if it leads to a node that has been reached earlier. He mentioned that, beside Bellman, Moore, Ford, and himself, also D. Gale and D.R. Fulkerson proposed shortest path methods, ‘in informal conversations’. Documenta Mathematica

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Dijkstra 1959 Dijkstra [1959] gave a concise and clean description of ‘Dijkstra’s method’, yielding an O(n2 )-time implementation. Dijkstra stated: The solution given above is to be preferred to the solution by L.R. Ford [3 ] as described by C. Berge [4 ], for, irrespective of the number of branches, we need not store the data for all branches simultaneously but only those for the branches in sets I and II, and this number is always less than n. Furthermore, the amount of work to be done seems to be considerably less. (Dijkstra’s references [3] and [4] are Ford [1956] and Berge [1958].) Dijkstra’s method is easier to implement (as an O(n2 ) algorithm) than Dantzig’s, since we do not need to store the information in lists: in order to find a next vertex v minimizing d(v), we can just scan all vertices. Later, using the more efficient data structures of heaps and Fibonacci heaps, one realized that Dijkstra’s method has implementations with running times O(m log n) and O(m + n log n) respectively, where m is the number of arcs (Johnson [1972] and Fredman and Tarjan [1987]). Moore 1959 At the International Symposium on the Theory of Switching at Harvard University in April 1957, Moore [1959] of Bell Laboratories, presented a paper “The shortest path through a maze”: The methods given in this paper require no foresight or ingenuity, and hence deserve to be called algorithms. They would be especially suited for use in a machine, either a special-purpose or a generalpurpose digital computer. The motivation of Moore was the routing of toll telephone traffic. He gave algorithms A, B, C, and D. First, Moore considered the case of an undirected graph G = (V, E) with no length function, in which a path from vertex A to vertex B should be found with a minimum number of edges. Algorithm A is: first give A label 0. Next do the following for k = 0, 1, . . .: give label k + 1 to all unlabeled vertices that are adjacent to some vertex labeled k. Stop as soon as vertex B is labeled. If it were done as a program on a digital computer, the steps given as single steps above would be done serially, with a few operations of the computer for each city of the maze; but, in the case of complicated mazes, the algorithm would still be quite fast compared with trial-and-error methods. Documenta Mathematica

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In fact, a direct implementation of the method would yield an algorithm with running time O(m). Algorithms B and C differ from A in a more economical labeling (by fewer bits). Moore’s algorithm D finds a shortest route for the case where each edge of the graph has a nonnegative length. This method is a refinement of Bellman’s method described above: (i) it extends to the case that not all pairs of vertices have a direct connection; that is, if there is an underlying graph G = (V, E) with length function; (ii) at each iteration only those di,j are considered for which ui has been decreased at the previous iteration. The method has running time O(nm). Moore observed that the algorithm is suitable for parallel implementation, yielding a decrease in running time bound to O(n∆(G)), where ∆(G) is the maximum degree of G. Moore concluded: The origin of the present methods provides an interesting illustration of the value of basic research on puzzles and games. Although such research is often frowned upon as being frivolous, it seems plausible that these algorithms might eventually lead to savings of very large sums of money by permitting more efficient use of congested transportation or communication systems. The actual problems in communication and transportation are so much complicated by timetables, safety requirements, signal-to-noise ratios, and economic requirements that in the past those seeking to solve them have not seen the basic simplicity of the problem, and have continued to use trial-and-error procedures which do not always give the true shortest path. However, in the case of a simple geometric maze, the absence of these confusing factors permitted algorithms A, B, and C to be obtained, and from them a large number of extensions, elaborations, and modifications are obvious. The problem was first solved in connection with Claude Shannon’s maze-solving machine. When this machine was used with a maze which had more than one solution, a visitor asked why it had not been built to always find the shortest path. Shannon and I each attempted to find economical methods of doing this by machine. He found several methods suitable for analog computation, and I obtained these algorithms. Months later the applicability of these ideas to practical problems in communication and transportation systems was suggested. Among the further applications of his method, Moore described the example of finding the fastest connections from one station to another in a given railroad timetable. A similar method was given by Minty [1958]. In May 1958, Hoffman and Pavley [1959] reported, at the Western Joint Computer Conference in Los Angeles, the following computing time for finding the distances between all pairs of vertices by Moore’s algorithm (with nonnegative lengths): Documenta Mathematica

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Alexander Schrijver It took approximately three hours to obtain the minimum paths for a network of 265 vertices on an IBM 704.

References [1956] M. Beckmann, C.B. McGuire, C.B. Winsten, Studies in the Economics of Transportation, Cowles Commission for Research in Economics, Yale University Press, New Haven, Connecticut, 1956. [1958] R. Bellman, On a routing problem, Quarterly of Applied Mathematics 16 (1958) 87–90. [1958] C. Berge, Th´eorie des graphes et ses applications, Dunod, Paris, 1958. [1976] N.L. Biggs, E.K. Lloyd, R.J. Wilson, Graph Theory 1736–1936, Clarendon Press, Oxford, 1976. [1958] F. Bock, S. Cameron, Allocation of network traffic demand by instant determination of optimum paths [paper presented at the 13th National (6th Annual) Meeting of the Operations Research Society of America, Boston, Massachusetts, 1958], Operations Research 6 (1958) 633–634. [1957] G.B. Dantzig, Discrete-variable extremum problems, Operations Research 5 (1957) 266–277. [1958] G.B. Dantzig, On the Shortest Route through a Network, Report P1345, The RAND Corporation, Santa Monica, California, [April 12] 1958 [Revised April 29, 1959] [published in Management Science 6 (1960) 187– 190]. [1959] E.W. Dijkstra, A note on two problems in connexion with graphs, Numerische Mathematik 1 (1959) 269–271. [1970] J. Edmonds, Exponential growth of the simplex method for shortest path problems, manuscript [University of Waterloo, Waterloo, Ontario], 1970. [1956] L.R. Ford, Jr, Network Flow Theory, Paper P-923, The RAND Corporation, Santa Monica, California, [August 14], 1956. [1987] M.L. Fredman, R.E. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms, Journal of the Association for Computing Machinery 34 (1987) 596–615. [1958] T. Gallai, Maximum-minimum S¨atze u ¨ber Graphen, Acta Mathematica Academiae Scientiarum Hungaricae 9 (1958) 395–434. [1959] W. Hoffman, R. Pavley, Applications of digital computers to problems in the study of vehicular traffic, in: Proceedings of the Western Joint Computer Conference (Los Angeles, California, 1958), American Institute of Electrical Engineers, New York, 1959, pp. 159–161. Documenta Mathematica

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[1955] E. Jacobitti, Automatic alternate routing in the 4A crossbar system, Bell Laboratories Record 33 (1955) 141–145. [1972] E.L. Johnson, On shortest paths and sorting, in: Proceedings of the ACM Annual Conference 25 (Boston, Massachusetts, 1972), The Association for Computing Machinery, New York, 1972, pp. 510–517. [1973a] D.B. Johnson, A note on Dijkstra’s shortest path algorithm, Journal of the Association for Computing Machinery 20 (1973) 385–388. [1973b] D.B. Johnson, Algorithms for Shortest Paths, Ph.D. Thesis [Technical Report CU-CSD-73-169, Department of Computer Science], Cornell University, Ithaca, New York, 1973. [1977] D.B. Johnson, Efficient algorithms for shortest paths in sparse networks, Journal of the Association for Computing Machinery 24 (1977) 1–13. [1939] T. Koopmans, Tanker Freight Rates and Tankship Building – An Analysis of Cyclical Fluctuations, Publication Nr 27, Netherlands Economic Institute, De Erven Bohn, Haarlem, 1939. [1942] Tj.C. Koopmans, Exchange ratios between cargoes on various routes (non-refrigerating dry cargoes), Memorandum for the Combined Shipping Adjustment Board, Washington D.C., 1942, 1–12 [first published in: Scientific Papers of Tjalling C. Koopmans, Springer, Berlin, 1970, pp. 77–86]. [1948] Tj.C. Koopmans, Optimum utilization of the transportation system, in: The Econometric Society Meeting (Washington, D.C., 1947; D.H. Leavens, ed.) [Proceedings of the International Statistical Conferences – Volume V], 1948, pp. 136–146 [reprinted in: Econometrica 17 (Supplement) (1949) 136–146] [reprinted in: Scientific Papers of Tjalling C. Koopmans, Springer, Berlin, 1970, pp. 184–193]. [1959] Tj.C. Koopmans, A note about Kantorovich’s paper, “Mathematical methods of organizing and planning production”, Management Science 6 (1959-60) 363–365. [1992] Tj.C. Koopmans, [autobiography] in: Nobel Lectures including presentation speeches and laureates’ biographies – Economic Sciences 1969–1980 (A. Lindbeck, ed.), World Scientific, Singapore, 1992, pp. 233–238. [1947] H.D. Landahl, A matrix calculus for neural nets: II, Bulletin of Mathematical Biophysics 9 (1947) 99–108. [1946] H.D. Landahl, R. Runge, Outline of a matrix algebra for neural nets, Bulletin of Mathematical Biophysics 8 (1946) 75–81. Documenta Mathematica

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[1957] M. Leyzorek, R.S. Gray, A.A. Johnson, W.C. Ladew, S.R. Meaker, Jr, R.M. Petry, R.N. Seitz, Investigation of Model Techniques – First Annual Report – 6 June 1956 – 1 July 1957 – A Study of Model Techniques for Communication Systems, Case Institute of Technology, Cleveland, Ohio, 1957. ´ Lucas, R´ecr´eations math´ematiques, deuxi`eme ´edition, Gauthier[1882] E. Villars, Paris, 1882–1883. [1950] R.D. Luce, Connectivity and generalized cliques in sociometric group structure, Psychometrika 15 (1950) 169–190. [1949] R.D. Luce, A.D. Perry, A method of matrix analysis of group structure, Psychometrika 14 (1949) 95–116. [1950] A.G. Lunts, Prilozhen ie matrichno˘ı bulevsko˘ı algebry k analizu i sintezu rele˘ıno-kontaktiykh skhem [Russian; Application of matrix Boolean algebra to the analysis and synthesis of relay-contact schemes], Doklady Akademii Nauk SSSR (N.S.) 70 (1950) 421–423. [1952] A.G. Lunts, Algebraicheskie metody analiza i sinteza kontaktiykh skhem [Russian; Algebraic methods of analysis and synthesis of relay contact networks], Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 16 (1952) 405–426. [1957] G.J. Minty, A comment on the shortest-route problem, Operations Research 5 (1957) 724. [1958] G.J. Minty, A variant on the shortest-route problem, Operations Research 6 (1958) 882–883. [1959] E.F. Moore, The shortest path through a maze, in: Proceedings of an International Symposium on the Theory of Switching, 2–5 April 1957, Part II [The Annals of the Computation Laboratory of Harvard University Volume XXX] (H. Aiken, ed.), Harvard University Press, Cambridge, Massachusetts, 1959, pp. 285–292. [1955] A. Orden, The transhipment problem, Management Science 2 (1955–56) 276–285. [1956] J.T. Robacker, Min-Max Theorems on Shortest Chains and Disjoint Cuts of a Network, Research Memorandum RM-1660, The RAND Corporation, Santa Monica, California, [12 January] 1956. [1956] L. Rosenfeld, Unusual problems and their solutions by digital computer techniques, in: Proceedings of the Western Joint Computer Conference (San Francisco, California, 1956), The American Institute of Electrical Engineers, New York, 1956, pp. 79–82. Documenta Mathematica

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[1951] A. Shimbel, Applications of matrix algebra to communication nets, Bulletin of Mathematical Biophysics 13 (1951) 165–178. [1953] A. Shimbel, Structural parameters of communication networks, Bulletin of Mathematical Biophysics 15 (1953) 501–507. [1955] A. Shimbel, Structure in communication nets, in: Proceedings of the Symposium on Information Networks (New York, 1954), Polytechnic Press of the Polytechnic Institute of Brooklyn, Brooklyn, New York, 1955, pp. 199–203. [1895] G. Tarry, Le probl`eme des labyrinthes, Nouvelles Annales de Math´ematiques (3) 14 (1895) 187–190 [English translation in: N.L. Biggs, E.K. Lloyd, R.J. Wilson, Graph Theory 1736–1936, Clarendon Press, Oxford, 1976, pp. 18–20]. [1952] D.L. Trueblood, The effect of travel time and distance on freeway usage, Public Roads 26 (1952) 241–250. [1873] Chr. Wiener, Ueber eine Aufgabe aus der Geometria situs, Mathematische Annalen 6 (1873) 29–30.

Alexander Schrijver CWI Science Park 123 1098 XG Amsterdam The Netherlands [emailprotected]

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On the History of the Transportation and Maximum Flow Problems Alexander Schrijver

Abstract. We review two papers that are of historical interest for combinatorial optimization: an article of A.N. Tolsto˘ı from 1930, in which the transportation problem is studied, and a negative cycle criterion is developed and applied to solve a (for that time) large-scale (10 × 68) transportation problem to optimality; and an, until recently secret, RAND report of T. E. Harris and F. S. Ross from 1955, that Ford and Fulkerson mention as motivation to study the maximum flow problem. The papers have in common that they both apply their methods to the Soviet railway network. 2010 Mathematics Subject Classification: 01A60, 05-03, 05C21, 05C85, 90C27 Keywords and Phrases: Maximum flow, minimum cut, transportation, algorithm, cycle cancelling, history

1

Transportation

The transportation problem and cycle cancelling methods are classical in optimization. The usual attributions are to the 1940’s and later1 . However, as early as 1930, A.N. Tolsto˘ı [1930]2 published, in a book on transportation planning issued by the National Commissariat of Transportation of the Soviet Union, an article called Methods of finding the minimal total kilometrage in cargo-transportation planning in space, in which he studied the transportation problem and described a number of solution approaches, including the, now well-known, idea that an optimum solution does not have any negative-cost 1 The transportation problem was formulated by Hitchco*ck [1941], and a cycle criterion for optimality was considered by Kantorovich [1942] (Kantorovich and Gavurin [1949]), Koopmans [1948] (Koopmans and Reiter [1951]), Robinson [1949, 1950], Gallai [1957, 1958], Lur’e [1959], Fulkerson [1961], and Klein [1967]. 2 Later, Tolsto˘ ı described similar results in an article entitled Methods of removing irrational transportations in planning [1939], in the September 1939 issue of Sotsialisticheski˘ı Transport.

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cycle in its residual graph3 . He might have been the first to observe that the cycle condition is necessary for optimality. Moreover, he assumed, but did not explicitly state or prove, the fact that checking the cycle condition is also sufficient for optimality. Tolsto˘ı illuminated his approach by applications to the transportation of salt, cement, and other cargo between sources and destinations along the railway network of the Soviet Union. In particular, a, for that time large-scale, instance of the transportation problem was solved to optimality. We briefly review the article. Tolsto˘ı first considered the transportation problem for the case where there are only two sources. He observed that in that case one can order the destinations by the difference between the distances to the two sources. Then one source can provide the destinations starting from the beginning of the list, until the supply of that source has been used up. The other source supplies the remaining demands. Tolsto˘ı observed that the list is independent of the supplies and demands, and hence it is applicable for the whole life-time of factories, or sources of production. Using this table, one can immediately compose an optimal transportation plan every year, given quantities of output produced by these two factories and demands of the destinations. Next, Tolsto˘ı studied the transportation problem in the case when all sources and destinations are along one circular railway line (cf. Figure 1), in which case the optimum solution is readily obtained by considering the difference of two sums of costs. He called this phenomenon circle dependency. Finally, Tolsto˘ı combined the two ideas into a heuristic to solve a concrete transportation problem coming from cargo transportation along the Soviet railway network. The problem has 10 sources and 68 destinations, and 155 links between sources and destinations (all other distances are taken to be infinite), as given in the following table. Tolsto˘ı’s heuristic also makes use of insight in the geography of the Soviet Union. He goes along all sources (starting with the most remote sources), where, for each source X, he lists those destinations for which X is the closest source or the second closest source. Based on the difference of the distances to the closest and second closest sources, he assigns cargo from X to the destinations, until the supply of X has been used up. (This obviously is equivalent to considering cycles of length 4.) In case Tolsto˘ı foresees a negative-cost cycle in the residual graph, he deviates from this rule to avoid such a cycle. No backtracking occurs. In the following quotation, Tolsto˘ı considers the cycles DzerzhinskRostov-Yaroslavl’-Leningrad-Artemovsk-Moscow-Dzerzhinsk and DzerzhinskNerekhta-Yaroslavl’-Leningrad-Artemovsk-Moscow-Dzerzhinsk. It is the sixth 3 The

residual graph has arcs from each source to each destination, and moreover an arc from a destination to a source if the transport on that connection is positive; the cost of the ‘backward’ arc is the negative of the cost of the ‘forward’ arc. Documenta Mathematica

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Agryz Aleksandrov Almaznaya Alchevskaya Baku Barybino Berendeevo Bilimbai Bobrinskaya Bologoe Verkhov’e Volovo Vologda Voskresensk V.Volochek Galich Goroblagodatskaya Zhlobin Zverevo Ivanovo Inza Kagan Kasimov Kinel’ Kovylkino Kyshtym Leningrad Likino Liski Lyuberdzhy Magnitogorskaya Mauk Moskva Navashino Nizhegol’ Nerekhta Nechaevskaya N.-Novgorod Omsk Orenburg Penza Perm’ Petrozavodsk Poltoradzhk Pskov Rostov/Don Rostov/Yarosl Rtishchevo Savelovo Samara San-Donato Saratov Sasovo Slavyanoserbsk Sonkovo Stalingrad St.Russa Tambov Tashkent Tula Tyumen’ Khar’kov Chelyabinsk Chishmy Shchigry Yudino Yama Yasinovataya Supply

709

135

1064 397

Dekonskaya

Iledzhk

693

566

549

44 85

52 93 55

543

65 114 1563 968

430 200

634

59 655 1398 678 757

663

1022 1353

1005 1343

882 227

890 235

661 740

1236 427 434 224

1056 434

380

196

259 735

1272 2445

2379

0 752 355

1208 421

1237

454

1667 223

1447 1213

159

709

1675

328 443

268

426 1074

411 932 398

288 12

378 78

50

678 136

405

818 1030

1022

333

316

349 92 32 1159

904

411

1040

1749 1394

1746 76 883

1739 1497 287 56

3085

880 1206 495 416

1072 1096 119 1337 624 1507 783 3051 840

193 558

584

1055 1079 115 607 1515 766 1775 848

329 251

511 773

1123 403

8.5

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·

757

100

863 1196 1406

157

504

11.5

1748 1505 296

454

711

5

1023

121

325

257

259 949 889

999

12

15

Demand 2 4 1.5 4 10 2 10 1 10 1 1 3 2 1 5 0.5 0.5 8 5 6 2 0.5 1 2 2 3 55 15 1 1 1 5 141 2 1 5 0.5 25 5 1.5 7 1 1 4 10 20 2 1 5 7 1 15 1 1.1 0.5 15.4 5 4 3 8 6 60 2 0.5 4 0.5 5 6

1180 81 106 1554 985

389

815

Artemovsk

Sverdlovsk

Kishert’

Dzerzhinsk

Balakhonikha

Murom

Yaroslavl’

Arkhangelsk

Table 1: Table of distances (in kilometers) between sources and destinations, and of supplies and demands (in kilotons). (Tolsto˘ı did not give any distance for Kasimov. We have inserted a distance 0 to Murom, since from Tolsto˘ı’s solution it appears that Kasimov is connected only to Murom, by a waterway.)

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Figure 1: Figure from Tolsto˘ı [1930] to illustrate a negative cycle step in his method, after the transports from the factories in Iletsk, Sverdlovsk, Kishert, Balakhonikha, and Murom have been set: 6. The Dzerzhinsk factory produces 100,000 tons. It can forward its production only in the Northeastern direction, where it sets its boundaries in interdependency with the Yaroslavl’ and Artemovsk (or Dekonskaya) factories.

Berendeevo Nerekhta Rostov

Aleksandrov Moscow

From Dzerzhinsk

From Yaroslavl’

430 km 349 ,, 454 ,,

135 km 50 ,, 56 ,,

From Dzerzhinsk

From Artemovsk

397 km 405 ,,

1,180 km 1,030 ,,

Difference to Dzerzhinsk −295 km −299 ,, −398 ,, Difference to Dzerzhinsk +783 km +625 ,,

The method of differences does not help to determine the boundary between the Dzerzhinsk and Yaroslavl’ factories. Only the circle dependency, specified to be an interdependency between the Dzerzhinsk, Yaroslavsl’ and Artemovsk factories, enables us to exactly determine how far the production of the Dzerzhinsk factory should be advanced in the Yaroslavl’ direction. Documenta Mathematica

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Suppose we attach point Rostov to the Dzerzhinsk factory; then, by the circle dependency, we get: Dzerzhinsk-Rostov Yaroslavl’- ,, Yaroslavl’-Leningrad Artemovsk- ,, Artemovsk-Moscow Dzerzhinsk- ,, Total

454 km 56 ,, 709 ,, 1,667 ,, 1,030 ,, 405 ,,

−398 km +958 ,, −625 ,, −65 km

Nerekhta 349 km −299 km ,, 50 ,, These points remain unchanged because only the quantity of production sent by each factory changes +34 km

Therefore, the attachment of Rostov to the Dzerzhinsk factory causes over-run in 65 km, and only Nerekhta gives a positive sum of differences and hence it is the last point supplied by the Dzerzhinsk factory in this direction. As a result, the following points are attached to the Dzerzhinsk factory: N. Novgorod Ivanova Nerekhta Aleksandrov Berendeevo Likino Moscow Total

25,000 6,000 5,000 4,000 10,000 15,000 35,000 100,000

tons ,, ,, ,, ,, ,, ,, tons

(remainder of factory’s production)

After 10 steps, when the transports from all 10 factories have been set, Tolsto˘ı “verifies” the solution by considering a number of cycles in the network, and he concludes that his solution is optimum: Thus, by use of successive applications of the method of differences, followed by a verification of the results by the circle dependency, we managed to compose the transportation plan which results in the minimum total kilometrage. The objective value of Tolsto˘ı’s solution is 395,052 kiloton-kilometers. Solving the problem with modern linear programming tools (CPLEX) shows that Tolsto˘ı’s solution indeed is optimum. But it is unclear how sure Tolsto˘ı could have been about his claim that his solution is optimum. Geographical insight probably has helped him in growing convinced of the optimality of his solution. On the other hand, it can be checked that there exist feasible solutions that have none of the negative-cost cycles considered by Tolsto˘ı in their residual graph, but that are yet not optimum4 . 4 The

maximum objective value of a feasible solution, whose residual graph does not contain any nonnegative-cost cycle of length 4, and not any of the seven longer nonnegativelength cycles considered by Tolsto˘ı (of lengths 6 and 8), is equal to 397,226. Documenta Mathematica

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The Soviet rail system also roused the interest of the Americans, and again it inspired fundamental research in optimization. In their basic paper Maximal Flow through a Network (published first as a RAND Report of November 19, 1954), Ford and Fulkerson [1954] mention that the maximum flow problem was formulated by T. E Harris as follows: Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other. In their 1962 book Flows in Networks, Ford and Fulkerson [1962] give a more precise reference to the origin of the problem:5 It was posed to the authors in the spring of 1955 by T. E. Harris, who, in conjunction with General F. S. Ross (Ret.), had formulated a simplified model of railway traffic flow, and pinpointed this particular problem as the central one suggested by the model [11]. Ford-Fulkerson’s reference 11 is a secret report by Harris and Ross [1955] entitled Fundamentals of a Method for Evaluating Rail Net Capacities, dated October 24, 19556 and written for the US Air Force. At our request, the Pentagon downgraded it to “unclassified” on May 21, 1999. As is known (Billera and Lucas [1976]), the motivation for the maximum flow problem came from the Soviet railway system. In fact, the Harris-Ross report solves a relatively large-scale maximum flow problem coming from the railway network in the Western Soviet Union and Eastern Europe (‘satellite countries’). Unlike what Ford and Fulkerson say, the interest of Harris and Ross was not to find a maximum flow, but rather a minimum cut (‘interdiction’) of the Soviet railway system. We quote: Air power is an effective means of interdicting an enemy’s rail system, and such usage is a logical and important mission for this Arm. As in many military operations, however, the success of interdiction depends largely on how complete, accurate, and timely is the commander’s information, particularly concerning the effect of his interdiction-program efforts on the enemy’s capability to move men and supplies. This information should be available at the time the results are being achieved. 5 There seems to be some discrepancy between the date of the RAND Report of Ford and Fulkerson (November 19, 1954) and the date mentioned in the quotation (spring of 1955). 6 In their book, Ford and Fulkerson incorrectly date the Harris-Ross report October 24, 1956.

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The present paper describes the fundamentals of a method intended to help the specialist who is engaged in estimating railway capabilities, so that he might more readily accomplish this purpose and thus assist the commander and his staff with greater efficiency than is possible at present. First, much attention is given in the report to modeling a railway network: taking each railway junction as a vertex would give a too refined network (for their purposes). Therefore, Harris and Ross propose to take ‘railway divisions’ (organizational units based on geographical areas) as vertices, and to estimate the capacity of the connections between any two adjacent railway divisions. In 1996, Ted Harris remembered (Alexander [1996]): We were studying rail transportation in consultation with a retired army general, Frank Ross, who had been chief of the Army’s Transportation Corps in Europe. We thought of modeling a rail system as a network. At first it didn’t make sense, because there’s no reason why the crossing point of two lines should be a special sort of node. But Ross realized that, in the region we were studying, the “divisions” (little administrative districts) should be the nodes. The link between two adjacent nodes represents the total transportation capacity between them. This made a reasonable and manageable model for our rail system. The Harris-Ross report stresses that specialists remain needed to make up the model (which is always a good tactics to get a new method accepted): The ability to estimate with relative accuracy the capacity of single railway lines is largely an art. Specialists in this field have no authoritative text (insofar as the authors are informed) to guide their efforts, and very few individuals have either the experience or talent for this type of work. The authors assume that this job will continue to be done by the specialist. The authors next dispute the naive belief that a railway network is just a set of disjoint through lines, and that cutting these lines would imply cutting the network: It is even more difficult and time-consuming to evaluate the capacity of a railway network comprising a multitude of rail lines which have widely varying characteristics. Practices among individuals engaged in this field vary considerably, but all consume a great deal of time. Most, if not all, specialists attack the problem by viewing the railway network as an aggregate of through lines. The authors contend that the foregoing practice does not portray the full flexibility of a large network. In particular it tends to gloss Documenta Mathematica

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Alexander Schrijver over the fact that even if every one of a set of independent through lines is made inoperative, there may exist alternative routings which can still move the traffic. This paper proposes a method that departs from present practices in that it views the network as an aggregate of railway operating divisions. All trackage capacities within the divisions are appraised, and these appraisals form the basis for estimating the capability of railway operating divisions to receive trains from and concurrently pass trains to each neighboring division in 24-hour periods.

Whereas experts are needed to set up the model, to solve it is routine (when having the ‘work sheets’): The foregoing appraisal (accomplished by the expert) is then used in the preparation of comparatively simple work sheets that will enable relatively inexperienced assistants to compute the results and thus help the expert to provide specific answers to the problems, based on many assumptions, which may be propounded to him. For solving the problem, the authors suggested applying the ‘flooding technique’, a heuristic described in a RAND Report of August 5, 1955 by A. W. Boldyreff [1955a]. It amounts to pushing as much flow as possible greedily through the network. If at some vertex a ‘bottleneck’ arises (that is, more trains arrive than can be pushed further through the network), the excess trains are returned to the origin. The technique does not guarantee optimality, but Boldyreff speculates: In dealing with the usual railway networks a single flooding, followed by removal of bottlenecks, should lead to a maximal flow. Presenting his method at an ORSA meeting in June 1955, Boldyreff [1955b] claimed simplicity: The mechanics of the solutions is formulated as a simple game which can be taught to a ten-year-old boy in a few minutes. The well-known flow-augmenting path algorithm of Ford and Fulkerson [1955], that does guarantee optimality, was published in a RAND Report dated only later that year (December 29, 1955). As for the simplex method (suggested for the maximum flow problem by Ford and Fulkerson [1954]) Harris and Ross remarked: The calculation would be cumbersome; and, even if it could be performed, sufficiently accurate data could not be obtained to justify such detail. Documenta Mathematica

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Figure 2: From Harris and Ross [1955]: Schematic diagram of the railway network of the Western Soviet Union and Eastern European countries, with a maximum flow of value 163,000 tons from Russia to Eastern Europe, and a cut of capacity 163,000 tons indicated as “The bottleneck”. The Harris-Ross report applied the flooding technique to a network model of the Soviet and Eastern European railways. For the data it refers to several secret reports of the Central Intelligence Agency (C.I.A.) on sections of the Soviet and Eastern European railway networks. After the aggregation of railway divisions to vertices, the network has 44 vertices and 105 (undirected) edges. The application of the flooding technique to the problem is displayed step by step in an appendix of the report, supported by several diagrams of the railway network. (Also work sheets are provided, to allow for future changes in capacities.) It yields a flow of value 163,000 tons from sources in the Soviet Union to destinations in Eastern European ‘satellite’ countries (Poland, Czechoslovakia, Austria, Eastern Germany), together with a cut with a capacity of, again, 163,000 tons. So the flow value and the cut capacity are equal, hence optimum. In the report, the minimum cut is indicated as ‘the bottleneck’ (Figure 2). While Tolsto˘ı and Harris-Ross had the same railway network as object, their objectives were dual.

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Acknowledgements. I thank Sasha Karzanov for his efficient help in finding Tolsto˘ı’s paper in the (former) Lenin Library in Moscow, Irina V. Karzanova for accurately providing me with an English translation of it, Kim H. Campbell and Joanne McLean at Air Force Pentagon for declassifying the Harris-Ross report, and Richard Bancroft and Gustave Shubert at RAND Corporation for their mediation in this. References [1996] K. S Alexander, A conversation with Ted Harris, Statistical Science 11 (1996) 150–158. [1976] L. J Billera, W. F Lucas, Delbert Ray Fulkerson August 14, 1924 – January 10, 1976, Mathematics of Operations Research 1 (1976) 299–310. [1955a] A. W Boldyreff, Determination of the Maximal Steady State Flow of Traffic through a Railroad Network, Research Memorandum RM-1532, The RAND Corporation, Santa Monica, California, [5 August] 1955 [published in Journal of the Operations Research Society of America 3 (1955) 443– 465]. [1955b] A. W Boldyreff, The gaming approach to the problem of flow through a traffic network [abstract of lecture presented at the Third Annual Meeting of the Society, New York, June 3–4, 1955], Journal of the Operations Research Society of America 3 (1955) 360. [1954] L. R Ford, D. R Fulkerson, Maximal Flow through a Network, Research Memorandum RM-1400, The RAND Corporation, Santa Monica, California, [19 November] 1954 [published in Canadian Journal of Mathematics 8 (1956) 399–404]. [1955] L. R Ford, Jr., D. R Fulkerson, A Simple Algorithm for Finding Maximal Network Flows and an Application to the Hitchco*ck Problem, Research Memorandum RM-1604, The RAND Corporation, Santa Monica, California, [29 December] 1955 [published in Canadian Journal of Mathematics 9 (1957) 210–218]. [1962] L. R Ford, Jr., D. R Fulkerson, Flows in Networks, Princeton University Press, Princeton, New Jersey, 1962. [1961] D. R Fulkerson, An out-of-kilter method for minimal-cost flow problems, Journal of the Society for Industrial and Applied Mathematics 9 (1961) 18–27. [1957] T. Gallai, Gr´ afokkal kapcsolatos maximum-minimum t´etelek (I. r´esz) [Hungarian; Maximum-minimum theorems for networks (part I)], A Magyar Tudom´ anyos Akad´emia Matematikai ´es Fizikai Tudom´ anyok Oszt´ aly´ anak K¨ ozlem´enyei 7 (1957) 305–338. Documenta Mathematica

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[1958] T. Gallai, Maximum-minimum S¨atze u ¨ber Graphen, Acta Mathematica Academiae Scientiarum Hungaricae 9 (1958) 395–434. [1955] T. E Harris, F. S Ross, Fundamentals of a Method for Evaluating Rail Net Capacities, Research Memorandum RM-1573, The RAND Corporation, Santa Monica, California, 1955. [1941] F. L Hitchco*ck, The distribution of a product from several sources to numerous localities, Journal of Mathematics and Physics 20 (1941) 224– 230. [1942] L. V Kantorovich, O peremeshchenii mass [Russian], Doklady Akademii Nauk SSSR 37:7-8 (1942) 227–230 [English translation: On the translocation of masses, Comptes Rendus (Doklady) de l’Acad´emie des Sciences de l’U.RS.S. 37 (1942) 199–201 [reprinted: Management Science 5 (1958) 1–4]]. [1949] L. V Kantorovich, M. K Gavurin, Primenenie matematicheskikh metodov v voprosakh analiza gruzopotokov [Russian; The application of mathematical methods to freight flow analysis], in: Problemy povysheniya effectivnosti raboty transporta [Russian; Collection of Problems of Raising the Efficiency of Transport Performance], Akademiia Nauk SSSR, MoscowLeningrad, 1949, pp. 110–138. [1967] M. Klein, A primal method for minimal cost flows with applications to the assignment and transportation problems, Management Science 14 (1967) 205–220. [1948] Tj. C. Koopmans, Optimum utilization of the transportation system, in: The Econometric Society Meeting (Washington, D.C., September 6– 18, 1947; D. H Leavens, ed.) [Proceedings of the International Statistical Conferences – Volume V], 1948, pp. 136–146 [reprinted in: Econometrica 17 (Supplement) (1949) 136–146] [reprinted in: Scientific Papers of Tjalling C. Koopmans, Springer, Berlin, 1970, pp. 184–193]. [1951] Tj.C. Koopmans, S. Reiter, A model of transportation, in: Activity Analysis of Production and Allocation – Proceedings of a Conference (Tj.C. Koopmans, ed.), Wiley, New York, 1951, pp. 222–259. [1959] A. L Lur’e, Methods of establishing the shortest running distances for freights on setting up transportation systems [in Russian], in: Primenenie matematiki v `ekonomicheskikh issledovaniyakh [Russian; Application of Mathematics in Economical Studies] (V. S Nemchinov, ed.), Izdatel’stvo ` Sotsial’no-Ekonomichesko˘ ı Literatury, Moscow, 1959, pp. 349–382 [English translation in: The Use of Mathematics in Economics (V. S Nemchinov, ed.), Oliver and Boyd, Edinburgh, 1964, pp. 323–355]. Documenta Mathematica

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[1949] J. Robinson, On the Hamiltonian Game (A Traveling Salesman Problem), Research Memorandum RM-303, The RAND Corporation, Santa Monica, California, 1949. [1950] J. Robinson, A Note on the Hitchco*ck–Koopmans Problem, Research Memorandum RM-407, The RAND Corporation, Santa Monica, California, 1950. [1930] A. N Tolsto˘ı, Metody nakhozhdeniya naimen’shego summovogo kilometrazha pri planirovanii perevozok v prostranstve [Russian; Methods of finding the minimal total kilometrage in cargo-transportation planning in space], in: Planirovanie Perevozok, Sbornik pervy˘ı [Russian; Transportation Planning, Volume I], Transpechat’ NKPS [TransPress of the National Commissariat of Transportation], Moscow, 1930, pp. 23–55. [1939] A. Tolsto˘ı, Metody ustraneniya neratsional’nykh perevozok pri planirovanii [Russian; Methods of removing irrational transportation in planning], Sotsialisticheski˘ı Transport 9 (1939) 28–51 [also published as ‘pamphlet’: Metody ustraneniya neratsional’nykh perevozok pri sostavlenii operativnykh planov [Russian; Methods of Removing Irrational Transportation in the Construction of Operational Plans], Transzheldorizdat, Moscow, 1941].

Alexander Schrijver CWI Science Park 123 1098 XG Amsterdam The Netherlands [emailprotected]

Original publication of this article: A. Schrijver, On the history of the transportation and maximum flow problems, Mathematical Programming, 91 (2002) 437-445. Documenta Mathematica

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Documenta Math.

Edmonds, Matching and the Birth of Polyhedral Combinatorics William R. Pulleyblank

2010 Mathematics Subject Classification: 05C70, 05C85, 90C10, 90C27, 68R10, 68W40 Keywords and Phrases: Matchings, factors, polyhedral combinatorics, nonbipartite matching, integer programming

1

Summer of 1961, a Workshop at RAND

In the summer of 1961, Jack Edmonds, a twenty-seven year old mathematician, was attending a high powered workshop on combinatorics at the Rand Corporation in Santa Monica, California. His participation had been arranged by Alan Goldman, his manager at the National Bureau of Standards (now NIST), supported by Edmonds’ Princeton mentor, A. W. Tucker. It seemed to Edmonds that every senior academician doing combinatorics was there. This included such luminaries as George Dantzig, Alan Hoffman, Ray Fulkerson, Claude Berge and Bill Tutte. The only “kids” participating were Michel Balinski, Larry Brown, Chris Witzgall, and Edmonds, who shared an office during the workshop. Edmonds was scheduled to give a talk on his research ideas. At that time, he was working on some big questions. He had become intrigued by the possibility of defining a class of algorithms which could be proven to run more efficiently than exhaustive enumeration, and by showing that such algorithms existed. This was a novel idea. At this time, people were generally satisfied with algorithms whose running times could be proved to be finite, such as Dantzig’s Simplex Algorithm for linear programming. In 1958, Ralph Gomory [14], [15] had developed an analogue of the Simplex Algorithm that he showed solved integer programs in finite time, similar to the Simplex Algorithm. Many people in the Operations Research community viewed a problem as “solved” if it could be formulated as an integer programming problem. However, unlike the Simplex Algorithm, Gomory’s integer programming algorithm seemed to take so long on some problems that it was often unusable in practice. At this time, the combinatorics community was not very interested in algorithms. Generally, graphs considered were finite and so most problems had Documenta Mathematica

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Jack Edmonds 1957 (courtesy Jeff Edmonds)

trivial finite solution methods. In 1963, Herb Ryser [23] published his monograph which noted that there were two general types of problems appearing in the combinatorics literature: existence problems (establish conditions characterising whether a desired structure exists) and enumeration problems (if a structure exists, determine how many of them there are). (A decade later, in 1972, Ryser, speaking at a conference on graph theory, added a third type of problem: develop an efficient algorithm to determine whether a desired object exists.) Earlier, in 1954, Dantzig, Fulkerson and Selmer Johnson [4] had published what proved to be a ground breaking paper. They showed that a traveling salesman problem, looking for a shortest tour visiting the District of Columbia plus a selected major city in each of the (then) 48 states, could be solved to provable optimality by combining the ideas of linear and integer programming. They did not make any claims as to the efficiency of their solution method. What they did show was that it was possible to present an optimal solution to an instance of a combinatorial optimization problem, and a proof of optimality, that required much less time to check than it would have taken to try all possible solutions. Through the 1950s, the world was seeing rapid development in the power and availability of digital computers. This provided another impetus to algorithmic development. Many combinatorial optimization problems were recognized as having practical applications. However even with the speed of the “high performance” computers of the day, it was recognized that improved algorithms were needed if problems of realistic size were to be solved in practice. Documenta Mathematica

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What Edmonds wanted was a specific concrete open example for which he could produce a better than finite algorithm and thereby illustrate the power and importance of his ideas. The perfect matching problem in a graph G = (V, E) is to determine whether there exists a set of edges meeting each node exactly once. If the graph is bipartite – its nodes can be partitioned into V1 ∪V2 and every edge joins a node in V1 to a node of V2 – then a rich theory had already been developed which not only characterized those bipartite graphs which had perfect matchings (Hall, [17]), but showed that this problem could be formulated as a small linear program. However, the more general case of nonbipartite graphs, graphs that contain odd cardinality cycles, seemed different. A necessary condition was that the number of nodes had to be even, but that was far from sufficient. Tutte [25] in 1947 had proved a generalization of Hall’s theorem to nonbipartite graphs. However, it did not seem to lead to an algorithm more efficient than simply trying all possible subsets of the edges in hope that one would be a perfect matching. A matching M in a graph G is a set of edges which meets each node at most once. M is perfect if it meets every node. Let U be the set of nodes not met by edges in M . An augmenting path with respect to M in G is a simple path joining two nodes of U whose edges are alternately not in M and in M . If an augmenting path exists, then a matching can be made larger – just remove the edges of the path that are in M and add to M the edges of the path not in M . In 1957 Claude Berge [1] showed that this characterized maximum matchings. Theorem 1 (Berge’s augmenting path theorem). A matching M in a graph G is of maximum size if and only if there exists no augmenting path. This result was not only simple to prove, but also applied both to bipartite and nonbipartite graphs. However, whereas there were efficient methods for finding such augmenting paths, if they existed, in bipartite graphs, no such algorithms were known for nonbipartite graphs. The night before his scheduled talk, Edmonds had an inspiration with profound consequences. A graph is nonbipartite if and only if it has an odd cycle. It seemed that it was the presence of these odd cycles that confounded the search for augmenting paths. But if an odd cycle was found in the course of searching for an augmenting path in a nonbipartite graph, the cycle could be shrunk to form a pseudonode. Thereby the problem caused by that odd cycle could be eliminated, at least temporarily. This simple and elegant idea was the key to developing an efficient algorithm for determining whether a nonbipartite graph had a perfect matching. Equally important, it gave Edmonds a concrete specific example of a problem that could illustrate the richness and the power of the general foundations of complexity that he was developing. This became the focal point of his talk the next day which launched some of the most significant research into algorithms and complexity over the next two decades. Alan Hoffman recounted an exchange during the discussion period following Edmonds’ lecture. Tutte’s published proof of his characterization of nonbiparDocumenta Mathematica

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tite graphs having perfect matchings was an ingenious application of matrix theory. Responding to a question, Edmonds ended a sentence by saying “using methods known only to Tutte and God”. Tutte rarely made comments at the end of another person’s lecture. There was a pause, at which point it was appropriate for Tutte to say something, but he said nothing. Hoffman intervened, asking “Would either of those authors care to comment?” Tutte did respond. 2

Context I: Bipartite graphs and the Hungarian method

The problem of determining whether a bipartite graph had a perfect matching had already been encountered in many different guises, and there were several equivalent characterizations of bipartite graphs having perfect matchings. See Schriver [24]. A node cover is a set C of nodes such that each edge is incident with at least one member of C. Each edge in any matching M will have to be incident with at least one member of C, and no member of C can be incident with more than one member of M . Therefore, the size of a largest matching provides a lower bound on the size of a smallest node cover. In 1931, D´enes K˝onig [18] had published a min-max theorem showing that these values are equal. Theorem 2 (K˝ onig’s Bipartite Matching Theorem). The maximum size of a matching in a bipartite graph G = (V, E) equals the minimum size of a node cover. In 1935, in the context of transversals of families of sets, Phillip Hall [17] proved the following: Theorem 3 (Hall’s Bipartite matching Theorem). A bipartite graph G = (V, E) has a perfect matching if and only if, for every X ⊆ V , the number of isolated nodes in G − X is at most |X|. These two theorems are equivalent, in that each can be easily deduced from the other. (Deducing Hall’s Theorem from K˝onig’s Theorem is easier than going the other direction.) If a bipartite graph G has no perfect matching, then either of these provides a guaranteed simple way of showing that this is the case. We can exhibit a node cover of size less than |V |/2 or exhibit a set X ⊆ V such that G − X has at least |X| + 1 isolated nodes. (For now, do not worry about the time that it takes to find the cover or the set X.) Note how these contrast with Berge’s augmenting path theorem. Berge’s theorem does suggest an approach for constructing a perfect matching if one exists, but if we wanted to use it to show that G had no perfect matching, we would have to start with a less-than-perfect matching in G and somehow prove that no augmenting path existed. How could this be done? In 1931, Jen˝ o Egerv´ ary [12] published an alternate proof and a weighted generalization of K˝ onig’s theorem. (See [24].) Suppose that we have a bipartite Documenta Mathematica

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graph G = (V, E) and a real edge weight cj for each j ∈ E. The weight of a matching is the sum of the weights of its edges. He proved a min-max theorem characterizing the maximum possible weight of a matching in G by showing that it was equal to the minimum weight of a weighted node cover of the edges of G. Theorem 4 (Egerv´ ary’s Theorem). Let G = (V, E) be a bipartite graph and let (cj : j ∈ E) be a vector ofP edge weights. The maximum weight of a matching in G equals the minimum of v∈V yv , where y = (yv : v ∈ V ) satisfies yu +yv ≥ cj for every j = {u, v} ∈ E. This implied that the existence of a perfect matching in a bipartite graph G = (V, E) could be determined by solving a linear system. For each edge j ∈ E, define a variable xj . Then x = (xj : j ∈ E) is a real vector indexed by the edges of G. Consider the following system of linear equations and (trivial) inequalities: X (xj : j ∈ E incident with v) = 1 for each node v ∈ V, (1) xj ≥ 0 for each j ∈ E.

(2)

If G has a perfect matching M , we can define x ˆj = 1 for j ∈ M and x ˆj = 0 for j ∈ E\M . Then x ˆ is a feasible solution to this linear system. Conversely, if we have an integer solution to this linear system, all variables will have value 0 or 1 and the edges with value 1 will correspond to the edges belonging to a perfect matching of G. Theorem 5. A bipartite graph G = (V, E) has a perfect matching if and only if the linear system (1), (2) has an integer valued solution. However, in general there also exist fractional solutions to this system. Could there exist fractional solutions to this linear system but no integer valued solutions? In this case, the solution to the linear system might not tell us whether the graph had a perfect matching. Egerv´ary’s Theorem showed that this was not the case. Egerv´ ary’s Theorem is not true in general for nonbipartite graphs. It already fails for K3 . In this case, the linear system has a solution obtained by setting xj = 1/2 for all three edges, but there is no integer valued solution. (The conditions of Hall’s and K˝ onig’s Theorems also fail to be satisfied for K3 .) Egerv´ ary’s Theorem showed that the maximum weight matching problem for bipartite graphs could be solved by solving the linear program of maximizing P (x · c : j ∈ E) subject to (1), (2). The dual linear program is to minimize j j P y , where y = (yv : v ∈ V ) satisfies yu +yv ≥ cj for every j = {u, v} ∈ E. v v∈V His proof showed how to find an integer x and (possibly) fractional y which were optimal primal and dual solutions. In 1955, Harold Kuhn [19] turned Egerv´ary’s proof of his theorem into an algorithm which would find a maximum weight matching in a bipartite graph. Documenta Mathematica

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The algorithm was guaranteed to stop in finite time. In 1957, James Munkres [20] showed that this algorithm, called “The Hungarian Method”, would terminate in time O(n4 ) for a simple bipartite graph with n vertices. 3

Context II: Tutte’s theorem and the Tutte–Berge formula

In 1947, William Tutte [25] had generalized Hall’s theorem to nonbipartite graphs. He proved that replacing “isolated nodes” by “odd cardinality components” yielded a characterization of which nonbipartite graphs have perfect matchings. Theorem 6 (Tutte’s matching Theorem). A (nonbipartite or bipartite) graph G = (V, E) has a perfect matching if and only if, for every X ⊆ V , the number of odd cardinality components of G − X is at most |X|. As in the case of Hall’s Theorem, the necessity of the condition is straightforward. If there exists a perfect matching M , then an edge of M must join some node of each odd component of G − X to a node of X, since it is impossible to pair off all the nodes of an odd component K using only edges with both ends in K. The important part of the theorem is the sufficiency, which asserts that if G does not have a perfect matching, then there exists an X whose removal creates more than |X| odd cardinality components. Hall’s Theorem does strengthen Tutte’s theorem in the bipartite case as follows. It shows that, in this case, we can restrict our attention to components of G − X which consist of single nodes, rather than having to consider all possible components. But Tutte’s theorem works for all graphs. For example, whereas Hall’s condition is not violated for K3 , Tutte’s Theorem shows that no perfect matching exists, by taking X = ∅. In 1958, Berge [2] noted that Tutte’s theorem implied a min-max theorem for ν(G), the size of a largest matching in a graph G = (V, E). For any X ⊆ V , we let odd(X) be the number of odd cardinality components of G − X. Theorem 7 (Tutte–Berge Formula). For any graph G = V, E, ν(G) =

1 (|V | − min(odd(X) − |X| : X ⊆ V )). 2

The formula shows that the smallest number of nodes which must be left unmet by any matching equals the largest possible difference between odd(X) and |X|. Here then were the challenges: Could the notion of “efficient” be made precise mathematically? Was it possible to develop an efficient algorithm for determining whether an arbitrary graph had a perfect matching? Given an arbitrary graph G = (V, E), could you either find a perfect matching or find a set X ⊆ V for which |X| < odd(X)? Documenta Mathematica

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Paths, Trees and Flowers; P and N P

Edmonds’ landmark paper [5], Paths, Trees and Flowers, evolved from the talk that he presented at Rand in 1961. His algorithm for determining whether a nonbipartite graph G = (V, E) has a perfect matching can be summarized as follows. Start with any matching M . If M is perfect, then the algorithm is done. If not, some node r is not met by any edge of M . In this case, grow an alternating search tree T rooted at r which will either find an augmenting path, enabling the matching to be made larger, or find a set X ⊆ V for which |X| < odd(X). The search tree initially consists of just the root node r. Each node v of T is classified as even or odd based on the parity of the length of the (unique) path in T from r to v. The algorithm looks for an edge j of G that joins an even node u of T to a node w which is not already an odd node of T . If such a j exists, there are three possibilities. 1. Grow Tree: If w is met by an edge k of M , then T is grown by adding j, k and their end nodes to T . 2. Augment M : If w is not met by an edge of M , then we have found an augmenting path from r to w. We augment M using this path, as proposed by Berge, and select a new r if the matching is not perfect. 3. Shrink: If w is an even node of T , then adding j to T creates a unique odd cycle C. Shrink C by combining its nodes to form a pseudonode. The pseudonode C will be an even node of the tree created by identifying the nodes of G belonging to C. If no such j exists, then let X be the set of odd nodes of T . Each even node w of T will correspond to an odd cardinality component of G − X. If w is a node of G, then the component consists of the singleton w. If w was formed by shrinking, then the set of all nodes of G shrunk to form w will induce an odd component of G. If G is bipartite, then the Shrink step will not occur and the algorithm reduces to a previously known matching algorithm for bipartite graphs. One point we skipped over is what happens to an augmenting path when it passes through a pseudo-node. It can be shown that by choosing an appropriate path through the odd cycle, an augmenting path in a graph obtained by shrinking can be extended to an augmenting path in the original graph. See Edmonds [5] or Cook et al [3] for details. Edmonds [5] presents his algorithm for the closely related problem of finding a maximum cardinality matching in an arbitrary graph. If the above algorithm terminates without finding a perfect matching, then he calls the search tree T Hungarian. He lets G′ be the graph obtained from G by deleting all vertices in T or contained in pseudonodes of T . He shows that a maximum matching of G′ , combined with a maximum matching of the subgraph of G induced by Documenta Mathematica

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the nodes belonging to T or contained in pseudonodes of T , forms a maximum matching of G. The second section of Edmonds [5] is entitled “Digression”. This section began by arguing that finiteness for an algorithm was not enough. He defined a good algorithm as one whose worse case runtime is bounded by a polynomial function of the size of the input. This criteria is robust, it is independent of the actual computing platform on which the algorithm was run. Also, it has the attractive feature that good algorithms can use other good algorithms as subroutines and still be good. He stressed that this idea could be made mathematically rigorous. The maximum matching algorithm, which Edmonds (conservatively) showed had run time O(|V |4 ), provided an initial case study. This was the first known algorithm for maximum matching in nonbipartite graphs with a running time asymptotically better than trying all possible subsets. The bound on the running time was about the same as the bound on solving the matching problem for a bipartite graph. One concern raised about Edmonds’ notion of a good algorithm was that a good algorithm with a high degree polynomial bound on its run times could still take too long to be practical. Edmonds stressed that his goal was to develop a mathematically precise measure of running times for algorithms that would capture the idea of “better than finite”. A second concern arose from the simplex algorithm for linear programming. This algorithm was proving itself to be very effective for solving large (at the time) linear programs, but no polynomial bound could be proved on its running time. (It would be almost two decades later that a good algorithm would be developed for linear programming.) So the concept of “good algorithm” was neither necessary nor sufficient to characterize “efficient in practice”. But there was a high degree of correlation, and this concept had the desired precision and concreteness to form a foundation for a study of worst case performance of algorithms. Part of the reason for the lasting significance of [5] is that the paper promoted an elegant idea – the concept of a good (polynomially bounded) algorithm. It also gave the first known such algorithm for the matching problem in nonbipartite graphs, a fundamental problem in graph theory. Edmonds also raised the question of whether the existence of theorems like Tutte’s Theorem or Hall’s Theorem – min-max theorems or theorems characterizing the existence of an object (a perfect matching in a bipartite graph) by prohibiting the existence of an obstacle ( a set X ⊂ V for which G − X has at least |X| + 1 isolated nodes) – could enable the construction of efficient algorithms for finding the objects if they existed. He had shown how this worked in the case of matchings in bipartite graphs and his algorithm had extended this to nonbipartite graphs. He called these sorts of theorems good characterizations. Some people argued that nobody could possibly check all subsets X and see how many isolated nodes existed in G − X. There were simply too many of them; the number grew exponentially with the size of G. What did this have to do with answering the original question? Documenta Mathematica

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But here was the point. Consider the question: does G have a perfect matching? If the answer is “Yes”, we can prove this by exhibiting a perfect matching M . If the answer is “No”, then we can prove this by exhibiting a single X ⊆ V for which G − X has at least |X| + 1 isolated nodes. This has not yet described an effective method for finding M or X, but at least it provided a polynomially bounded proof for either alternatives. It gave a stopping criterion for an algorithm. A decade later, these concepts were essential ideas embodied in the classes P and N P. The question Edmonds asked relating the existence of good characterizations to the existence of good algorithms became what is now recognized as the most important open question in theoretical computer science: Is P = N P? 5

Weighty matters

Edmonds quickly generalized his nonbipartite matching algorithm to the corresponding edge weighted problem (Edmonds [6]). (Recall, each P edge j is given a cost cj and the algorithm constructs a matching M for which (cj : j ∈ M ) is maximum.) He did this by an elegant extension of Egerv´ary’s approach that had worked for bipartite graphs. He showed how to use the primal-dual method for linear programming and the operation of shrinking to extend the cardinality case to the weighted case. Edmonds began by formulating the maximum weight matching problem as a linear programming problem: X Maximize (cj xj : j ∈ E) subject to

X

X

j∈E

(xj : j ∈ E incident with v) ≤ 1 for each node v ∈ V,

(3)

(xj : j has both ends in S) ≤ (|S| − 1)/2 for each S ⊆ V such that |S| ≥ 3 is odd,

(4)

xj ≥ 0 for each j ∈ E.

(5)

This was really an audacious idea. The number of inequalities (4) grows exponentially with the number of nodes of G. No available linear programming code could read and store the set of constraints for a moderate sized weighted matching problem, let alone solve the problem. However Edmonds’ idea was this: the real value of linear programming for a problem like weighted matching is not the simplex algorithm. It is that linear duality theory provides a method of giving a short proof of optimality. His algorithm constructed a vector x = (xj : j ∈ E) which was the (0-1)incidence vector of a matching in G. It also constructed a feasible solution to the dual linear program to maximizing c · x subject to (3), (4) and (5). Moreover, x and the dual solution would satisfy the complementary slackness conditions of linear programming which established their optimality. Documenta Mathematica

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The algorithm had essentially the same bound on its run time as the maximum cardinality algorithm. There was a minor complication. The bound had to take into account the complexity of arithmetic operations on the costs cj . These operations were addition, subtraction, comparison andPdivision by 2. This required either the introduction in the bound of a factor j∈E log(cj ) or else a “fixed word” assumption that all costs were within some bounded range. 6

Generality and extensions

Soon after this, Ellis L. Johnson, a recent Berkeley PhD student of Dantzig, began to work with Edmonds. They wanted to see how much they could generalize this theory of matchings in general graphs, in the context of linear and integer programming. They extended the algorithm to accommodate the following extensions (see [8]): 6.1

General degree constraints

Generalize the constraints (3) to X (xj : j ∈ E incident with v) ≤ bv for each node v ∈ V,

(6)

where, for each v ∈ V , bv is a nonnegative integer. This extends the graph theoretic idea of a matching to a vector x = (xj : j ∈ E) of nonnegative integers such that, for each v ∈ V , the sum of the xj on the edges j is at most bv . Such a vector x is called a b-matching. If bv = 1 for all v ∈P V , then a b-matching is the incidence vector of a matching. Let b(V ) denote v∈V bv . Tutte [26] had already shown that this problem could be transformed into a matching problem in which bv = 1 for all v ∈ V by replacing each vertex for which bv > 1 by |bv | new vertices, and each edge j = {u, v} with a complete bipartite graph joining the sets of new vertices corresponding to u and P v. For a b matching x, the deficiency d(x, v) of x at vertex v is defined as b − (xj : v P j ∈ E, j incident with v). The deficiency D(x) of x is defined as v∈V d(x, v). The Tutte–Berge Formula generalizes to b-matchings as follows: For each X ⊆ V , let K 0 (X) be the nodes belonging to one node components of G − X; let odd(X) be the P number of components K of G − X having at least three nodes for which i∈V (K) bi is odd. Theorem 8 (Tutte–Berge Formula for b-matchings). For any graph G = V, E and any vector b = (bv : v ∈ V ) of nonnegative integers, min (D(x) : x is a b − matching of G) X = max(

v∈K 0 (X)

bv + odd(X) −

X

bv : X ⊆ V ).

v∈X

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primal/dual framework to reduce the weighted problem to a cardinality problem. It startedPwith an arbitrary b-matching x ¯ and defined a node v to be unsaturated if (¯ xj : j ∈ E incident with v) < bv . Now an augmenting path became a path in G joining two unsaturated nodes such that for each even edge j in the path, x ¯j > 0. This would enable an augmentation to be made by increasing x ¯j for the odd edges in the path and decreasing x ¯j for the even edges. Similar to before, the algorithm grew an alternating search tree T rooted at an unsaturated node r. If it found an unsaturated even node of T other than r, it augmented the b-matching. If an edge j was found joining two even nodes of T , then it had found an odd cycle which it shrunk. But in this case any nodes of the tree joined to the odd cycle by paths in the tree for which every edge j had x ¯j > 0 were also shrunk with the odd cycle. Set bv = 1 for the resulting pseudonode v. Let x ¯ be the initial b-matching. This algorithm had worst case running time of O(D(¯ x) · |V |2 ). The bound came from the fact that each augmentation reduced the sum of the deficiencies by at least 2, and the time taken to find an augmentation, if one existed, was O(|V |2 ). If we started with x ¯ = 0, then the bound was O(b(V ) · |V |2 ). This created a potential problem. The length of a binary encoding of the inP put was polynomial in |V | and log bv . However, b(V ) grows exponentially v∈V P with v∈V log bv and so the bound on the run time was growing exponentially with the size of a “natural” encoding of the input. How could it be made into a good algorithm? Creating a good algorithm for finding a maximum (or minimum) weight perfect b-matching required three ideas. First, for each v ∈ V , let ˆbv be the largest even integer no greater than bv . The resulting ˆb-matching problem can be transformed into a network flow problem in a bipartite directed graph G′ having 2|V | nodes. For each node v ∈ V , create two nodes v ′ and v ′′ in G′ and for each edge {u, v} in G, create two directed arcs (u′ , v ′′ ) and (v ′ , u′′ ) in G′ . Let b′v = bv /2 and let b′′v = −bv /2. Edmonds and Richard Karp [11] created a good algorithm for finding a maximum flow in G′ having maximum cost. By adding together the flows in the arcs (u′ , v ′′ ) and (v ′ , u′′ ) for each edge {u, v} of G, we get a ˆb-matching x ¯ of G having minimum deficiency with respect to ˆb. Second, use x ¯ as a starting matching to find a maximum weight b-matching in G. The third idea was to show that the deficiency of x ¯ cannot be too large. let R be the set of nodes v for which bv is odd. By the Tutte-Berge formula for b-matchings, if the deficiency of x ¯ is greater than |R|, then G does not have a perfect b-matching. Otherwise, the weighted b-matching algorithm performs at most |R| augmentations, so the bound on the running time becomes O(|R|·|V |2 ) and we have a good algorithm. See Gerards [13]. Documenta Mathematica

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For each edge j ∈ E, let uj be an integral upper bound and let lj be an integral lower bound on the value of xj for the edge j. That is, the inequalities (5) are replaced with lj ≤ xj ≤ uj for each j ∈ E. (7) The constraints (3) and (5) of the original weighted matching problem forced every edge j to have a value 0 or 1. However we now permit xj to be any integer in the range [lj , uj ]. If we add this to the b-matching problem, we obtain the capacitated b-matching problem. In the special case that lj = 0 and uj = 1 for all j ∈ E, we obtain a factor problem. Now we want to find a maximum weight subset of the edges that meet each vertex v at most bv times. We have now gone to a significantly more general set of linear constraints on our problem. The case bv = 2 for all v ∈ V and cj = 1 for all j ∈ E is particularly interesting. This is the maximum 2-factor problem – find a set of vertex disjoint cycles in a graph that contain the maximum possible number of vertices. 6.3

Bidirected graphs

Edmonds and Johnson recognized that they could develop a unified model that included matching in general undirected graphs as well as network flow problems in directed graphs by introducing the idea of bidirected graphs. Each edge of the graph will have one or two ends. Each end will be either a head or a tail. Some edges will have a head and a tail. These are called directed edges. Some will have two heads or two tails. These are called links. An edge with one end is called a slack and that end can be either a head or a tail. The constraints (6) are now changed to the following: X (xj : j ∈ E, j has a head incident with v) X − (xj : j ∈ E, j has a tail incident with v) = bv for every node v ∈ V.

If all edges are links with both ends heads, then this becomes the capacitated b-matching problem. If all edges are directed, then this becomes a network flow problem. However, allowing a mixture of links, slacks and arcs provides a mixture of the two models, plus more. Note that by allowing slacks, all degree constraints can be turned into equations. Combining these extensions, Edmonds and Johnson had developed a good algorithm for integer programming problems, maximize cx subject to Ax = b l≤x≤u Documenta Mathematica

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where b, l, and u are integral, A is a matrix all of whose entries are 0, 1, −1, 2, −2 and, for each column of A, the sum of the absolute values of the entries is at most 2. 6.4

Parity constraints

Edmonds and Johnson [9] also extended the idea of capacitated b-matching to allow so called parity constraints at the nodes. For each v ∈ V, bv = 0 or 1. The constraints (6) became: X

(xj : j ∈ E incident with v) ≡ bv mod 2 for each node v ∈ V.

This enabled the so-called Chinese Postman Problem or T -join problem to be formulated as a capacitated b-matching problem. They provided both a direct algorithm and a reduction to this problem. See also Gr¨otschel and Yuan [16]. At this time, Edmonds, Johnson and Scott Lockhart [10] developed a FORTRAN computer code for the weighted capacitated b-matching problem in bidirected graphs. This showed convincingly that this algorithm was a practical way to solve very large matching problems. It also provided a concrete instantiation of the algorithm which enabled precise calculation of an upper bound on its running time as a function of the input size. Part of the motivation for doing this appeared in Section 2 of [5]. The described FORTRAN machine was an alternative to a Turing machine, a widely adopted model of computation for theoretical computing science. The FORTRAN machine was very close to the machine architectures of the day, and there existed a good algorithm for a FORTRAN machine if and only if there existed a good algorithm for a Turing machine. Also, the upper bound of the run time on a FORTRAN machine was much lower than for a Turing machine. Edmonds and Johnson [8] also described reductions that enabled these extensions to be transformed to weighted matching problems in larger graphs. 7

Combinatorial polyhedra

In the early 1960s, it was recognized that a great many combinatorial optimization problems could be formulated as integer linear programs. It was also known that an integer linear program could be transformed into a linear program by adding a sufficient set of additional inequalities, called cuts, that trimmed the polyhedron of feasible solutions so that all vertices were integer valued, without removing any feasible integer solutions. Gomory’s algorithm for integer programming gave a finite procedure for solving any integer program by successively adding cuts and re-solving until an optimum solution was found which was integer valued. His algorithm seemed to be a simple extension of the simplex algorithm for linear programming. However it had already been observed empirically that whereas the simplex algorithm was very successful for linear programs, Gomory’s algorithm often failed to obtain a solution to an Documenta Mathematica

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integer program in an acceptable amount of time. The only bound on the number of cuts that might be generated was exponential. This supported Edmonds’ view that “finite was not good enough”. There were classes of integer programs for which no cuts needed to be added, for example, network flow problems and maximum weighted matching in bipartite graphs. Most of these classes of problems had total unimodularity at the core. A matrix A = (aij : i ∈ I, j ∈ J) is totally unimodular if for any square submatrix M of A, det(M ) = 0, 1, or − 1. Note that this implies that all entries of A have value 0, 1, or − 1. Suppose that A is totally unimodular and b is integral valued. It follows directly from Cramer’s rule that, for any c, if the linear program maximize cx subject to Ax = b, x ≥ 0 has an optimum solution, then it has one that is integer valued. It was well known that if G was a bipartite graph, then the matrix A defined by (1) is totally unimodular, so a maximum matching in a bipartite graph could be obtained by solving the linear program of maximizing cx subject to (3) and (2). If A was the node-arc incidence matrix of a directed graph, then the maximum flow problem could be formulated as a linear program with a totally unimodular matrix implying that if the node demands and arc capacities were integral, then there existed an integral optimal flow. See Cook et al [3]. It was well known that the weighted matching problem could P be formulated as the integer linear programming problem of maximizing (cj xj : j ∈ E) subject to (3) and xj ≥ 0, integer for all j ∈ E. Edmonds had shown that the weighted matching algorithm correctly solved the problem by showing that it gave an integerP valued optimum solution to the linear programming problem of maximizing (cj xj : j ∈ E) subject to (3), (4) and (5). That is, he had shown that the integrality constraint could be replaced by adding the cuts (4). This was the first known example of a general combinatorial problem which could be formulated as a linear programming problem by adding an explicitly given set of cuts to a natural integer programming formulation. Dantzig et al [4] had shown that a particular instance of a traveling salesman problem could be solved starting from an integer programming formulation by adding a small set of cuts. What Edmonds had shown was that for any maximum weight matching problem, by adding the cuts (4), the integer program could be transformed to a linear program. He and Johnson had also shown for all the extensions in the previous section that the same paradigm worked. They gave explicit sets of cuts that, when added, transformed the problem to a linear programming problem. This motivated further research on other problems amenable to this approach. It worked in many cases (for example, matroid optimization, matroid intersection, optimum branchings, triangle-free 2-matchings) but there are still many natural problems for which no explicit set of cuts is known. The matching polyhedron M (G) is the convex hull of the incidence vectors of the matchings of a graph G = (V, E). Edmonds showed that M (G) = {x ∈ ℜE : x satisfies (3), (4) and (5)}. This problem of finding a linear system sufficient to define a polyhedron defined by a combinatorial optimization problem – or Documenta Mathematica

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equivalently, formulating the problem as a linear program – became a very active area of research through the 1970s, building on the successes obtained with matching problems. The fundamental role of shrinking in solving nonbipartite matching problems had another interesting consequence. In general, not all constraints (4) are necessary to obtain a linear system sufficient to define M (G). For example, if |S| is odd, but G[S], the subgraph of G induced by S, is not connected, then the constraint (4) corresponding to S is unnecessary. It is implied by these constraints for the nodesets of the odd cardinality connected components of G[S]. Edmonds and Pulleyblank [22] showed that the essential constraints (4) for M (G) correspond to those sets S ⊆ V for which G[S] is 2-connected and is shrinkable. Shrinkable means that G[S] will be reduced to a single pseudonode if the maximum matching algorithm is applied to it. Equivalently, a graph G[S] is shrinkable if and only if G[S] has no perfect matching, but for every node v ∈ S, the graph obtained from G[S] by deleting v and all incident edges does have a perfect matching. The generalizations to b-matching appeared in Pulleyblank’s PhD thesis [21], prepared under the supervision of Edmonds. The problem of determining the essential inequalities to convert an integer program to a linear program is called facet determination. This became an active research area over the 1970s and 1980s – determining the facets of combinatorially defined polyhedra. Acknowledgements. I am grateful to Kathie Cameron, Bill Cunningham, Alan Hoffman and, especially, Jack Edmonds for assistance with the primary source research for this chapter. References [1] C. Berge, Two theorems in graph theory, Proc. Nat. Academy of Sciences (U.S.A.) 43 (1957) 842–844. [2] C. Berge, Sur le couplage maximum d’un graphe, Comptes Rendu de l’Acad´emie des Sciences Paris, series 1, Math´ematique 247 (1958), 258– 259. [3] W.J. Cook, W.H. Cunningham, W.R. Pulleyblank and A. Schrijver, Combinatorial Optimization, Wiley-Interscience (1998). [4] G. Dantzig, D.R, Fulkerson and S. Johnson, Solution of a large scale traveling salesman problem, Operations Reserach 2 (1954) 393–410. [5] J. Edmonds, Paths, trees and flowers, Canadian J. of Math. 17 (1965) 449–467. [6] J. Edmonds, Maximum matching and a polyhedron with 0,1 vertices, J. Res. Nat’l. Bureau of Standards 69B (1965) 125–130. Documenta Mathematica

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[7] J. Edmonds, A glimpse of heaven, in History of Mathematical Programming: A collection of Personal Reminiscences (J.K. Lenstra, A.H.G. Rinnoy Kan and A. Schrijver eds.), North-Holland (1991), pp. 32–54. [8] J. Edmonds and E.L. Johnson, Matchings: a well solved class of integer linear programs, in Combinatorial Structures and their Applications (R.K. Guy, H. Hanani, N. Sauer and J. Sch¨onheim eds.), Gordon and Breach, New York (1970), pp. 89–92. [9] J. Edmonds and E.L. Johnson, Matchings, Euler tours and the Chinese Postman, Mathematical Programming 5 (1973) 88–124. [10] J. Edmonds, E.L. Johnson and S.C. Lockhart, Blossom I, a code for matching, unpublished report, IBM T.J. Watson Research Center, Yorktown Heights, New York (1969) [11] J. Edmonds and R.M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems, J. of the ACM 19 (1972) 248–264. [12] J. Egerv´ ary, Matrixok kombinatorius tulajdons´agair´ol, (in Humgarian)(On combinatorial properties of matrices), Matematikai ´es Fizikai Lapok 38 (1931) 16–28. [13] A.M.H. Gerards, Matching, Chapter 3 in M.O. Ball et al eds., Handbooks in OR and MS Vol. 7 (1995) pp. 135–224. [14] R.E. Gomory, Outline of an algorithm for integer solutions to linear programs, Bulletin of the American Mathematical Society 64 (1958), 275–278. [15] R.E. Gomory, Solving linear programming problems in integers, in Combinatorial Analysis (R. Bellman and M. Hall Jr. eds.), American Mathematical Society (1960), pp. 211–215. [16] M. Gr¨ otschel and Ya-Xiang Yuan, Euler, Mei-Ko Kwan, K¨onigsberg, and a Chinese Postman, this volume, Chapter 7 (2012). [17] P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), 26–30. [18] D. K˝ onig, Graphok ´es matrixok, Matematikai ´es Fizikai Lapok 38 (1931) 116–119. [19] H.W. Kuhn, The Hungarian method for the assignment problem, Naval Research Logistics Quarterly 2 (1955) 83–97. [20] J. Munkres, Algorithms for the assignment and transportation problems, J. of Soc. for Industrial and Applied Mathematics 5 (1957) 32–38. [21] W.R. Pulleyblank, Faces of Matching Polyhedra, PhD Thesis, University of Waterloo (1973). Documenta Mathematica

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[22] W.R. Pulleyblank and J. Edmonds, Facets of 1-matching polyhedra, in Hypergraph Seminar (C. Berge and D. Ray-Chaudhuri, eds.) Springer, Berlin (1974) pp. 214–242. [23] H.J. Ryser, Combinatorial Mathematics, Math. Assoc. of America, John Wiley and Sons, Inc. (1963). [24] A. Schrijver, Combinatorial Optimization, Springer Verlag (2003). [25] W.T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947) 107–111. [26] W.T. Tutte, A short proof of the factor theorem for finite graphs, Canadian J. of Math. 6 (1954) 347–352.

William R. Pulleyblank Department of Mathematical Sciences United States Military Academy, West Point West Point, NY 10996, USA [emailprotected]

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Flinders Petrie, the Travelling Salesman Problem, and the Beginning of Mathematical Modeling in Archaeology ¨ tschel Thomas L. Gertzen and Martin Gro

Abstract. This article describes one of the first attempts to use mathematical modeling and optimization in archaeology. William Matthew Flinders Petrie (1853–1942), eminent British archaeologist, excavating a large graveyard at Naqada in Upper Egypt suggested in his article “Sequences in Prehistoric Remains” [17] to employ a “distance function” to describe the “closeness of graves in time”. Petrie’s grave distance is known today as Hamming metric, based on which he proposed to establish the chronology of the graves, i.e., the correct sequence of points in time when the graves were built (briefly called seriation). He achieved this by solving a graph theoretic problem which is called weighted Hamiltonian path problem today and is, of course, equivalent to the symmetric travelling salesman problem. This paper briefly sketches a few aspects of Petrie’s biographical background and evaluates the significance of seriation. 2010 Mathematics Subject Classification: 01A55, 05-03, 90-03, 90C27 Keywords and Phrases: Travelling salesman problem, seriation, Hamming metric, archaeology

Introduction When the second author of this article wrote his PhD thesis on the travelling salesman problem (TSP) more than thirty-five years ago, he came across two articles by D. G. Kendall [12] and A. M. Wilkinson [23], respectively investigating the TSP in connection with archaeological seriation. Since he was interested in solving large-scale TSP instances (and in archaeology), he tried to find the original data of the Naqada-graves, based upon which W. M. Flinders Petrie established the prehistoric chronology of Egypt. His search was unsuccessful. In 2011, planning this Optimization Stories book, the second author approached the director of the German Archaeological Institute in Cairo, S. Seidlmayer. He suggested contacting the first author, who had recently finished Documenta Mathematica

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his Master thesis, entitled: “Methodik und wissenschaftsgeschichtlicher Hintergrund der Arbeiten von Sir W. M. Flinders Petrie”. Asking for the original Petrie papers on Naqada, the second author learned from the first that these materials, according to E. Baumgartel, referring to a conversation with M. Murray, were no longer existent: She answered that when they had to give up the Egyptian Department, one room [. . . ] was filled from top to bottom with Petrie’s papers. She had worked through them with some students who showed her the papers. She said ‘published, destroy, unpublished keep.’ Well, Naqada was published. (See [2, p. 6].) In order to be absolutely sure, the first author contacted the curator of the Petrie Museum London, S. Quirke, who informed him that certain Petrie materials had been rediscovered within the archives of the museum recently, among others, the original “Naqada-slips”, to be explained below. The Petrie Museum staff kindly provided digitized images of the material in spring 2012. Originally, the two authors planned to jointly reprocess Petrie’s data, in order to determine optimum solutions for his seriation problems and to publish their results in this article. However, it turned out that Petrie’s materials only represent a rough sketch and show certain inconsistencies, which require careful additional archaeological investigation and also a certain amount of science historical interpretation. This time consuming work is currently carried out and is going to be published in the near future. Instead, this paper briefly outlines Petrie’s modeling concept and the method he applied to solve the mathematical problem he formulated. This very much resembles the engineering approach to combinatorial optimization still prevailing in industry today: Employ experience/knowledge based heuristics skillfully. The beginning of mathematical modeling in archaeology Archeology originally was a field dominated by art historians and linguists. The use of methods from the natural sciences and mathematics began slowly. One of the pioneers of this approach to archaeology was Petrie, one of the most eminent Egyptologists of the late 19th century. To sequence graves in Naqada he developed a mathematical “Ansatz” which has led to mathematical objects such as matrices with the consecutive ones property, Petrie-matrices, the travelling salesman problem, and data mining. Petrie outlined his approach in archaeological terms and made no formal mathematical definitions or investigation, but he was aware that he was utilizing mathematical techniques. He already introduced and employed concepts, such as the Hamming distance, before they were formally defined in other areas of mathematics and the information sciences and which have completely different applications nowadays. Documenta Mathematica

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The travelling salesman problem There is an almost infinite number of articles on the travelling salesman problem, many of these describe some of the origins of the TSP and its great variety of applications. (We recommend Chapters 1 and 2 of [1] for an excellent survey of these two topics.) Since the TSP is usually introduced as the task to find a shortest round trip through a given number of cities, the TSP applications are often associated with vehicle routing, optimal machine control, and the like. One “origin” of the TSP that is often forgotten in overviews is archaeology. That is why we highlight here the independent invention of the TSP in this field. In fact, Petrie also invented a distance measure between graves, which constitutes what we call Hamming metric today. The Hamming metric In mathematics, the Hamming distance of two vectors in some vector space is equal to the number of components where the two vectors have different entries. This distance function is clearly non-negative, symmetric, zero only when the two vectors are identical, and obeys the triangle inequality. In other words, it is a metric. A computer scientist would say that the Hamming distance between two strings of symbols is the number of positions at which the corresponding symbols disagree. This distance is named after Richard Hamming, who introduced it in his fundamental paper [5] on what we now call Hamming codes. The Hamming distance is, e.g., used in communication to count the number of flipped bits in a transmitted word (in order to estimate errors occurring), and plays an important role in information and coding theory, and cryptography. Sir William Matthew Flinders Petrie The excellent biography [3] provides a detailed account of the life and the achievements of Petrie who was born in 1853 near London, died 1942 in Jerusalem and held the first chair of Egyptology (at the University College London) in the United Kingdom. We provide only a few details relevant for the topic addressed here. Petrie, a grandson of Matthew Flinders, surveyor of the Australian coastline, was tutored at home and had almost no formal education. His father William Petrie, an engineer who held several patents and had great interest in science, taught his son to survey accurately, laying the foundation for his career in archaeology. William Matthew Flinders Petrie is described by many as a “brilliant” extraordinary individual, one of the leading Egyptologists of his time. Notwithstanding his archaeological discoveries, the fact that he set new standards in painstaking recording of excavations and care of artifacts – thereby inaugurating what might be correctly termed as ‘modern’ archaeology –, high honors such as a knighthood bestowed upon him and honorary memberships in innumerable Documenta Mathematica

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c Courtesy of the Egypt Figure 1: Sir William Matthew Flinders Petrie ( Exploration Society, London)

British and international learned societies, Petrie remains a controversial figure due to his right-wing views on social topics and his belief in eugenics, see [19]. Upon his death, he donated his skull to the Royal College of Surgeons London, in particular, to be investigated for its high intellectual capacity in the field of mathematics, see [21]. Petrie and mathematics William Petrie wrote about his son when Matthew was not yet ten: He continues most energetically studying [. . . ] chemicals and minerals. [. . . ] we gave him a bit of garden ground to cultivate, to induce him not to spend too long a time in reading his chemical books and making – considering his age – very deep arithmetical calculations . . . . (See [3, p. 17].) Matthew’s scientific approach and mathematical mind, basically self-taught, except for two university courses in algebra and trigonometry – but only at the age of twenty-four –, shaped his archaeological career. Having, already at the age of 19, made attempts to understand the geometry of Stonehenge, Petrie applied the same techniques in his 1880–1882 survey of the Pyramids at Giza. His report on his measurements and his analysis of the architecture of the pyramids are till today a prime example of adequate methodology and Documenta Mathematica

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accuracy. The results of the work published in [14]; [15], and [16] helped to refute a number of mysticism theories linked to ancient monuments. Petrie’s work on the relative chronological ordering of archaeological artifacts showed already a deep understanding of the mathematics behind the seriation problem and was praised in [12, p. 213] as follows: While his writings are not easy to follow, they make fascinating reading for a mathematician, [. . . ], and in my view Petrie should be ranked with the great applied mathematicians of the nineteenth century. [. . . ] his writings contain what must surely be the first ‘mathematical model’ [. . . ] in the literature of archaeology. Seriation If in some old country mansion one room after another had been locked up untouched at the death of each successive owner, then on comparing all the contents it would easily be seen which rooms were of consecutive dates; and no one could suppose a Regency room to belong between Mary and Anne, or an Elizabethan room to come between others of George III. The order of rooms could be settled to a certainty on comparing all the furniture and objects. Each would have some links of style in common with those next to it, and much less connection with others which were farther from its period. And we should soon frame the rule that the order of the rooms was that in which each variety or article should have as short a range of date as it could. Any error in arranging the rooms would certainly extend the period of a thing over a longer number of generations. This principle applies to graves as well as rooms, to pottery as well as furniture. (Petrie, 1899 quoted in [3, p. 254]) Below we review and comment Petrie’s fundamental publication [18] of 1901. All quotes (written in italic) are from this paper. Being confronted with the task of establishing a prehistoric chronology of Egypt, based on the finds from his excavations at Naqada, Petrie had to find a way of dealing “simultaneously with records of some hundreds of graves” from the cemeteries. He therefore developed a method of abstract classification of objects – mainly ceramics. The pottery was divided into nine distinct categories, subdivided into several type-variations. Fig. 2 shows an example of such a classification. This typology was recorded in alphanumerical codes. The inventory of the graves Petrie excavated was subsequently written on a separate slip of card for each [individually numbered] tomb. [. . . ] All the slips were ruled in nine columns, one of each kind of pottery. Every form of pottery found in a given tomb was then expressed by writing the number of that form in the column of that kind of pottery. Documenta Mathematica

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Figure 2: Types of pottery [18] http://archive.org/stream/ diospolisparvac01macegoog#page/n8/mode/2up Figure 3 shows the scan of such slips, provided by the Petrie Museum. The first slip is the “header slip”. The first entry indicates that in all “tomb slips” Documenta Mathematica

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c Courtesy of the Petrie Museum, London) Figure 3: Paper-slips (

the first entry is the individual alphanumerical code of the tomb represented by the slip. The following nine entries of the header slip contain the abbreviated names of Petrie’s classification of pottery. Documenta Mathematica

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The second slip of Fig. 3 records the inventory of the grave encoded by B 130 (first entry). Six of the following nine entries of the slip are void, indicating that no objects of these six pottery categories were found. The other three entries show that tomb number B 130 contains B(black-topped), F(fancy formed) and N(incised black) pottery, tomb number U 115 contains no N but P(polished red) pottery as well. The entry in column B of row B 130 records the types 22a, 25c and 25f. What we see here is a data structure which we would call today “sparse matrix representation” or “linked list”. Petrie explains that he came up with this representation in order to avoid producing large tables with many empty entries. One can interpret Petrie’s data structure as an implicitly defined “gravepottery type incidence matrix”. Each row of this matrix represents a grave. The nine columns B, F, P, . . . , L of his slips have to be expanded so that each column corresponds to one type variation of the nine pottery categories. The entry aij of such an incidence matrix A is equal to “1” if the grave represented by row i contains the pottery type variation represented by column j. In this way every grave is represented by a 0/1-vector describing its pottery contents. Grave B 130, for instance, would have a coefficient “1” in the components representing the pottery type variations B22a, B25c, B25f, F14, N34, and N37, all other components are “0”. In order to pre-arrange the material, Petrie sorted the slips according to stylistic criteria: The most clear series of derived forms is that of the wavy-handled vases [W]. Beginning almost globular, [. . . ] they next become more upright, then narrower with degraded handles, then the handle becomes a mere wavy line, and lastly an upright cylinder with an arched pattern or a mere cord line around it Petrie also knew that: “there is a class [. . . ] we have seen to be later [L] than the rest, as it links on to the forms of historic age.” and arranged his slips accordingly. After this first arrangement of material (modern algorithmic term: knowledge based preprocessing), Petrie considered the other types of pottery, trying to establish a rough relative chronological order, according to the principles of the Hamming metric, cited above: This rough placing can be further improved by bringing together as close as may be the earliest and the latest examples of any type; as it is clear that any disturbance of the original order will tend to scatter the types wider, therefore the shortest range possible for each type is the probable truth. Looking at what Petrie has actually done, one can conclude that this constitutes the simultaneous introduction of the Hamming metric and the TSP. In his chronological arrangement, Petrie considered the closeness of two graves as Documenta Mathematica

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c Courtesy of the Petrie Figure 4: Petrie’s arrangement of slips, partial view. ( Museum, London)

the number of different entries in the 0/1-vector encoding the graves, which is exactly the Hamming distance of the two grave incidence vectors. Moreover, he claimed that finding an overall arrangement of all graves such that the sum of the Hamming distances between two consecutive graves is as small as possible, would solve his chronological ordering (seriation) problem. And this is nothing but the formulation of the TSP in archaeological terms. Petrie was aware that the available data are imprecise, and that hence the mathematically obtained chronological ordering is only approximate (“probable truth”) so that further archaeological “post processing” is necessary. Having come up with this mathematical model of chronological ordering, Petrie noticed that the amount of data would be outside of his computational capacities. So he applied data reduction and decreased the number of graves according to their statistical relevance: “In this and all the later stages only graves with at least five different types of pottery were classified, as poorer instances do not give enough ground for study.” And thus he began to arrange the 900 remaining paper-slips according to the relative order of appearance of different types of pottery and determined a heuristic solution of a “900-city-TSP”. He succeeded in a “satisfactory” arrangement of 700 slips and subsequently made: “a first division into fifty equal stages, numbered 30 to 80, termed sequence dates or S.D. and then [made] a list of all the types of pottery, stating the sequence date of every example that occurs in theses graves.” By this he was able to provide a relative chronology, without having to name absolute chronological dates. In other words: Petrie made 49 “cuts” into the list of 700 graves, thereby defining 50 time-periods without giving absolute dates, that are identified by the simultaneous appearance of very similar pottery. This also enabled him to introduce and indicate in his publications periods of appearance of certain pottery types. “Now on the basis of the list made [. . . ] we incorporate all the other graves which contain enough pottery to define their position.” In modern TSP-terminology Petrie did the following: He started out with a large number of cities and dispensed those who were irrelevant for the problem, due to insufficient data, to reduce the TSP-instance to a manageable size. (We call this data reduction today). Then he identified a certain subset of cities for Documenta Mathematica

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which he was able to identify a satisfactory solution (identification of important cities for which a good solution can be found). After that he used a clusteringbased insertion-method to produce a feasible and hopefully good solution of the overall problem. A piece of the final sequence of graves (TSP solution) is shown in Fig. 4. Final remarks Petrie’s sequence dates, which are an outcome of his TSP-approach to seriation, constitute a true paradigm change within the field of archaeology, rendering a scholarly subject, dominated by art historians and linguists, a veritable “scientific” discipline. Pioneering as it was, Petrie’s method had and has been further developed and complemented by later archaeologists. Mathematically speaking, other researchers suggested to replace the Hamming distance by weighted versions and other metrics, taking for instance into account spatial distribution, by dissimilarity coefficients, obtained from statistical analysis of grave contents, and so on. In most of these cases the result was a mathematical model that is equivalent to the TSP with an objective function describing some grave-relationship. A brief survey of these and other approaches, the definition of Petrie matrices, and related concepts can be found in [20]. Literature and further reading [1] D. L. Applegate, R. E. Bixby, V. Chv´atal and W. J. Cook , The Traveling Salesman Problem: A Computational Study, Princeton University Press, Princeton, 2006. [2] E. J. Baumgartel, Petrie’s Naqada Excavation. A Supplement, London, 1970. [3] M. Drower, Flinders Petrie. A Life in Archaeology, 2nd edition, University of Wisconsin Press, Madison, 1996. [4] M. K. H. Eggert, Pr¨ ahistorische Arch¨ aologie. Konzepte und Methoden, A. Francke Verlag, T¨ ubingen und Basel, 2001. [5] R. W. Hamming, Error detecting and error correcting codes, Bell System Technical Journal 29 (1950) 147–160. [6] S. Hendrickx, The Relative Chronology of the Naqada Culture. Problems and Possibilities, in J. Spencer(ed.) Aspects of Early Egypt, London, 1999, 36–69. [7] S. Hendrickx, La Chronologie de la pr´ehistoire tardive et des d´ebuts de l’histoire de l’Egypte, Arch´eo-Nil 9 (1999) 13–33. Documenta Mathematica

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[8] F. R. Hodson, D. G. Kendall and P. Tautu (eds.), Mathematics in the Archaeological and Historical Sciences. Edinburgh University Press, Edinburgh, 1971. ¨ [9] W. Kaiser, Studien zur Vorgeschichte Agyptens, 1955. [10] W. Kaiser, Stand und Probleme der ¨agyptischen Vorgeschichtsforschung, ¨ Zeitschrift f¨ ur Agyptische Sprache und Altertumskunde 81 (1956) 87–109. [11] W. Kaiser, Zur inneren Chronologie der Naqadakultur, Archaeologia Geographica 6 (1957) 69–77. [12] D. G. Kendall, Seriation from abundance matrices, Mathematics in the Archaeological and Historical Sciences, in [7], pp. 215–252. [13] M. J. O’ Brien and R. L. Lyman, Seriation, Stratigraphy and Index Fossils. The Backbone of Archaeological Dating, Kluwer Academic/Plenum Publishers, New York, 1999. [14] W. M. Flinders Petrie, Researches on the Great Pyramid, Or Fresh Connections: Being a Preliminary Notice of some Facts, London, 1874. [15] W. M. Flinders Petrie, Inductive Metrology. Or the Recovery of Ancient Measures from the Monuments, London, 1877. [16] W. M. Flinders Petrie, Stonehenge. Plans, Desriptions and Theories, London, 1880. [17] W. M. Flinders Petrie, Sequences in Prehistoric Remains, The Journal of the Anthropological Institute of Great Britain and Ireland 29 (1899) 295–301. [18] W. M. Flinders Petrie, Diospolis Parva. The Cemeteries of Abadiyeh and Hu, London, 1901. [19] N. A. Silberman, Petrie’s Head. Eugenics and Near Eastern Archaeology, in: A. B. Kehoe and M. B. Emmerichs (eds.), Assembling the Past: Studies in the Professionalization of Archaeology, University of New Mexico Press, Albuquerque, 1999, pp. 69–79. [20] A. Shuchat, Matrix and network Models in Archaeology, Mathematics Magazine, 57 (1984) 3–14. [21] P. J. Ucko, The Biography of a Collection. The Sir Flinders Petrie Palestinian Collection and the Role of University Museums. Appendix A: Donnation of Remains of Sir William Flinders Petrie by Simon Chaplin, Museum Management and Curatorship 17 (1998) 391–394. [22] E. M. Wilkinson, Archaeological Seriation and the Travelling Salesman Problem, in [7], pp. 276–283. Documenta Mathematica

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¨ tschel Thomas L. Gertzen and Martin Gro Thomas Gertzen Wehnertstraße 3 12277 Berlin-Marienfelde Germany

Martin Gr¨otschel Konrad-Zuse-Zentrum f¨ ur Informationstechnik Berlin (ZIB) Takustraße 7 14195 Berlin Germany

[emailprotected]

[emailprotected]

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D. Ray Fulkerson and Project Scheduling ¨ hring Rolf H. Mo

2010 Mathematics Subject Classification: 90B35, 90B36, 05C21 Keywords and Phrases: Stochastic scheduling, underestimation error, time-cost tradeoff

1

Introduction

D. Ray Fulkerson (1922–1976) made fundamental and lasting contributions to combinatorial mathematics, optimization, and operations research [2]. He is probably best known for his work on network flows and in particular for the famous max flow–min cut theorem, stating that the maximum amount of a flow from a node s to a node t in a directed graph equals the minimum capacity of a cut separating s from t. Less known is the fact that he also made important contributions to project scheduling. One deals with time-cost tradeoff analysis of project networks, which he solved with min-cost flow techniques. This method has meanwhile entered standard text books such as [1] (often as an exercise of application of flow methods) and will not be discussed here. The much less known contribution concerns project planning when the individual job times are random variables. Fulkerson was one of the first to

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c Figure 2: Polaris A-3 at Cape Canaveral ( Wikimedia Commons) recognize the deficiency of the then state-of-the-art operations research techniques, and he developed a method for better analysis that has started a whole stream of research on risk analysis in project planning. This chapter tells the story of this contribution. 2

The background [10, 3]

During the Cold War, around the late fifties and early sixties, Lockheed Corporation developed and built the first version of the Polaris missile for the United States Navy as part of the United States arsenal of nuclear weapons. It was a two-stage solid-fuel nuclear-armed submarine-launched intercontinental ballistic missile with a range of 4.600 km that replaced the earlier cruise missile launch systems based on submarines [3]. The complexity of this and similar projects required new planning tools that could deal with research and development programs for which time is an uncertain but critical factor. To support the Polaris project, the Navy’s Special Projects Office developed the Program Evaluation and Review Technique (PERT), which still is applied as a decision-making tool in project planning. Willard Fazar, Head of the Program Evaluation Branch of the Special Projects Office [4] recalls: The Navy’s Special Projects Office, charged with developing the Polaris-Submarine weapon system and the Fleet Ballistic Missile capability, has developed a statistical technique for measuring and Documenta Mathematica

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forecasting progress in research and development programs. This Program Evaluation and Review Technique (code-named PERT) is applied as a decision-making tool designed to save time in achieving end-objectives, and is of particular interest to those engaged in research and development programs for which time is a critical factor. The new technique takes recognition of three factors that influence successful achievement of research and development program objectives: time, resources, and technical performance specifications. PERT employs time as the variable that reflects planned resourceapplications and performance specifications. With units of time as a common denominator, PERT quantifies knowledge about the uncertainties involved in developmental programs requiring effort at the edge of, or beyond, current knowledge of the subject – effort for which little or no previous experience exists. [. . . ] The concept of PERT was developed by an operations research team staffed with representatives from the Operations Research Department of Booz, Allen and Hamilton; the Evaluation Office of the Lockheed Missile Systems Division; and the Program Evaluation Branch, Special Projects Office, of the Department of the Navy. I will explain the main idea underlying PERT in the next section. Fulkerson noticed that PERT makes a systematic error, as it generally underestimates the expected makespan of a project. He worked at the RAND Cooperation at that time and wrote in research memorandum RM-3075-PR prepared for the United States Air Force [6] and later published in slightly revised form in [5]: The calculation of project duration times and project cost by means of network models has become increasingly popular within the last few years. These models, which go by such names as PERT (Program Evaluation Review Technique), PEP (Program Evaluation Procedure), Critical Path Scheduling, Project Cost Curve Scheduling, and others, have the common feature that uncertainties in job times are either ignored or handled outside the network analysis, usually by replacing each distribution of job times by its expected value. He continues his criticism of PERT in the follow-up report RM-3075-PR [7]: The PERT model of a project usually assumes independent random variables for job times, instead of deterministic times [. . . ]. But the usual practice has been to replace these random variables by their expected values, thereby obtaining a deterministic problem. The solution of this deterministic problem always provides an optimistic estimate of the expected length of the project. Documenta Mathematica

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Although the analysis of a PERT model, with fixed job times, is trivial from the mathematical point of view, the model itself appears to be a useful one, judging from its widespread acceptance and use throughout industry today. But it should be added that it is difficult to assess the usefulness of PERT on this basis alone, since the model has been the subject of much hard-sell advertising and exaggerated claims. Fulkerson instead suggests an algorithm that uses discrete random job times and calculates a much better lower bound on the expected project makespan than the one obtained by the PERT. It was published in 1962 [5] and has become one of the fundamental papers in the area of project risk analysis. I will outline some of the underlying mathematics of this development in the next section. Part of that exposition is taken from [11]. 3

Coping with uncertainty in scheduling: The math

In real-life projects, it usually does not suffice to find good schedules for fixed deterministic processing times, since these times mostly are only rough estimates and subject to unpredictable changes due to unforeseen events such as weather conditions, obstruction of resource usage, delay of jobs and others. In order to model such influences, PERT assumes that the processing time of a job j ∈ V = {1, . . . , n} is assumed to be a random variable pj . Then p = (p1 , p2 , . . . , pn ) denotes the (random) vector of job processing times, which is distributed according to a joint probability distribution Q. This distribution Q is assumed to be known, though sometimes, also partial information may suffice. In general, Q may contain stochastic dependencies, but most methods require that the job processing times are stochastically independent. (Fulkerson allows some dependencies in his method, see below.)) Jobs are subject to precedence constraints given by a directed acyclic graph G = (V, E). We refer to G also as the project network. Now consider a particular realization p = (p1 , . . . , pn ) of the random processing time vector p = (p1 , p2 , . . . , pn ). Since there are no resource constraints, every job j can complete at its earliest possible completion time Cj = Cj (p), which is equal to the length of a longest path in G that ends with j, where the length of a job j is its processing time pj . The earliest project completionPor makespan for the realization p is then Cmax (p) := maxj Cj (p) = maxP j∈P pj , where P ranges over all inclusionmaximal paths of G. Since the processing times pj are random, the makespan P Cmax is also a random variable, and it may be written as Cmax = maxP j∈P pj , i.e., as the maximum of sums over subsets of a common set of random variables. An example is given in Figure 3. The main goal of project risk analysis is to obtain information about the distribution of this random variable Cmax . Documenta Mathematica

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Cmax = max { p1 + p3 + p6 ,

p1 + p5 + p7 ,

p2 + p4 + p6 ,

p2 + p4 + p7 ,

p2 + p5 + p7 }

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Figure 3: An example project network and its makespan Cmax

Fulkerson noticed the systematic underestimation Cmax E(p1 ), . . . , E(pn ) ≤ E Cmax (p1 , . . . , pn ) when one compares the “deterministic makespan” Cmax (E(p1 ), . . . , E(pn )) obtained from the expected processing times E(pj ) with the expected makespan E(Cmax (p)). This error may in fact become arbitrarily large with increasing number of jobs or increasing variances of the processing times [9]. Equality holds if and only if there is one path that is the longest with probability 1, see Theorem 1 below. The error becomes even worse if one compares the deterministic value Cmax (E(p1 ), . . . , E(pn )) with quantiles tq such that P rob{Cmax (p) ≤ tq } ≥ q for large values of q (say q = 0.9 or 0.95). A simple example is given in Figure 4 for a project with n parallel jobs that are independent and uniformly distributed on [0,2]. Then the deterministic makespan Cmax (E(p1 ), . . . , E(pn )) = 1, while P rob(Cmax ≤ 1) → 0 for n → ∞. Similarly, all quantiles tq → 2 for n → ∞ (and q > 0). This is the reason why good practical planning tools should incorporate stochastic methods.

Prob(Cmax≤ t) 1 q

t 0

1

2

Figure 4: Distribution function of the makespan for n = 1, 2, 4, 8 parallel jobs that are independent and uniformly distributed on [0,2]. Documenta Mathematica

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Theorem 1. Let G = (V, E) be a project network with random processing time vector p. Then Cmax E(p1 ), . . . , E(pn ) ≤ E(Cmax p1 , . . . , pn ) .

Equality holds iff there is one path that is the longest with probability 1.

Proof. Since Cmax is the maximum of sums of processing times, it is obviously a convex function of p. Thus the inequality is a special case of Jensen’s inequality for convex functions. We give here an elementary proof for Cmax . Let P1 , . . . , Pk be the inclusion-maximal paths of G and let Y1 , . . . , Yk denote P their (random) length, i.e., Yi := j∈Pi pj . Then Cmax = maxi Yi , and Cmax (E(p)) = max i

X

E(pj ) = max E( i

j∈Pi

= E(Yi0 )

X

j∈Pi

pj ) = max E(Yi ) i

assume that the maximum is attained at i0

≤ E(max Yi ) since Yi0 ≤ max Yi as functions of p i

i

= E(Cmax (p)). Now assume that Y1 is the longest path with probability 1. Then, with probability 1, Cmax = Y1 ≥ Yi . Hence E(Cmax ) = E(Y1 ) ≥ E(Yi ) and the above calculation yields Cmax (E(p)) = maxi E(Yi ) = E(Y1 ) = E(Cmax ). In the other direction assume that E(Cmax (p)) = Cmax (E(p)). Let w.l.o.g. P1 be the longest path w.r.t. expected processing times E(pj ). Then E(Y1 ) = E(Cmax (p)) and 0 = E Cmax (p) − Cmax E(p) = E max Yi − max E(Yi ) i Z = E(max E(Yi ) − Y1 ) = max E(Yi ) − Y1 dQ. Since the integrand in non-negative, it follows that it is 0 with probability 1. Hence Y1 = max E(Yi ) = Cmax with probability 1. The probabilistic version of PERT is based on the second statement of this theorem. It only analyzes the distribution of the path with the longest expected path length. It thus fails when there are many paths that are critical with high probability. The algorithm of Fulkerson uses the arc diagram of the precedence graph G, which is common also to PERT. It considers jobs of a project as arcs of a directed graph instead of vertices. This construction uses a directed acyclic graph D = (N, A) with a unique source s and a unique sink t. Every job j of G is represented by an arc of D such that precedence constraints are preserved, i.e., if (i, j) is an edge of G, then there is a path from the end node of i to the start node of j in D. Figure 5 gives an example. Such a representation is called an arc diagram (sometimes also PERT network ) of the project. In Documenta Mathematica

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3

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4 2

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Node diagram jobs are nodes of digraph G

Arc diagram jobs are arcs of digraph D dummy arcs may be necessary

Figure 5: Arc diagram of the project network of Figure 3 general, one needs additional arcs (so-called dummy arcs) to properly represent the precedence constraints. Arc diagrams are thus not unique, but as dummy arcs obtain processing time 0, this ambiguity has no influence on the makespan. Fulkerson assumes that stochastic dependencies may only occur in job bundles, where a bundle consists of all jobs with the same end node in the arc diagram. His algorithm then computes for each node v a value tv that is iteratively obtained along a topological sort of the arc diagram as tv = EQv max {tu + p(u,v) } , (u,v)∈E

where Qv is the joint distribution of the processing times in the bundle of jobs ending in v, and the maximum is taken over all arcs in this bundle. A simple inductive argument shows that this gives indeed a lower bound on the expected makespan. Fulkerson applies this to discrete job processing times, and so his algorithm is exponential in the maximum size of a bundle. He already noticed that it is computationally difficult to compute the exact value of the expected makespan, which was later mathematically confirmed by Hagstrom [8]. Hagstrom considers the following two problems: Mean: Given a project network with discrete, independent processing times pj , compute the expected makespan E(Cmax (p)). DF: Given a project network with discrete, independent processing times pj and a time t, compute the probability P rob{Cmax (p) ≤ t} that the project finishes by time t. She shows that DF and the 2-state versions of Mean, in which every processing time pj has only two discrete values, are #P-complete. The complexity status of the general version of Mean is open (only the 2-state version, which has a short encoding, is known to be #P-complete). Documenta Mathematica

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If the processing times pj may take more than 2 values, the problem has a longer encoding that in principle could admit a polynomial algorithm for solving Mean. However, Hagstrom provides some evidence that problems with a long encoding may still be difficult, since Mean and DF cannot be solved in time polynomial in the number of values of Cmax (p) unless P = N P. These results show that efficient methods for calculating the expected makespan or quantiles of the distribution function of the makespan are very unlikely to exist, and thus justify the great interest in approximate methods such as bounds, simulation etc. that started with the work of Fulkerson. The search for “expected completion time” +network in Google Scholar currently shows more than 1,500 results. References [1] Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network Flows. Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs, NJ, 1993. [2] Robert G. Bland and James B. Orlin. IFORS’ Operational Research Hall of Fame: Delbert Ray Fulkerson. Intl. Trans. in Op. Res., 12:367–372, 2005. [3] Grant R. Cates. Improving Project Management With Simulation And Completion Distribution Functions. PhD thesis, University of Florida, Orlando, Florida, 2004. [4] Willard Fazar. Program evaluation and review technique. The American Statistician, 13(2):10, 1959. [5] D. R. Fulkerson. Expected critical path lengths in PERT networks. Oper. Res., 10:808–817, 1962. [6] D. Ray Fulkerson. Expected critical path lengths in PERT networks. Technical Report RM-3075-PR, RAND Cooperation, March 1962. [7] D. Ray Fulkerson. Scheduling in project networks. Technical Report RM4137-PR, RAND Cooperation, June 1964. [8] Jane N. Hagstrom. Computational complexity of PERT problems. Networks, 18:139–147, 1988. [9] Ursula Heller. On the shortest overall duration in stochastic project networks. Methods Oper. Res., 42:85–104, 1981. [10] J. J. Moder and C. R. Phillips. Project management with CPM and PERT. Reinhold, New York, 1964. Documenta Mathematica

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[11] Rolf H. M¨ ohring. Scheduling under uncertainty: Bounding the makespan distribution. In Helmut Alt, editor, Computational Discrete Mathematics: Advanced Lectures, volume 2122 of Lecture Notes in Computer Science, pages 79–97. Springer-Verlag, 2001.

Rolf H. M¨ohring Technische Universit¨at Berlin Institut f¨ ur Mathematik Straße des 17. Juni 136 10623 Berlin [emailprotected]

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The Ongoing Story of Gomory Cuts ´rard Cornue ´jols Ge

2010 Mathematics Subject Classification: 90C10, 90C11 Keywords and Phrases: Gomory cut, GMI cut The story of Gomory cuts is characterized by swings between great acclaim in the early days, near oblivion for decades and an amazing come back in the last 20 years. These cuts have been described as “elegant”, “disappointing” and “the clear winner” at various times over the last 55 years. This essay retraces that roller coaster. Ralph Gomory’s paper “Early Integer Programming” recounts his discovery of fractional cuts. It is a few years after he wrote his doctoral dissertation on nonlinear differential equations that he heard of linear programming for the first time. He was working for the Navy at the time. In one particular instance, it would have been preferable to have solutions in integers. Gomory thought that, somehow, one should be able to accomplish this. Within a few days he had invented fractional cuts. His approach was to first solve the linear program and then, using appropriate integer linear forms, to generate valid linear inequalities cutting off the undesirable fractional solution. By adding these cuts to the linear program, solving again using the simplex algorithm and iterating, Gomory could solve by hand any small integer linear program that he tried. However, he did not have a finiteness proof yet. At this point, he happened to run into Martin Beale in the halls of Princeton University in late 1957 and mentioned that he could solve linear programs in integers. When Beale immediately responded “but that’s impossible”, Gomory realized that he was not the first to think about this problem. As it turns out, Dantzig, Fulkerson, and Johnson had pioneered the cutting plane approach in a seminal paper published in 1954. They devised special-purpose cuts for the traveling salesman problem and, as a result, were able to solve to optimality an instance with 48 cities. However, Gomory’s goal was different and more ambitious. His fractional cuts were general-purpose cuts that applied to all integer linear programs. In his reminiscences “Early Integer Programming”, Gomory recounts the excitement that followed his encounter with Beale. During the exciting weeks that followed, I finally worked out a finiteness proof and then programmed the algorithm on the E101, a pin board computer that was busy during the day but that I could use Documenta Mathematica

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´rard Cornue ´jols Ge late at night. The E101 had only about 100 characters of memory and the board held only 120 instructions at a time, so that I had to change boards after each simplex maximization cycle and put it in a new board that generated the cut, and then put the old board back to remaximize. It was also hard work to get the simplex method down to 120 E101 instructions. But the results were better and more reliable than my hand calculations, and I was able to steadily and rapidly produce solutions to four- and five-variable problems.

When Gomory presented his results in early 1958, the impact was enormous and immediate. Gomory had achieved the impossible: reducing integer linear programming to a sequence of linear programs. This was a great theoretical breakthrough. The next logical step was to try turning this work into a practical algorithm. In the summer of 1958, Gomory programmed his fractional cutting plane algorithm in FORTRAN (a new computer language at the time). He says Most of the problems ran quickly but one went on and on . . . it was the first hint of the computational problems that lay ahead . . . In the summer of 1959, I joined IBM Research and was able to compute in earnest . . . We started to experience the unpredictability of the computational results rather steadily. In 1960, Gomory [6] extended his approach to mixed-integer linear programs (MILPs), inventing the “mixed-integer cuts”, known today as GMI cuts (the acronym stands for Gomory mixed-integer cuts). GMI cuts are remarkable on at least two counts: 1) They are stronger than the fractional cuts when applied to pure integer programs; 2) They apply to MILPs, a crucial feature when generating cutting planes in an iterative fashion because pure integer programs typically turn into MILPs after adding cuts. Three years later, in 1963, Gomory [7] states that these cuts are “almost completely computationally untested.” Surprisingly, Gomory does not even mention GMI cuts in his reminiscences in 1991. In the three decades from 1963 to 1993, Gomory cuts were considered impractical. Several quotes from the late 80s and early 90s illustrate this widely held view. Williams [11]: “Although cutting plane methods may appear mathematically fairly elegant, they have not proved very successful on large problems.” Nemhauser and Wolsey [9]: “They do not work well in practice. They fail because an extremely large number of these cuts frequently are required for convergence.” Padberg and Rinaldi [10]: These cutting planes have poor convergence properties . . . classical cutting planes furnish weak cuts . . . A marriage of classical cutting planes and tree search is out of the question as far as the solution of large-scale combinatorial optimization problems is concerned. Documenta Mathematica

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By contrast, the Dantzig, Fulkerson, Johnson strategy of generating specialpurpose cuts had gained momentum by the early 90s. Padberg and Rinaldi [10] obtained spectacular results for the traveling salesman problem using this approach. It was applied with a varying degree of success to numerous other classes of problems. The effectiveness of such branch-and-cut algorithms was attributed to the use of facets of the integer polyhedron. Was this view of cutting planes justified? Despite the bad press Gomory cuts had in the research community and in textbooks, there was scant evidence in the literature to justify this negative attitude. Gomory’s quote from thirty years earlier was still current: GMI cuts were “almost completely computationally untested.” In 1993 I convinced Sebastian Ceria, who was a PhD student at Carnegie Mellon University at the time, to experiment with GMI cuts. The computational results that he obtained on MIPLIB instances were stunning [1]: By incorporating GMI cuts in a branch-and-cut framework, he could solve 86 % of the instances versus only 55 % with pure branch and bound. For those instances that could be solved by both algorithms, the version that used GMI cuts was faster on average, in a couple of cases by a factor of 10 or more. This was a big surprise to many in the integer programming community and several years passed before it was accepted. In fact, publishing the paper reporting these results, which so strongly contradicted the commonly held views at the time, was an uphill battle (one referee commented “there is nothing new” and requested that we add a theoretical section, another so distrusted the results that he asked to see a copy of the code. The associate editor recommended rejection, but in the end the editor overruled the decision, and the paper [1] was published in 1996). Our implementation of Gomory cuts was successful for three main reasons: • We added all the cuts from the optimal LP tableau (instead of just one cut, as Gomory did). • We used a branch-and-cut framework (instead of a pure cutting plane approach). • LP solvers were more stable by the early 1990s. Commercial solvers for MILPs, such as Cplex, started incorporating GMI cuts in 1999. Other cutting planes were implemented as well and solvers became orders of magnitude faster. Bixby, Fenelon, Gu, Rothberg and Wunderling [3] give a fascinating account of the evolution of the Cplex solver. They view 1999 as the transition year from the “old generation” of Cplex to the “new generation”. Their paper lists some key features of a 2002 “new generation” solver and compares the speedup in computing time achieved by enabling one feature versus disabling it, while keeping everything else unchanged. The table below summarizes average speedups obtained for each feature on a set of 106 instances. Documenta Mathematica

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224 Feature

Speedup factor

Cuts Presolve Variable selection Heuristics

54 11 3 1.5

The clear winner in these tests was cutting planes. In 2002 Cplex implemented eight types of cutting planes. Which were the most useful? In a similar experiment disabling only one of the cut generators at a time, Bixby, Fenelon, Gu, Rothberg and Wunderling obtained the following degradation in computing time. Cut type

Factor

GMI MIR Knapsack cover Flow cover Implied bounds Path Clique GUB cover

2.5 1.8 1.4 1.2 1.2 1.04 1.02 1.02

Even when all the other cutting planes are used in Cplex (2002 version), the addition of Gomory cuts by itself produces a solver that is 2.5 times faster! As Bixby and his co-authors conclude “Gomory cuts are the clear winner by this measure”. Interestingly the MIR (Mixed Integer Rounding) cuts, which come out second in this comparison, turn out to be another form of GMI cuts! However, that’s not the end of the story of Gomory cuts. More work is needed on how to generate “safe” Gomory cuts: The textbook formula for generating these cuts is not used directly in open-source and commercial software due to the limited numerical precision in the computations; solvers implement additional steps in an attempt to avoid generating invalid cuts. Despite these steps, practitioners are well aware that the optimal solution is cut off once in a while. More research is needed. Another issue that has attracted attention but still needs further investigation is the choice of the equations used to generate GMI cuts: Gomory proposed to generate cuts from the rows of the optimal simplex tableau but other equations can also be used. Balas and Saxena [2], and Dash, G¨ unl¨ uk and Lodi [4] provide computational evidence that MILP formulations can typically be strengthened very significantly by generating Gomory cuts from a well chosen set of equations. But finding such a good family of equations “efficiently” remains a challenge. Documenta Mathematica

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Acknowledgement This work was supported in part by NSF grant CMMI 1024554 and ONR grant N00014-12-10032. References [1] E. Balas, S. Ceria, G. Cornu´ejols and N. Natraj, Gomory cuts revisited, Operations Research Letters 19 (1996) 1–9. [2] E. Balas and A. Saxena, Optimizing over the split closure, Mathemtical Programming 113 (2008) 219–240. [3] R.E. Bixby, M. Fenelon, Z. Gu, Ed Rothberg and R. Wunderling, MixedInteger Programming: A Progress Report, in The Sharpest Cut: The Impact of Manfred Padberg and His Work, edited by Martin Gr¨otschel, MPS-SIAM Series on Optimization (2004) 309–325. [4] S. Dash, O. G¨ unl¨ uk and A. Lodi, On the MIR closure of polyhedra, 12th International IPCO Conference, Ithaca, NY, June 2007, (M. Fischetti and D.P. Williamson eds.) LNCS 4513 (2007) 337–351. [5] R. Gomory, Outline of an Algorithm for Integer Solutions to Linear Programs, Bulletin of the American Mathematical Society 64 (1958) 275–278. [6] R. Gomory, An algorithm for the mixed integer problem, Technical Report RM-2597, The Rand Corporation (1960). [7] R. Gomory, An algorithm for integer solutions to linear programs, in R.L. Graves and P. Wolfe eds., Recent Advances in Mathematical Programming, McGraw-Hill, New York (1963) 269–302. [8] R. Gomory, Early integer programming, in J.K. Lenstra, A.H.G. Rinnooy Kan and A. Schrijver eds., History of Mathematical Programming, A Collection of Personal Reminiscences, North-Holland, Amsterdam (1991) 55–61. [9] G.L. Nemhauser and L.A. Wolsey, Integer Programming, in G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd eds., Handbook in Operations Research and Management Science 1: Optimization, North-Holland, Amsterdam (1989) 447–527. [10] M. Padberg and G. Rinaldi, A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems, SIAM Review 33 (1991) 60–100. [11] H.P. Williams, Model Building in Mathematical Programming, Wiley, New York (1985). Documenta Mathematica

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´rard Cornue ´jols Ge G´erard Cornu´ejols Tepper School of Business Carnegie Mellon University Pittsburgh Pennsylvania 15213-3890 USA [emailprotected]

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Markowitz and Manne + Eastman + Land and Doig = Branch and Bound

William Cook

2010 Mathematics Subject Classification: 90C57, 90C10 Keywords and Phrases: Branch and bound, integer programming, traveling salesman problem The branch-and-bound method consists of the repeated application of a process for splitting a space of solutions into two or more subspaces and adopting a bounding mechanism to indicate if it is worthwhile to explore any or all of the newly created subproblems. For example, suppose we need to solve an integerprogramming (IP) model. A bounding mechanism is a computational technique for determining a value B such that each solution in a subspace has objective value no larger (for maximization problems) than B. For our IP model, the objective value of any dual feasible solution to the linear-programming (LP) relaxation provides a valid bound B. We can compute such a bound with any LP solver, such as the simplex algorithm. The splitting step is called branching. In our IP example, suppose a variable xi is assigned the fractional value t in an optimal solution to the LP relaxation. We can then branch by considering separately the solutions having xi ≤ ⌊t⌋ and the solutions having xi ≥ ⌊t⌋ + 1, where ⌊t⌋ denotes t rounded down to the nearest integer. The two newly created subproblems need only be considered for further exploration if their corresponding bound B is greater than the value of the best known integer solution to the original model. Branch and bound is like bread and butter for the optimization world. It is applied routinely to IP models, combinatorial models, global optimization models, and elsewhere. So who invented the algorithm? A simple enough question, but one not so easy to answer. It appears to have three origins, spread out over four years in the mid to late 1950s. As the starting point, the notion of branch and bound as a proof system for integer programming is laid out in the 1957 Econometrica paper “On the solution of discrete programming problems” by Harry Markowitz and Alan Manne [17]. Their description of the components of branch and bound is explicit, but they note in the paper’s abstract that the components are not pieced together into an algorithm. Documenta Mathematica

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William Cook We do not present an automatic algorithm for solving such problems. Rather we present a general approach susceptible to individual variations, depending upon the problem and the judgment of the user.

The missing algorithmic glue was delivered several years later by Ailsa Land and Alison Doig in their landmark paper “An automatic method of solving discrete programming problems” [12], published in the same journal in 1960. The Land-Doig abstract includes the following statement. This paper presents a simple numerical algorithm for the solution of programming problems in which some or all of the variables can take only discrete values. The algorithm requires no special techniques beyond those used in ordinary linear programming, and lends itself to automatic computing. Their proposed method is indeed the branch-and-bound algorithm and their work is the starting point for the first successful computer codes for integer programming. There is a further historical twist however. Sandwiched in between Markowitz-Manne and Land-Doig is the 1958 Harvard Ph.D. thesis of Willard Eastman titled Linear Programming with Pattern Constraints [5]. Eastman designed algorithms for several classes of models, including the traveling salesman problem (TSP). Page 3–5 of his thesis gives the following concise description of the heart of his technique. It is useful, however, to be able to establish lower-bounds for the costs of solutions which have not yet been obtained, in order to permit termination of any branch along which all solutions must exceed the cost of some known feasible solution. His methods, too, are early implementations of branch and bound. So Markowitz-Manne or Eastman or Land-Doig? Fortunately there is no need to make a choice: we can give branch-and-bound laurels to each of these three groups of researchers. 1

Markowitz and Manne (1957)

The Markowitz-Manne paper is one of the earliest references dealing with general integer programming. The paper was published in Econometrica in 1957, but an earlier version appeared as a 1956 RAND research paper [16], where the order of the authors is Manne-Markowitz. Even further, George Dantzig’s 1957 paper [1] cites the Manne-Markowitz report as having been written on August 1, 1955. This is indeed at the beginning of the field: Dantzig, Fulkerson, and Johnson’s classic paper on the TSP is typically cited as the dawn of integer programming and it appeared as a RAND report in April 1954 [2]. Markowitz-Manne, or Manne-Markowitz, discuss in detail two specific applications: a production-planning problem and an air-transport problem. A Documenta Mathematica

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Left: Harry Markowitz, 2000 (Photograph by Sue cl*tes). Right: Alan Manne (Stanford University News). fascinating thing is their inclusion of two appendices, one for each of the models, having subsections labeled “Proof” and “Verification” respectively. The “Proofs” consist of branch-and-bound subproblems and the “Verifications” explain why steps taken in the creation of the subproblems are valid. The general mixed IP model considered by Markowitz-Manne is to maximize a linear function π over a set D(0) wherein some or all variables take on integral values. For a nonempty set S in the same space as D(0), π(S) is defined to be max(π(X) : X ∈ S) if the maximum exists and otherwise π(S) ≡ ∞. Quoting from their paper, Markowitz-Manne lay out the following branch-and-bound framework. At each step s we have: (a) a best guess X(s) (b) one or more sets D1 (s), . . . , DK (s) such that D(0) ⊃ Dk (s)

k = 1, . . . , K,

π D(0) = π D1 (s) ∪ D2 (s) · · · ∪ DK (s) ∪ X(s) and (c) polyhedral sets Lk (s), such that Lk (s) ⊃ Dk (s)

k = 1, . . . , K

Clearly π ∪k Lk (s) ∪ X(s) = max π L1 (s) , . . . , π LK (s) , π X(s) ≥ π D(0) ≥ π X(s) .

The general strategy is to reduce the size of the sets ∪Dk and ∪Lk , and to bring together the lower and upper bounds on π(D(0)). Documenta Mathematica

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The “best guess” is the currently best-known solution X(s) ∈ D(0). If X(s) is itself not optimal, then the union of the sets Dk (s) is known to contain an optimal solution. The sets Lk (s) are LP relaxations of the discrete sets Dk (s), thus the upper bound max π L1 (s) , . . . , π LK (s) , π X(s)

on the IP objective can be computed via a sequence of LP problems. In just a few lines, Markowitz-Manne summed up much of the branch-andbound theory we use today! Indeed, they incorporate the idea of improving the LP relaxations Lk (s) from one step to the next, as is now done in sophisticated branch-and-cut algorithms. Moreover, their steps to create subregions Dk (s) involve the concept of branching on hyperplanes, that is, splitting a k − 1 level subregion into a number of k-level subregions by enforcing linear equations c(X) = ti for appropriate values of ti . The “Proof” subsections consist of explicit listings of the sets Dk (s) and Lk (s) used at each level in the example models, and the “Verifications” subsections explain why the adopted cutting planes are valid and how hyperplanes are used to subdivide subregions into further subregions. These appendices are amazingly complete formal proofs of the optimality of proposed solutions to the two applied problems. It would be beautiful if we could somehow recapture such formal correctness in current computational claims for optimal solutions to large-scale IP models. Julia Robinson and the TSP Markowitz and Manne carried out their work at the famed RAND Corporation, home in the 1950s of what was far and away the world’s top center for the study of mathematical optimization. They introduce their general branch-and-bound framework as follows [17]. Our procedure for handling discrete problems was suggested by that employed in the solution of the ‘traveling-salesman’ problem by Dantzig, Fulkerson, and Johnson. We have already mentioned that the 1954 TSP work of Dantzig et al. is viewed as the dawn of IP research. Their LP-approach to the TSP actually goes back a bit further, to the 1949 RAND report by Julia Robinson [23] and important follow-up studies in the early 1950s by Isidor Heller [8] and Harold Kuhn [9]. Robinson studied an algorithm for the assignment-problem relaxation of the TSP while Heller and Kuhn began investigations of linear descriptions of the convex hull of TSP tours, considering tours as characteristic vectors of their edge sets. In notes from a George Dantzig Memorial Lecture delivered in 2008 [10], Kuhn writes the following concerning his TSP study. We were both keenly aware of the fact that, although the complete set of faces (or constraints) in the linear programming formulation of Documenta Mathematica

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the Traveling Salesman Problem was enormous, if you could find an optimal solution to a relaxed problem with a subset of the faces that is a tour, then you had solved the underlying Traveling Salesman Problem. It is clear the researchers knew that LP relaxations could be a source of lower bounds for the TSP, but neither Heller nor Kuhn consider the bounding problem as a means to guide a search algorithm such as in branch and bound. In the case of Robinson’s work, it is tempting to read between the lines and speculate that she must have had some type of enumerative process (like branch and bound) in mind. Why else would she use the title “On the Hamiltonian game (a traveling salesman problem)” for a paper covering a solution method for the assignment problem? It is difficult to guess what she had in mind, but the introduction to the paper suggests she was trying for a direct solution to the TSP rather than an enumerative method through bounding. An unsuccessful attempt to solve the above problem led to the solution of the following . . . The “problem” in the quote is the TSP and the “following” is a description of the assignment problem. Thus, it appears that early TSP researchers had bounding techniques at their disposal, but were hopeful of direct solution methods rather than considering a branch-and-bound approach. Bounds and reduced-cost fixing by Dantzig-Fulkerson-Johnson Dantzig et al. began their study of the TSP in early 1954. Their successful solution of a 49-city instance stands as one of the great achievements of integer programming and combinatorial optimization. But the main body of work did not make use of the LP relaxation as a bounding mechanism. Indeed, the preliminary version [2] of their paper describes their process as follows, where C1 denotes the solution set of the LP relaxation, Tn denotes the convex hull of all tours through n cities, and dij is the cost of travel between city i and city j. What we do is this: Pick a tour x which looks good, and consider it as an extreme point of C1 ; use the simplex algorithm to move to an adjacent extreme point e in C1 which gives a smaller value of the functional; either e is a tour, in which case start again with this new tour, or there exists a hyperplane separating e from the convex of tours; in the latter case cut down C1 by one such hyperplane that passes through x, obtaining a new convex C2 with x as an extreme point. Starting with x again, repeat the process until a tour x ˆ and a convex C ⊃ T are obtained over which x ˆ gives a minimum of m n P dij xij .

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algorithm. Thus they do not have in hand a lower bound until the process has actually found the optimal TSP tour. In a second part of their paper, however, they work out a method that can take possibly infeasible values for the LP dual variables and create a lower bound B on the cost of an optimal tour. They accomplish this by taking advantage of the fact that the variables in the TSP relaxation are bounded between 0 and 1. The explicit variable bounds correspond to slack and surplus variables in the dual, allowing one to convert any set of dual values into a dual feasible solution by raising appropriately either the slack or surplus for each dual constraint. Dantzig et al. use this lower bound to eliminate variables from the problem by reduced-cost fixing, that is, when the reduced cost of a variable is greater than the difference between the cost of a best known tour and the value of B then the variable can be eliminated. During the early stages of the computation, E may be quite large and very few links can be dropped by this rule; however, in the latter stages often so many links are eliminated that one can list all possible tours that use the remaining admissible links. A general method for carrying out this enumeration of tours is not given, but in [4] an example is used to describe a possible scheme, relying on forbidding subtours. Their description is not a proper branch-and-bound algorithm, however, since the bounding mechanism is not applied recursively to the examined subproblems. Nonetheless, it had a direct influence on Dantzig et al.’s RAND colleagues Markowitz and Manne. 2

Eastman (1958)

It is in the realm of the TSP where we find the first explicit description of a branch-and-bound algorithm, namely Eastman’s 1958 Ph.D. thesis. The algorithm is designed for small instances of the asymmetric TSP, that is, the travel cost between cities i and j depends on the direction of travel, either from i to j or from j to i. The problem can thus be viewed as finding a minimum cost directed circuit that visits each city. In Eastman’s algorithm, the lower bound on the cost of a TSP tour is provided by the solution to a variant of the assignment problem that provides a minimum cost collection of circuits such that each city is in exactly one of the circuits in the collection. If there is only one circuit in the collection, then the assignment problem solves the TSP. Otherwise, Eastman chooses one of the circuits having, say, m edges, then in a branching step he creates m new subproblems by setting to 0, one at a time, each of the variables corresponding to the edges in the circuit. Eastman describes and illustrates his process as a search tree, where the nodes of the tree are the subproblems. Documenta Mathematica

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Willard Eastman (Photograph courtesy of Willard Eastman)

This process can be illustrated by a tree in which nodes correspond to solutions and branches to excluded links. The initial solution (optimal for the unrestricted assignment problem) forms the base of the tree, node 1. Extending from this node are m branches, corresponding to the m excluded links, and leading to m new nodes. Extending from each of these are more branches, corresponding to links excluded from these solutions, and so forth. This is very similar to how branch-and-bound search is usually viewed today: we speak of the size of the search tree, the number of active tree nodes, etc. Eastman clearly has a full branch-and-bound algorithm for the TSP and he illustrates its operation on a ten-city example. He also applies his framework to other combinatorial problems, including a transportation model with nonlinear costs and a machine-scheduling model. His work does not include general integer programming, but it is an important presentation of branch-and-bound techniques. 3

Land and Doig (1960)

General mixed integer programming, where only some of the variables are required to take on integer values, is the domain of Land and Doig. Their branchand-bound paper played a large role in the rapid rise of mixed IP as an applied tool in the 1960s and 70s. Documenta Mathematica

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Left: Ailsa Land, Banff, 1977 (Photograph courtesy of Ailsa Land). Right: Alison Doig, The Sun, October 21, 1965. (Courtesy of Alison (Doig) Harcourt) The methods of Markowitz-Manne and Land-Doig are on opposite sides of the algorithmic spectrum: whereas Markowitz-Manne is best viewed as a flexible proof system, Land-Doig is a detailed algorithm designed for immediate implementation. In a memoir [13] published in 2010, Land and Doig write the following. We were very well aware that the solution of this type of problem required electronic computation, but unfortunately LSE at that time did not have any access to such a facility. However, we had no doubt that using the same approach to computing could be achieved, if rather painfully, on desk computers, which were plentifully available. We became quite skillful at doing vector operations by multiplying with the left hand, and adding and subtracting with the right on another machine! Storage of bases and intermediate results did not present a limitation since it was all simply recorded on paper and kept in a folder. The reference to “bases” is indicative of the details given in the paper: the description of the general flow of the algorithm is intertwined with its implementation via the simplex algorithm, where the variables taking on fractional values in a solution are known to lie within the set of basic variables in the final simplex iteration. The Land-Doig algorithm follows the quick outline for IP branch and bound we mentioned in the introduction to this article: use the LP relaxation as a bounding mechanism and a fractional-valued variable as the means to create subproblems. The algorithm differs, however, in the manner in which it searches the space of solutions. Indeed, Land-Doig considers subproblems created with equality constraints xi = k, rather than inequality constraints, at the expense of possibly building a search tree with nodes having more than two child nodes, Documenta Mathematica

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that is, corresponding to a range of potential integer values k for the branching variable xi . Besides the nicely automated method, a striking thing about the paper is the computational tenacity of the authors. Although they worked with hand calculators, Land and Doig explored numerous disciplines for running their algorithm, including a variable selection rule that is similar in spirit to current “strong-branching” techniques. Land was also involved in very early work on the TSP, writing a paper with George Morton in 1955 [19], but combinatorial problems are not considered in the Land-Doig paper. In an email letter from June 9, 2012, Land confirmed that at the time she had not considered the application of branch and bound to the TSP. I only got involved in applying B&B to the TSP when Takis Miliotis was doing his PhD under my supervision. The thesis work of Miliotis [18] was carried out in the 1970s and Land herself authored a computational TSP paper in 1979 [11], but there is no direct connection between Eastman’s work at Harvard and the Land-Doig algorithm for general integer programming. 4

Coining the term branch and bound

The concise and descriptive name “branch and bound” has likely played a role in unifying the many diverse implementations of the algorithmic framework. On this point, however, our three pioneering teams cannot take credit. Markowitz and Manne modestly refer to their process as “a general approach” or “our method”. Eastman called his algorithm “the method of link exclusion” in reference to the fact that his branches are determined by excluding certain edges, that is, by setting the corresponding variables to the value zero. Land and Doig provide the following discussion of their procedure’s name [13]. We did not initially think of the method as ‘branch and bound’, but rather in the ‘geometrical’ interpretation of exploring the convex feasible region defined by the LP constraints. We are not sure if ‘branch and bound’ was already in the literature, but, if so, it had not occurred to us to use that name. We remember Steven Vajda telling us that he had met some French people solving ILPs by ‘Lawndwa’, and realizing that they were applying a French pronunciation to ‘Land-Doig’, so we don’t think they knew it as branch and bound either. It was John Little, Katta Murty, Dura Sweeney, and Caroline Karel who in 1963 coined the now familiar term. Here are the opening lines from the abstract to their TSP paper [15]. Documenta Mathematica

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William Cook A ‘branch and bound’ algorithm is presented for solving the traveling salesman problem. The set of all tours (feasible solutions) is broken up into increasingly small subsets by a procedure called branching. For each subset a lower bound on the length of the tours therein is calculated. Eventually, a subset is found that contains a single tour whose length is less than or equal to some lower bound for every tour.

In a recent note [20], Murty further pinpointed the naming of the algorithm, giving credit to his coauthor Sweeney. Later in correspondence John Little told me that one of his students at MIT, D. Sweeney, suggested the name “Branch and Bound” for the method . . . So while the origin of the algorithm is complicated, the origin of the name is at least clear! 5

Branch-and-cut algorithms

The Markowitz-Manne framework includes the idea of improving an LP relaxation Lk (s) of a subproblem by the addition of linear inequalities satisfied by all solutions in Dk (s). This incorporates into branch and bound the technique that was so successful in the Dantzig et al. TSP study. In fact, the MarkowitzManne paper may contain the first published use of the term “cutting plane” to refer to such valid linear inequalities. We refer to (3.7) as a cutting line (when N > 2, a cutting plane). Cutting planes, of course, appear in the starring role in the 1958 integerprogramming algorithm of Ralph Gomory [6], but the idea did not work its way into the Land-Doig computational procedure. Concerning this, Ailsa Land and Susan Powell [14] make the following remark in a 2007 paper. While branch and bound began to be built into computer codes, the cutting plane approach was obviously more elegant, and we spent a great deal of time experimenting with it. (. . . ) Work was done, but it was not published because as a method to solve problems branch and bound resoundingly won. The combination of branch-and-bound and cutting planes, as outlined in Markowitz-Manne, eventually became the dominant solution procedure in integer programming and combinatorial optimization. The first big successes were the 1984 study of the linear-ordering problem by Martin Gr¨otschel, Michael J¨ unger, and Gerhard Reinelt [7] and the late 1980s TSP work by Manfred Padberg and Giovanni Rinaldi [21, 22], It was Padberg and Rinaldi who coined the term branch and cut for the powerful combination of the two competing algorithms. Land and Powell conclude Documenta Mathematica

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their 2007 paper with the fitting statement “It is gratifying that the combination, ‘branch and cut’, is now often successful in dealing with real problems.” References [1] Dantzig, G. B. 1957. Discrete-variable extremum problems. Operations Research 5, 266–277. [2] Dantzig, G., R. Fulkerson, S. Johnson. 1954. Solution of a large scale traveling salesman problem. Technical Report P-510. RAND Corporation, Santa Monica, California, USA. [3] Dantzig, G., R. Fulkerson, S. Johnson. 1954. Solution of a large-scale traveling-salesman problem. Operations Research 2, 393–410. [4] Dantzig, G. B., D. R. Fulkerson, S. M. Johnson. 1959. On a linearprogramming, combinatorial approach to the traveling-salesman problem. Operations Research 7, 58–66. [5] Eastman, W. L. 1958. Linear Programming with Pattern Constraints. Ph.D. Thesis. Department of Economics, Harvard University, Cambridge, Massachusetts, USA. [6] Gomory, R. E. 1958. Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society 64, 275–278. [7] Gr¨ otschel, M., M. J¨ unger, G. Reinelt. 1984. A cutting plane algorithm for the linear ordering problem. Operations Research 32, 1195–1220. [8] Heller, I. 1953. On the problem of the shortest path between points. I. Abstract 664t, Bulletin of the American Mathematical Society 59, 551. [9] Kuhn, H. W. 1955. On certain convex polyhedra. Abstract 799t, Bulletin of the American Mathematical Society 61, 557–558. [10] Kuhn, H. W. 2008. 57 years of close encounters with George. George Dantzig Memorial Site. INFORMS. Available at http://www2.informs. org/History/dantzig/articles_kuhn.html. [11] Land, A. 1979. The solution of some 100-city travelling salesman problems. Technical Report. London School of Economics, London, UK. [12] Land, A. H., A. G. Doig. 1960. An automatic method of solving discrete programming problems. Econometrica 28, 497–520. [13] Land, A. H., A. G. Doig. 2010. Introduction to An automatic method of solving discrete programming problems. In: J¨ unger et al., eds. 50 Years of Integer Programming 1958–2008. Springer, Berlin. 105–106. Documenta Mathematica

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[14] Land, A. H., S. Powell. 2007. A survey of the operational use of ILP models. K. Spielberg, M. Guignard-Spielberg, eds. History of Integer Programming: Distinguished Personal Notes and Reminiscences. Annals of Operations Research 149, 147–156. [15] Little, J. D. C., K. G. Murty, D. W. Sweeney, C. Karel. 1963. An algorithm for the traveling salesman problem. Operations Research 11, 972–989. [16] Manne, A. S., H. M. Markowitz. 1956. On the solution of discrete programming problems. Technical Report P-711. RAND Corporation, Santa Monica, California, USA. [17] Markowitz, H. M., A. S. Manne. On the solution of discrete programming problems. Econometrica 25, 84–110. [18] Miliotis, P. 1978. Using cutting planes to solve the symmetric travelling salesman problem. Mathematical Programming 15, 177–188. [19] Morton, G., A. H. Land. 1955. A contribution to the ‘travelling-salesman’ problem. Journal of the Royal Statistical Society, Series B, 17, 185–194. [20] Murty, K. G. 2012. The branch and bound approach: a personal account. Available at http://www-personal.umich.edu/~murty/B&BHistory.pdf. [21] Padberg, M., G. Rinaldi. 1987. Optimization of a 532-city symmetric traveling salesman problem by branch and cut. Operations Research Letters 6, 1–7. [22] Padberg, M., G. Rinaldi. 1991. A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Review 33, 60–100. [23] Robinson, J. 1949. On the Hamiltonian game (a traveling salesman problem). Research Memorandum RM-303. RAND Corporation, Santa Monica, California, USA.

William Cook School of Industrial and Systems Engineering Georgia Institute of Technology USA [emailprotected]

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Abstract. This chapter highlights fundamental contributions made by Ron Graham in the area of online optimization. In an online problem relevant input data is not completely known in advance but instead arrives incrementally over time. In two seminal papers on scheduling published in the 1960s, Ron Graham introduced the concept of worst-case approximation that allows one to evaluate solutions computed online. The concept became especially popular when the term competitive analysis was coined about 20 years later. The framework of approximation guarantees and competitive performance has been used in thousands of research papers in order to analyze (online) optimization problems in numerous applications. 2010 Mathematics Subject Classification: 68M20, 68Q25, 68R99, 90B35 Keywords and Phrases: Scheduling, makespan minimization, algorithm, competitive analysis

An architect of discrete mathematics Born in 1935, Ron Graham entered university at age 15. Already at that time he was interested in a career in research. He first enrolled at the University of Chicago but later transferred to the University of California at Berkeley, where he majored in electrical engineering. During a four-year Air Force service in Alaska he completed a B.S. degree in physics at the University of Alaska, Fairbanks, in 1958. He moved back to the University of California at Berkeley where he was awarded a M.S. and a Ph.D. degree in mathematics in 1961 and 1962, respectively. Immediately after graduation Ron Graham joined Bell Labs. Some friends were afraid that this could be the end of his research but, on the contrary, he built the labs into a world-class center of research in discrete mathematics and theoretical computer science. Ron Graham rose from Member of Technical Staff to Department Head and finally to Director of the Mathematics Center Documenta Mathematica

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Figure 1: Ron Graham at work and at leisure. Pictures taken in New Jersey in the late 1060s and mid 1970s, respectively. Printed with the permission of Ron Graham. at Bell Labs. After establishment of AT&T Labs Research he served as the first Vice President of the Information Sciences Research Lab and later became the first Chief Scientist of AT&T Labs. After 37 years of dedicated service he retired from AT&T in 1999. Since then he has held the Jacobs Endowed Chair of Computer and Information Science at the University of California at San Diego. Ron Graham is a brilliant mathematician. He has done outstanding work in Ramsey Theory, quasi-randomness, the theory of scheduling and packing and, last not least, computational geometry. The “Graham scan” algorithm for computing the convex hull of a finite set of points in the plane is standard material in algorithms courses. His creativity and productivity are witnessed by more than 300 papers and five books. Ron Graham was a very close friend of Paul Erd˝ os and allowed to look not only after his mathematical papers but also his income. Together they have published almost 30 articles. Ron Graham is listed in the Guinness Book of Records for the use of the largest number that ever appeared in a mathematical proof. He has many interests outside mathematics and, in particular, a passion for juggling. It is worth noting that he served not only as President of the American Mathematical Society but also as President of the International Jugglers’ Association. Ron Graham has received numerous awards. He was one of the first recipients of the P´ olya Prize awarded by the Society for Industrial and Applied Mathematics. In 2003 he won the Steele Prize for Lifetime Achievement awarded by the American Mathematical Society. The citation credits Ron Graham as “one of the principle architects of the rapid development worldwide of discrete mathematics in recent years” [2]. Documenta Mathematica

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Scheduling and performance guarantees The technical results presented in this chapter arose from extensive research on scheduling theory conducted at Bell Labs in the mid 1960s. Even today they exhibit some remarkable features: (1) They can be perfectly used to teach the concepts of provably good algorithms and performance guarantees to nonspecialists, e.g., high school students or scientists from other disciplines. (2) The specific scheduling strategies are frequently used as subroutines to solve related scheduling problems. (3) The results stimulate ongoing research; some major problems are still unresolved. Consider a sequence σ = J1 , . . . , Jn of jobs that must be scheduled on m identical machines operating in parallel. Job Ji has a processing time of pi , 1 ≤ i ≤ n. The jobs of σ arrive one by one. Each job Ji has to be assigned immediately and irrevocably to one of the machines without knowledge of any future jobs Jk , k > i. Machines process jobs non-preemptively: Once a machine starts a job, this job is executed without interruption. The goal is to minimize the makespan, i.e. the maximum completion time of any job in the schedule constructed for σ. The scheduling problem defined above is an online optimization problem. The relevant input arrives incrementally. For each job Ji an algorithm has to make scheduling decisions not knowing any future jobs Jk with k > i. Despite this handicap, a strategy should construct good solutions. Graham [5] proposed a simple greedy algorithm. The algorithm is also called List scheduling, which refers to the fact that σ is a list of jobs. Algorithm List: Schedule each job Ji on a machine that currently has the smallest load. The load of a machine is the sum of the processing times of the jobs presently assigned to it. A natural question is, what is the quality of the solutions computed by List. Here Graham introduced the concept of worst-case approximation. For any job sequence σ, compare the makespan of the schedule constructed by List to that of an optimal schedule for σ. How large can this ratio grow, for any σ? Formally, let List(σ) denote the makespan of List’s schedule for σ. Furthermore, let OPT(σ) be the makespan of an optimal schedule for σ. We would like to determine List(σ) c = sup , σ OP T (σ) which gives a worst-case performance guarantee for List. In online optimization such a guarantee is called competitive ratio. Following Sleator and Tarjan [8], an online algorithm A is c-competitive if, for any input, the cost of the solution computed by A is at most c times that of an optimal solution for that input. Graham [5] gave an elegant proof that List is (2 − 1/m)-competitive, i.e. remarkably List achieves a small constant performance ratio. For the proof, fix an arbitrary job sequence σ and consider the schedule computed by List. Without Documenta Mathematica

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Machines 1

Ji0 0

List(σ)

Time

Figure 2: Analysis of List

loss of generality, number the machines in order of non-increasing load. Hence machine 1 is one having the highest load and defines the makespan. Figure 2 depicts an example. In the time interval [0, List(σ)) machine 1 continuously processes jobs. Any other machine j, 2 ≤ j ≤ m, first processes jobs and then may be idle for some time. Let Ji0 be the job scheduled last on machine 1. We observe that in List’s schedule Ji0 does not start later than the finishing time of any machine j, 2 ≤ j ≤ m, because List assigns each job to a least loaded machine. This implies that the idle time on any machine j, 2 ≤ j ≤ m, cannot be higher than pi0 , the processing time of Ji0 . Considering the time interval [0, List(σ)) on all the m machines we obtain n X

mList(σ) ≤

pi + (m − 1)pi0 .

i=1

Dividing by m and taking into account that pi0 ≤ max1≤i≤n pi , we obtain n

List(σ) ≤

1 1 X pi + (1 − ) max pi . m i=1 m 1≤i≤n

A final argument is that the optimum makespan OP T (σ) cannot be smaller Pn 1 than m p , which is the average load on the m machines. Moreover, i i=1 obviously OP T (σ) ≥ max1≤i≤n pi . We conclude that List(σ) ≤ OP T (σ) + (1 − 1/m)OP T (σ) = (2 − 1/m)OP T (σ). Graham [5] also showed that the above analysis is tight. List does not achieve a competitive ratio smaller than 2 − 1/m. Consider the specific job sequence σ consisting of m(m − 1) jobs of processing time 1 followed by a large job having a processing time of m. List distributes the small jobs evenly among the m machines so that the final job cause a makespan of m − 1 + m = 2m − 1. On the other hand the optimum makespan is m because an optimal schedule will reserve one machine for the large job and distribute the small jobs evenly among the remaining m − 1 machines. Figure 3 shows the schedules by List and OPT. The above nemesis job sequence motivated Graham to formulate a second algorithm. Obviously List’s performance can degrade if large jobs arrive at Documenta Mathematica

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m

m

Machines 1

1 Time 0

m−1

2m − 1

Time 0

m

Figure 3: The worst-case performance of List. Online schedule (left) and an optimal schedule (right).

the end of the input sequence. Why not sort the jobs initially? Graham [6] proposed a Sorted List algorithm that first sorts the jobs in order of nonincreasing processing time and then applies List scheduling. Of course Sorted List is not an online algorithm because the entire job sequence must be known and rearranged in advance. Graham [6] proved that Sorted List achieves a worst-case approximation ratio of 4/3 − 1/(3m). The analysis is more involved than that of List but the global idea can be described in one paragraph: Consider an arbitrary sorted job sequence σ and assume without loss of generality that the last job of σ defines Sorted List’s makespan. If this is not the case, then one can consider the prefix sequence σ ′ such that the last job of σ ′ defines Sorted List’s makespan for σ ′ and σ. It suffices to consider two cases. (1) If the last job Jn of σ has a processing time pn of at most OP T (σ)/3, then using the same arguments as above one can establish a performance factor of 4/3 − 1/(3m). (2) If pn > OP T (σ)/3, then all jobs of σ have a processing time greater than OP T (σ)/3. Hence in an optimal schedule each machine can contain at most two jobs and n ≤ 2m. Assume for simplicity n = 2m. One can show that there exists an optimal schedule that pairs the largest with the smallest job, the second largest with the second smallest job and so on. That is, the pairing on the m machines is (J1 , J2m ), (J2 , J2m−1 ), . . . , (Jm , Jm+1 ). If n = 2m − k, for some k ≥ 1, then there is an optimal schedule that is identical to the latter pairing except that J1 , . . . , Jk are not combined with any other job. Sorted List produces a schedule that is no worse than this optimal assignment, i.e., in this case the performance ratio is equal to 1. The above results led to a considerable body of further research. It was open for quite some time if online algorithms for makespan minimization can attain a competitive ratio smaller than 2 − 1/m. It turned out that this is indeed possible. Over the past 20 years the best competitiveness of deterministic online strategies was narrowed down to [1.88, 1.9201]. More specifically, there exists a deterministic online algorithm that is 1.9201-competitive, and no deterministic online strategy can attain a competitive ratio smaller than 1.88. If job preemption is allowed, i.e., the processing of a job may be stopped and resumed Documenta Mathematica

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later, the best competitiveness drops to e/(e − 1) ≈ 1.58. The book chapter [7] contains a good survey of results. During the last few years researchers have explored settings where an online algorithm is given extra information or ability to serve the job sequence. For instance, on online algorithm might be able to migrate a limited number of jobs or alternatively might know the total processing time of all jobs in σ. In these scenarios significantly improved performance guarantees can be achieved. Using limited job migration, the competitiveness reduces to approximately 1.46. The recent manuscript [1] points to literature for these extended problem settings. Nonetheless a major question is still unresolved. What is the best competitive ratio that can be achieved by randomized online algorithms? It is known that no randomized online strategy can attain a competitiveness smaller than e/(e − 1). However, despite considerable research interest, no randomized online algorithm that provably beats deterministic ones, for general m, has been devised so far. Finally, as mentioned above, the design and analysis of online algorithms has become a very active area of research. We refer the reader to two classical books [3, 4] in this field. References [1] S. Albers and M. Hellwig. On the value of job migration in online makespan minimization. Proc. 20th European Symposium on Algorithms, Springer LNCS 7501, 84–95, 2012. [2] AMS document about the 2003 Steele Prize. Accessible at http://en. wikipedia.org/wiki/Ronald_Graham. [3] A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, 1998. [4] A. Fiat and G.J. Woeginger (eds). Online Algorithms: The State of the Art. Springer LNCS 1442, 1998. [5] R.L. Graham. Bounds for certain multi-processing anomalies. Bell System Technical Journal, 45:1563–1581, 1966. [6] R.L. Graham. Bounds on multiprocessing timing anomalies. SIAM Journal of Applied Mathematics, 17(2):416–429, 1969. [7] K. Pruhs, J. Sgall and E. Torng. Online scheduling. Handbook on Scheduling, edited by J. Y-T. Leung. Chapman & Hall / CRC. Chapter 15, 2004. [8] D.D. Sleator and R.E. Tarjan. Amortized efficiency of list update and paging rules. Communications of the ACM, 28:202–208, 1985. Documenta Mathematica

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Ronald Graham: Foundations of Online Optimization Susanne Albers Department of Computer Science Humboldt-Universit¨at zu Berlin Unter den Linden 6 10099 Berlin Germany [emailprotected]

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Continuous Optimization Stories

Sometimes mathematicians coin terms and define relations between them that are “somewhat inconsistent”. Nonlinear programming is one such ill-defined term, since linear programming is considered a special case of nonlinear programming. Even someone not exceling in logic may find this strange. I therefore try to use continuous optimization instead of nonlinear programming, although I am aware that combinatorial optimization can be viewed as nonlinear programming but not necessarily as continuous optimization. Optimization may, in fact, be in need of a new consistent nomenclature. The Mathematical Programming Society has already made a small step by renaming itself into Mathematical Optimization Society in order to avoid confusions with computer programming. My original intention for the contents of this chapter was to highlight the contributions to optimization of mathematicians that have some relation to Berlin. Due to intensive discussions with potential authors, the section developed differently and now contains wonderful survey articles on a wide range of exciting developments in continuous optimization. It begins with the history of the gradient method, discusses the origins of the KKT theorem, the Nelder-Mead simplex algorithm, various aspects of subgradient techniques and nonsmooth optimization, updating techniques, the influence of the Cold War on the maximum principle, and the arrival of infinite-dimensional optimization. As the ISMP 2012 takes place in Berlin, I feel obliged, however, to provide at least some condensed information about mathematics and mathematicians who contributed to optimization and spent some time in Berlin. (I also use this opportunity to thank my wife for providing me with many of the details. She wrote a book [1], directed at a non-mathematical readership, that covers the history of all aspects of mathematics in Berlin.) We have already encountered Gottfried Wilhelm Leibniz. Mathematics in Berlin began with him. He initiated the foundation of the predecessor of what is today called Berlin-Brandenburgische Akademie der Wissenschaften (BBAW). The academy was officially opened in 1700 and has experienced many name changes in its more than 300 years of existence. Leibniz was the first academy president. Optimization would not exist without his development of calculus (there were other founders as well) and, in particular, his notational inventions. The integral sign is one of these. Documenta Mathematica

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Pierre Louis Moreau de Maupertuis came to Berlin in 1740, stayed for 17 years and was also president of the academy. Maupertuis developed a “principle of least action” that states that in all natural phenomena a quantity called ‘action’ tends to be minimized. His work was highly controversial, though. Leonhard Euler is the star of mathematics of the 18th century. Born in 1707 in Basel, he accepted an offer from the academy in St. Petersburg in 1727 and came to Berlin in 1741, he stayed until 1766 to return to St. Petersburg, where he died in 1783. Most of his gigantic mathematical production was carried out at the academy in Berlin. Another giant of optimization, Joseph Louis Lagrange, whose name is encoded in many terms of our field, spent 20 of his most productive years in Berlin. In 1766 he became Euler’s successor as the director of the mathematical class of the academy. Carl Gustav Jacobi, a mathematical household name, whom we encountered in this book in connection with the Hungarian method (Introduction to Discrete Optimization Stories), was born in 1804 in Potsdam, was privately tutored until the age of 12 and graduated at age 13. In 1821 he was allowed to start studying at Berlin University, passed his teacher examination at the age of 19 and obtained his PhD and habilitation in 1825. He became professor in K¨ onigsberg in 1826 and returned to Berlin in 1844 as a member of the academy. He died in 1851 in Berlin. Johann Peter Gustav Lejeune Dirichlet was mentioned in this book in the discussion of the LLL algorithm (Introduction to Linear Programming Stories). He was the first outstanding mathematician at Berlin University whose foundation in 1810 was initiated by Wilhelm von Humboldt. This university carried the name Friedrich-Wilhelms-Universit¨at from 1828 to 1945 and was renamed Humboldt-Universit¨ at in 1949, after the brothers Wilhelm and Alexander von Humboldt. Dirichlet was born in D¨ uren in 1805, came to Berlin in 1827 and stayed until 1855 when he accepted an offer from G¨ottingen to succeed Gauss. He died in 1859. Karl Theodor Weierstraß (1815–1897), also written Weierstrass, was one of the dominating figures of the 19th century mathematics in Berlin. He is known to every mathematician for bringing highest standards of rigor to analysis (e.g., the (ǫ, δ)-definition of continuity); many theorems carry his name. Every calculus student learns a result formulated by Weierstraß, namely, that every continuous function from a compact space to the real numbers attains its maximum and minimum. The Weierstraß Institut f¨ ur Angewandte Analysis und Stochastik is named after him. His grave is shown in Fig. 2. My wife and I live close to Waldfriedhof Heerstraße, a beautiful cemetery near the Olympic Stadium in Berlin. One day, my wife showed me the joint grave of Hermann Minkowski (1864–1909) and his brother Oskar (1858–1931). I was very astonished that the Minkowski brothers had an Ehrengrab (honorary grave maintained by the city of Berlin), see Fig. 1. I knew that Minkowski had studied in Berlin (under Weierstraß and Kummer) and had worked in K¨ onigsberg, Z¨ urich, and finally in G¨ottingen where he died. (Minkowski is Documenta Mathematica

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Figure 1: Minkowski’s grave c Iris Gr¨ ( otschel)

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Figure 2: Weierstrass’ grave c Iris Gr¨otschel) (

my academic great great grandfather.) Minkowski will forever be known as the person who coined the name spacetime, but for optimizers his work on convexity that arose via his studies of the geometry of numbers, an area he created, is of particular importance. This work is excellently surveyed in [2] and in chapter 0 (and several other chapters) of the handbook [3]. The idea to edit this book on optimization history, in fact, arose when my wife and I tried to find out more about Minkowski’s grave. One remark only: The city of Berlin decided on March 22, 1994 to declare the graves of Karl Weierstraß and Hermann Minkowski as honorary graves. Martin Gr¨otschel References [1] I. Gr¨ otschel, Das mathematische Berlin, Berlin Story Verlag, 2nd edition, 2011. [2] P. M. Gruber and J. M. Wills (eds.), Handbook of Convex Geometry, Vol. A and B, North Holland, 1993. [3] T. H. Kjeldsen, History of Convexity and Mathematical Programming: Connections and Relationships in Two Episodes of Research in Pure and Applied Mathematics of the 20th Century, in: R. Bhatia (ed.) et al., Proceedings of the International Congress of Mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. IV: Invited lectures, World Scientific, Hackensack; Hindustan Book Agency, New Delhi, 2011, pp. 3233–3257.

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Cauchy and the Gradient Method ´chal Claude Lemare

2010 Mathematics Subject Classification: 65K05, 90C30 Keywords and Phrases: Unconstrained optimization, descent method, least-square method Any textbook on nonlinear optimization mentions that the gradient method is due to Louis Augustin Cauchy, in his Compte Rendu ` a l’Acad´emie des Sciences of October 18, 18471 (needless to say, this reference takes a tiny place amongst his fundamental works on analysis, complex functions, mechanics, etc. Just have a look at http://mathdoc.emath.fr/cgi-bin/oetoc?id=OE_CAUCHY_1_ 10: a paper every week). Cauchy is motivated by astronomic calculations which, as everybody knows, are normally very voluminous. To compute the orbit of a heavenly body, he wants to solve not the differential equations, but the [algebraic] equations representing the motion of this body, taking as unknowns the elements of the orbit themselves. Then there are six such unknowns.2 . Indeed, a motivation related with operations research would have been extraordinary. Yet, it is interesting to note that equation-solving has always formed the vast majority of optimization problems, until not too long ago. To solve a system of equations in those days, one ordinarily starts by reducing them to a single one by successive eliminations, to eventually solve for good the resulting equation, if possible. But it is important to observe that 1◦ in many cases, the elimination cannot be performed in any way; 2◦ the resulting equation is usually very complicated, even though the given equations are rather simple.3 Something else is wanted. Thus consider a function u = f (x, y, z, . . .) 1 “M´ ethode

g´ en´ erale pour la r´ esolution des syst` emes d’´ equations simultan´ ees” plus aux ´ equations diff´rentielles, mais aux ´ equations finies qui repr´ esentent le mouvement de cet astre, et en prenant pour inconnues les ´ el´ ements mˆ emes de l’orbite. Alors les inconnues sont au nombre de six. 3 on commence ordinairement par les r´ eduire ` a une seule, ` a l’aide d’´ eliminations successives, sauf ` a r´ esoudre d´ efinitivement, s’il se peut, l’´ equation r´ esultante. Mais il importe d’observer, 1◦ que, dans un grand nombre de cas, l’´ elimination ne peut s’effectuer en aucune mani` ere ; 2◦ que l’´ equation r´ esultante est g´ en´ eralement tr` es-compliqu´ ee, lors mˆ eme que les ´ equations donn´ ees sont assez simples. 2 non

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Augustin Louis Cauchy, 1789–1857 (Wikimedia, Cauchy Dibner-Collection Smithsonian Inst.) of several variables, which never becomes negative, and stays continuous. To find the values of x, y, z, . . . satisfying the equation u = 0, it will suffice to let indefinitely decrease the function u, until it vanishes.4 Start from particular values x, y, z, . . . of the variables x, y, z; call u the corresponding value of u and X = fx′ , Y = fy′ , Z = fz′ , . . . the derivatives.5 Let α, β, γ, . . . be small increments given to the particular values x, y, z, . . .; then there holds approximately f (x + α, y + β, z + γ, · · · ) = u + Xα + Yβ + Zγ + · · · . Taking θ > 0 and α = −θX, β = −θY, γ = −θZ, . . . , we obtain approximately f (x − θX, y − θY, z − θZ, . . .) = u − θ(X2 + Y2 + Z2 + · · · ) . 4 Pour

(1)

trouver les valeurs de x, y, z, . . ., qui v´ erifieront l’´ equation u = 0, il suffira de faire d´ ecroˆıtre ind´ efiniment la fonction u, jusqu’` a ce qu’elle s’´ evanouisse. 5 Already in those times, one carefully distinguishes a function from a value of this function. Observe also that Cauchy cares about continuity but not differentiability . . . Documenta Mathematica

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It is easy to conclude that the value Θ of u, given by the formula Θ = f (x − θX, y − θY, z − θZ, . . .)

(2)

will become smaller than u if θ is small enough. If, now, θ increases and if, as we assumed, the function f (x, y, z, · · · ) is continuous, the value Θ of u will decrease until it vanishes, or at least until it coincides with a minimal value, given by the univariate equation6 Θ′θ = 0 .

(3)

One iteration of the gradient method is thus stated, with two variants: (2) (Armijo-type line-search) or (3) (steepest descent). A third variant, valid when u is already small, is defined by equating (1) to 0: θ=

X2

+

Y2

u . + Z2 + · · ·

Other remark: when good approximate values are already obtained, one may switch to Newton’s method. Finally, for a system of simultaneous equations u = 0, v = 0, w = 0, . . . , just apply the same idea to the single equation7 u2 + v 2 + w 2 + · · · = 0 .

(4)

Convergence is just sloppily mentioned: If the new value of u is not a minimum, one can deduce, again proceeding in the same way, a third value still smaller; and, so continuing, smaller and smaller values of u will be found, which will converge to a minimal value of u. If our function u, assumed not to take negative values, does take null values, these will always be obtained by the above method, provided that the values x, y, z, . . . are suitably chosen.8 According to his last words, Cauchy does not seem to believe that the method always finds a solution; yet, he also seems to hope it: see the excerpt of footnote 4. Anyway a simple picture reveals that the least-squares function in (4) 6 Il est ais´ e d’en conclure que la valeur Θ de u d´ etermin´ ee par la formule (2), deviendra inf´ erieure ` a u, si θ est suffisamment petit. Si, maintenant, θ vient ` a croˆıtre, et si, comme nous l’avons suppos´ e, la fonction f (x, y, z, . . .) est continue, la valeur Θ de u d´ ecroˆıtra jusqu’` a ce qu’elle s’´ evanouisse, ou du moins jusqu’` a ce qu’elle co¨ıncide avec une valeur minimum, d´ etermin´ ee par l’´ equation ` a une seule inconnue (3). 7 Here we have an additional proposal: least squares, which is some 50 years old. Incidentally, its paternity provoked a dispute between Legendre and Gauss (who peremptorily concluded: I did not imagine that Mr Legendre could feel so strongly about such a simple idea; one should rather wonder that nobody had it 100 years earlier ). 8 Si la nouvelle valeur de u n’est pas un minimum, on pourra en d´ eduire, en op´ erant toujours de la mˆ eme mani` ere, une troisi` eme valeur plus petite encore ; et, en continuant ainsi, on trouvera successivement des valeurs de u[sic] de plus en plus petites, qui convergeront vers une valeur minimum de u[sic]. Si la fonction u, qui est suppos´ ee ne point admettre de valeurs n´ egatives, offre des valeurs nulles, elles pourront toujours ˆ etre d´ etermin´ ees par la m´ ethode pr´ ec´ edente, pouru que l’on choisisse convenablement les valeurs de x, y, z, . . ..

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may display positive local minima, playing the role of “parasitic” solutions. On the other hand, he seems convinced that, being decreasing, the sequence of u-values has to converge to a (local) minimum, or at least a stationary point. Thus, the above excerpt is fairly interesting, coming from a mathematician among the most rigorous of his century. Admittedly, Cauchy has not given deep thought to the problem: I’ll restrict myself here to outlining the principles underlying [my method ], with the intention to come again over the same subject, in a paper to follow.9 However, the “paper to follow” does not seem to exist. Let us bet that he has underestimated the difficulty and eventually not been able to crack this tough nut. In fact, we are now aware that some form of uniformity is required from the objective’s continuity – not mentioning the choice of a “small enough” θ, which is also delicate. References [1] A. Cauchy. M´ethode g´en´erale pour la r´esolution des syst`emes d’´equations simultan´ees. C. R. Acad. Sci. Paris, 25:536–538, 1847.

Claude Lemar´echal INRIA 655 avenue de l’Europe Montbonnot 38334 Saint Ismier France [emailprotected]

9 Je me bornerai pour l’instant ` a indiquer les principes sur lesquels elle se fonde, me proposant de revenir avec plus de d´ etails sur le mˆ eme sujet, dans un prochain M´ emoire.

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William Karush and the KKT Theorem

Richard W. Cottle

2010 Mathematics Subject Classification: 01, 90, 49 Keywords and Phrases: Biography, nonlinear programming, calculus of variations, optimality conditions

1

Prologue

This chapter is mainly about William Karush and his role in the Karush-KuhnTucker theorem of nonlinear programming. It tells the story of fundamental optimization results that he obtained in his master’s thesis: results that he neither published nor advertised and that were later independently rediscovered and published by Harold W. Kuhn and Albert W. Tucker. The principal result – which concerns necessary conditions of optimality in the problem of minimizing a function of several variables constrained by inequalities – first became known as the Kuhn–Tucker theorem. Years later, when awareness of Karush’s pioneering work spread, his name was adjoined to the name of the theorem where it remains to this day. Still, the recognition of Karush’s discovery of this key result left two questions unanswered: why was the thesis not published? and why did he remain silent on the priority issue? After learning of the thesis work, Harold Kuhn wrote to Karush stating his intention to set the record straight on the matter of priority, and he did so soon thereafter. In his letter to Karush, Kuhn posed these two questions, and Karush answered them in his reply. These two letters are quoted below. Although there had long been optimization problems calling for the maximization or minimization of functions of several variables subject to constraints, it took the advent of linear programming to inspire the name “nonlinear programming.” This term was first used as the title of a paper [30] by Harold W. Kuhn and Albert W. Tucker. Appearing in 1951, the paper contained many results, but interest focused on the one declaring conditions that must be satisfied by a solution of the Maximum Problem. To find an x0 that maximizes g(x) constrained by F x ≥ 0, x ≥ 0. Documenta Mathematica

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In this formulation of the problem, F x denotes a mapping from Rn to Rm with component functions fi , i = 1, . . . , m. The function g and the fi were all assumed to be differentiable. A further assumption was immediately imposed. Kuhn and Tucker called it the constraint qualification. The precise statement of the Kuhn-Tucker constraint qualification is somewhat complicated, but it’s purpose is easy enough to understand. It is used in assuring the existence of the nonnegative Lagrange multipliers, u1 , . . . , um , which appear in the theorem statement. A simpler constraint qualification is the condition that the gradients of the active constraints at x0 be linearly independent. Citing a paper of Fritz John [16] at this point, Kuhn and Tucker then went ahead and constructed the Lagrangian function φ(x, u) = g(x) + u′ F x in which u denotes a vector of nonnegative Lagrange multipliers. With these assumptions in place, and the symbols φ0x and φ0u denoting the partial gradients of φ at (x0 , u0 ) with respect to x and u, their result was Theorem 1. In order that x0 be a solution of the maximum problem, it is necessary that x0 and some u0 satisfy conditions φ0x ≤ 0,

0 φ0′ x x = 0,

x0 ≥ 0

(1)

φ0u ≥ 0,

0 φ0′ u u = 0,

u0 ≥ 0

(2)

′

for φ(x, u) = g(x) + u F x. The equations and inequalities stated in (1) and (2) became known as the Kuhn–Tucker conditions for the stated maximum problem while the result itself became known as the Kuhn–Tucker theorem. Unbeknownst to Kuhn and Tucker, their theorem and several others in their paper had been establlshed in 1939 by William Karush in his master’s degree thesis [18]. At that time, Karush was a graduate student at the University of Chicago mathematics department which was noted for its preoccupation with a topic called the calculus of variations. The fundamental problem in the calculus of variations is to find a function ϕ(x) belonging to an admissible set of functions that minimizes the integral Z X¯ I= F x, ϕ(x), ϕ′ (x) dx (3) X

¯ Y¯ with X < X ¯ are given real numbers, such that ϕ(X) = Y , where X, Y, X, ¯ ¯ ϕ(X) = Y , and F (x, y, z) is a given function of three independent variables. With each admissible function ϕ(x) there is an associated real number I. Accordingly, when ϕ is regarded as an independent variable, I is a functional: a numerical-valued function of ϕ. (See Pars [34].) Much of the research in the calculus of variations concentrated on necessary and sufficient conditions for relative minima in (specializations of) these problems. Karush’s master’s thesis dealt with a truly finite-dimensional version Documenta Mathematica

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of this class of problems. He called the work “Minima of Functions of Several Variables with Inequalities as Side Conditions.” In stating the problems he proposed to analyze, Karush first made reference to those of the familiar Lagrangian type where a point x = (x1 , x2 , . . . , xn ) satisying a system of equations gα (x) = 0 (α = 1, 2, . . . , m) is to be found so as to minimize a given function f (x1 , x2 , . . . , xn ). Saying that the necessary and sufficient conditions for a relative minimum in this equality-constrained minimization problem had already been satisfactorily treated, Karush then announced This paper proposes to take up the corresponding problem in the class of points x satisfying the inequalities gα (x) ≥ 0

(α = 1, 2, . . . , m)

where m may be less than, equal to, or greater than n. Karush’s minimization problem is clearly one of nonlinear programming in the sense of Kuhn and Tucker. It takes only a little bit of elementary manipulation and notation changing to cast the Kuhn–Tucker maximization problem in the form of a minimization problem studied by Karush. One slight (and insignificant) difference between the two papers is that Karush seems to assume his functions are of class C 1 (or C 2 for second-order results). The precursor of (Kuhn and Tucker’s) Theorem 1 appears in Karush’s thesis as Theorem 3.2. Both the Kuhn–Tucker paper and the Karush paper point out the importance of the gradients of the active constraints (those satisfied as equations) at a relative maximum or minimum, respectively. Both papers make use of the notion of admissible arcs, both papers make use of linear inequality theory (even Farkas’s lemma), and both papers address the need for a constraint qualification. Where the papers differ is that the Kuhn–Tucker paper was published and Karush’s was not submitted for publication. Instead, it remained almost totally unknown for close to 30 years. This article tells more of the story about William Karush, his master’s thesis, and its place in optimization. 2

Introduction

For roughly four decades, the result originally known as the Kuhn–Tucker (KT) Theorem has been called the Karush-Kuhn–Tucker (KKT) Theorem in recognition of the fact that in 1939 William Karush had produced the same result in his Master of Science degree thesis [18] at the mathematics department of the University of Chicago.1 The Kuhn–Tucker paper [30] containing the eponymous theorem was published in 1951 having been presented the preceding year 1 Actually, both the thesis and the KT paper contain separate theorems on first-order and second-order necessary conditions and sufficient conditions for local optimality.

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at the Symposium on Mathematical Statistics and Probability held at the University of California, Berkeley. Nearly every textbook covering nonlinear programming relates this fact but gives no more information than what is stated above. There are, however, publications that give a much more specific account of this history. For instance, Harold Kuhn (coauthor of the Kuhn–Tucker paper [30]) has written at least three others [27], [28], and [29] in which he “sets the record straight” about the earlier work by Karush in his master’s thesis. In these three articles2 Kuhn relates that he first became aware of Karush’s earlier work from Akira Takayama’s 1974 monograph Mathematical Economics [36]. Kuhn has much more to say than just this. He gives a brief overview of publications prior to 1974 that cite the work of Karush. These include Pennisi [35], El-Hodiri [10], [11], and Fiacco and McCormick [13]. Both Takayama [36, pages 61 and 101], [37, pages 65 and 105], and Kuhn [27, pp. 10–11] present the key points regarding literature that very well could have influenced Karush. Moreover, it is worth reiterating a point already plain by Kuhn: namely, that Karush’s MS thesis also contains what we know today as Fritz John’s Theorem, a result that appeared in a 1948 paper [16] later cited by Kuhn and Tucker [30] but not actually declared there because it was inserted when the paper was in galley proof. John makes no mention of Karush’s work even though his research might be viewed as close to the mathematical school of thought from which Karush emerged. Kuhn [27, p. 15] tells the interesting story of John’s experience in the process of attempting to publish his paper. The three cited papers by Kuhn are very informative, yet somewhat limited in scope. There is more to say on how Takayama became aware of Karush’s Master of Science thesis – and about the thesis itself. I am grateful to Professor Kuhn for introducing me to the writings of Professor Tinne Hoff Kjeldsen, a professor of mathematics and historian of mathematical science at the University of Roskilde in Roskilde, Denmark. I wrote to her at once. She soon replied and kindly sent me a batch of her papers [23], [24], [25], and [26] on this subject. For most people, the most easily found of these papers is certain to be the rewarding journal article [24]. Professor Kjeldsen provided something else of enormous interest. In February 1975, as Harold Kuhn was preparing for his first historic effort to set the priority record straight, he wrote to William Karush announcing this intention. Copies of their correspondence were given to Kjeldsen when she visited Kuhn at Princeton to gather information for her doctoral dissertation. In 2012, when I came along requesting copies of this correspondence, they were no longer in Kuhn’s possession, having been discarded in the process of vacating his mathematics department office at Princeton. Fortunately, Professor Kjeldsen had copies of this valuable correspondence and graciously shared them with me. On March 7, 2012 I returned them (electronically) to Professor Kuhn. Among 2 Except

for their typesetting method and their Introductions, the first two of these articles are very much alike; the third is more autobiographical in nature. Here, for reasons of brevity and historical precedence, the earliest one [27] will be used for most citations. Documenta Mathematica

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other things, this correspondence addresses two questions that virtually all observers would ask: why didn’t Karush publish his MS thesis and why didn’t he make its existence known after the appearance of the Kuhn–Tucker paper, some 11 or 12 years later? Kuhn covers the main facts on this story in [27]. Karush’s answers to these and other questions from Kuhn are revealed below.3 What else does this chapter have to offer? In light of the widely known and available literature on nonlinear programming and the herein repeatedly cited historical papers by Kuhn and Kjeldsen, it seems unnecessary to spell out all the Karush-Kuhn–Tucker theorems with an analysis of whose paper had what, especially because Kuhn has so usefully reproduced the similar content of Karush’s thesis in [27]. And because the published Kuhn–Tucker paper can be found in many university libraries as well as online at https:// projecteuclid.org, I have chosen to concentrate on a few other aspects of Karush’s MS thesis. To obtain a proper appreciation of this work, one must consider it as a product of the milieu in which it was created, namely the research of the University of Chicago mathematicians devoted to the calculus of variations. Some of this has been done in [36], [27], and [24]. In truth, the exposition given here is much briefer than it could be. Quite a lot has been written about the careers of Harold W. Kuhn and Albert W. Tucker (see, for example, [24, p. 342], [2, Chapters 29 and 6], and a multitude of web sites including [38]), what then remains to be given is a bio-sketch of William Karush. Even this can be found on the web, but primarily in thumbnail form. The bio-sketch of Karush in this paper includes his image (which cannot ordinarily be seen elsewhere). As a bonus, the paper also exhibits an image of Fritz John (one can be found on the web). While both the biographical information and the concluding reference list provided here are necessarily condensed, they may prove to be the main contributions of this article and provide an incentive to explore this subject in greater depth. 3

On Karush’s Master’s Thesis

Dated December, 1939, the body of William Karush’s master’s thesis is a 25page document centered between two pages of front matter (the title page and table of contents) and two pages of back matter (the list of references and a half-page vita). In the vita Karush provides information on his date and place of birth, his prior education, and the (sur)names of ten faculty members under whom he studied at the University of Chicago. He acknowledges them all for “the helpful part they played in his mathematical development” and then singles out Professor Lawrence M. Graves, thanking him “for his guidance as a teacher and in the writing of this dissertation.” The work is composed of six sections, of which the first is an introduction to the class of problems under investigation, and the second presents preliminary results on systems of linear inequalities (about eight pages in all). The remaining four sections take up 3 Kjeldsen

[24, pp. 337–338] quotes a portion of this correspondence as well.

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necessary conditions and sufficient conditions involving only first derivatives and then the same issues involving second derivatives. Karush’s results are given in the Appendix of Kuhn’s paper [27]. Not given, however, is Karush’s list of references. The following is a replica thereof. LIST OF REFERENCES 1.

Bliss, G. A., Normality and Abnormality in the Calculus of Variations, Transactions of the American Mathematical Society, vol. 43 (1938), pp. 365-376.

2.

Dines, L. L., Systems of Linear Inequalities, Annals of Mathematics, vol. 23 (1922), p. 212.

3.

Dines and McCoy, On Linear Inequalities, Transactions of the Royal Society of Canada, vol. 27 (1933), pp. 37-70.

4.

Farkas, J. I., Theorie der einfachen Ungleichungen, Crelle, vol. 124 (1902), p. 1.

Stylistic inconsistency aside, three aspects of this list are peculiar. The first is that it contains only one publication from the calculus of variations. To a slight extent, this topic will be discussed in another section of this article. The second is that W.B. Carver, not L.L. Dines, is the author of the paper listed as Reference 2. The third (very minor) oddity is the insertion of a middle initial on the name of Farkas. His forename is given as “Julius” on the original German language paper, though in his native Hungary it would have been “Gyorgy.” And speaking of names, “Crelle” is a common nickname used for “Journal f¨ ur die reine und angewandte Mathematik” which in 1826 was founded and edited by August Leopold Crelle in Berlin. As stated above, the questions of why the thesis was not published and why its author remained silent on the subject after the publication of the Kuhn– Tucker paper were discussed in very cordial correspondence between Harold Kuhn and William Karush. I now take the liberty of quoting from some (almost the entirety) of it. On February 4, 1975 Kuhn wrote: In March I am talking at an AMS Symposium on “Nonlinear Programming - A Historical View.” Last summer I learned through reading Takayama’s Mathematical Economics of your 1939 Master’s Thesis and have obtained a copy. First, let me say that you have clear priority on the results known as the Kuhn–Tucker conditions (including the constraint qualification). I intend to set the record as straight as I can in my talk. You could help me if you would be kind enough to give me whatever details you remember regarding the writing of your thesis. Of special interest to me would be answers to the following questions: Who was your advisor (or other faculty influences)? Who set the problem? Why was the thesis never published? (One possibility would be to attempt (at least partial) publication as an appendix to my survey.) Documenta Mathematica

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William Karush and the KKT Theorem Dick Cottle, who organized the session, has been told of my plans to rewrite history and says “you must be a saint” not to complain about the absence of recognition. Al Tucker remembers you from RAND, wonders why you never called this to his attention and sends his best regards, In his friendly reply, dated February 10, 1975, Karush said: Thank you for your most gracious letter. I appreciate your thoughtfulness in wanting to draw attention to my early work. If you ask why I did not bring up the matter of priority before, perhaps the answer lies in what is now happening – I am not only going to get credit for my work, but I am going to crowned a “saint”! I wrote my master’s thesis at the University of Chicago under Lawrence M. Graves, who also proposed the problem. Those were the final years of the school of classical calculus of variations at the University and I suppose that the problem was given to me as a finite-dimensional version of research going on in the calculus of variations with inequalities as side conditions. Gilbert A. Bliss was chairman of the department, and Magnus R. Hestenes was a young member of the faculty; both of these men influenced me, and in fact I wrote my doctoral thesis later under Hestenes on isoperimetric problems and index theorems in the calculus of variations (this work was published after the war). The thought of publication never occurred to me at the time I wrote the master’s thesis. I was a struggling graduate student trying to meet the requirements for going on to my Ph.D. and Graves never brought up the question of publication. I imagine nobody at that time anticipated the future interest in the problem, That does not answer the question of why I did not point to my work in later years when nonlinear programming took hold and flourished. The thought of doing this did occur to me from time to time, but I felt rather diffident about that early work and I don’t think I have a strong necessity to be “recognized”. In any case, the master’s thesis lay buried until a few years ago when Hestenes urged me to look at it again to see if it shouldn’t receive its proper place in history – he expressed an interest in setting the record straight in some publication of his own. So I did look at the thesis again, and I looked again at your work with Tucker. I concluded that you two had exploited and developed the subject so much further than I, that there was no justification for announcing to the world, “Look what I did, first.” I expressed my feelings to Magnus Hestenes in the past year and that closed the matter as far as I was concerned. I will tell Magnus of your AMS Symposium talk and I am sure he will be glad of it. Documenta Mathematica

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This refreshing exchange of letters would seem to represent the last word on the subject. In the period from 1939 to 1942: Karush was, as he testified, busy working on a doctoral thesis and WWII broke out. It has been asserted that publication was curtailed during the war due to a shortage of paper. In any case, [18] was just a master’s thesis, part of the degree requirements, and was a finitedimensional version of results already in print. As Kjeldsen’s contextualized historical analysis [24] of the matter emphasizes, it was a little ahead of its time, particularly of the post-WWII period. There remains the question: How did Takayama learn of Karush’s work? Takayama’s book [36], and subsequently Kuhn’s papers [27], [28], and [29] suggest how this happened. Takayama heard about it from Mohamed A. ElHodiri [12] who (in 1963) had found a reference to [18] in a paper by Louis L. Pennisi [35]. El-Hodiri related this information to Leo Hurwicz among others and incorporated the Karush/John/Kuhn–Tucker results into his own writings [10], [11]. Strangely missing from the literature of the 1960s is a reference to Karush’s MS thesis (and the KT paper) in the book [14] by Magnus Hestenes. Nine years later, Hestenes’s book [15] gave Karush his due. 4

The Chicago School

William Karush began his undergraduate education in Chicago at Central Y.M.C.A. College.4 He spent two years there after which he transferred to the University of Chicago, receiving the Bachelor of Science degree there in June, 1938. His graduate studies began there in October that same year. The mathematics department at the University of Chicago was known as a bastion of the study of the calculus of variations. The history of the department and this powerful tradition have been chronicled in numerous articles, many available online. For our purposes, the works of Kuhn [27] and Kjeldsen [24] are more than adequate starting points, relating directly as they do to our subject. Kjeldsen’s article in particular goes into greater detail about the history and reputation of the department. She reports how it was thought (even by some Chicago mathematicians) to be exceptionally narrow with its concentration on the calculus of variations. Nevertheless, the Chicago mathematics department maintained a grand heritage. It is instructive (one might say fruitful) to trace a portion of the mathematical tree that leads to William Karush’s master’s thesis. As stated above, the problem was set Lawrence M. Graves, and the work was carried out under his supervision. Graves’s Ph.D. thesis advisor was Gilbert A. Bliss who was Chairman of the mathematics department at the time. Bliss was a powerful figure in the study of calculus of variations. He supervised the Ph.D. theses of many other mathematicians who are well known in mathematical programming circles today. They include, Lloyd Dines, Magnus Hestenes, Alston Householder, Edward McShane, and Frederick Valentine (who was advised 4 In

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by Graves in addition to Bliss). Bliss’s Ph.D. thesis was supervised by Oskar Bolza whose Ph.D. was obtained in G¨ottingen under the supervision of C. Felix Klein. Three more such steps lead us from Klein to Julius Pl¨ ucker and Rudolf Lipschitz (jointly) to Christian Ludwig Gerling to Carl Friedrich Gauß. This impressive lineage can be reconstructed using the Mathematics Genealogy Project [33]. Returning now to the master’s thesis of Karush, it is important to note that the results have been described by Takayama [36, pages 61] as a finitedimensional versions of counterparts from Valentine’s doctoral dissertation [40] completed in 1937. Indeed, even Karush (in his previously quoted letter to Kuhn) said, “I suppose that the problem was given to me as a finite-dimensional version of research going on in the calculus of variations with inequalities as side conditions.” Pennisi was, it seems, the first to cite Karush’s thesis, albeit briefly. In [35, section 3] which is called “The problem with a finite number of variables”, Pennisi asserts For the normal case, which is the only one we consider, our results are more general than those of Karush. Pennisi refers to Valentine’s Ph.D. thesis [40], but does not speak of [18] as a finite-dimensional version of it, Nonetheless, it is interesting to note that Valentine, Karush, and Pennisi were all supervised by Graves at the University of Chicago. The title of Valentine’s doctoral dissertation “The Problem of Lagrange with Differential Inequalities as Added Side Conditions” uses some rather common terminology of the time. Many research papers focused on “The Problem of Lagrange.” Another commonly treated subject was “The Problem of Bolza.” The phrase “added side conditions” is how these contemporary mathematicians spoke of what we call “constraints.” This kind of terminology is found in the title of Fritz John’s paper as well. More broadly the introduction of inequalities as “side conditions” had been going on for some time at the University of Chicago and elsewhere, and not just by Fritz John. In the calculus of variations literature, one finds inequalities as side conditions in Bolza’s 1913 paper [7]. Moreover, as noted by Kuhn [27], the type of modified Lagrangian function that we associate with Fritz John had been used by Bliss [5] many years earlier. In fact, Bliss himself used it well before 1938, for example, in the notes for his lectures [3] given in the Summer Quarter of 1925. Before that, Courant and Hilbert [9, p. 143] used this type of Lagrangian function and commented that if the multiplier associated with the minimand (objective function) is nonzero, then the conventional Lagrangian function can be recovered. 5

A Biographical Sketch of William Karush

William Karush was born in Chicago, Illinois on March 1, 1917. His parents Sam and Tillie (formerly Shmuel and Tybel) were fairly recent immigrants, Documenta Mathematica

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William Karush, circa 1987

Fritz John at NYU, circa 1987

Harold Kuhn and Albert Tucker, 1980 at von Neumann Prize presentation (Printed with permission of Larry Karush; NYU; Harold Kuhn and Alan Tucker.)

having come to the United States from Bialystok which was then under Russian control. (It is now in Poland.) As a child, William was known as “Willie;” his older brother Fred was called “Freddie” [39]. They eventually had two younger siblings, Jack and Esther. Of the four, only Esther is still living. Willie outgrew this diminutive name and became known as “Will.” He attended public schools in Chicago, graduating from Murray F. Tuley High School in June, 1934. From that point on, his Bachelor of Science, Master of Science, and Doctor of Philosophy were all earned at the University of Chicago in 1938, 1939, and 1942, respectively [18]. Based on an entry in the 17th Edition of American Men & Women of Science [1, p. 215], the table below gives a summary of the positions held by William Karush. The table does not make explicit the fact that during World War II, Karush worked on the Manhattan Project which culminated in the atomic Documenta Mathematica

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Table 1: Employment Chronology of William Karush [1] Year

Position

Employer

1942–43

Mathematician

1943–45

Physicist

1945–56

Instructor to Associate Professor Member, Senior Staff Sr. Operations Research Scientist Principal Scientist Professor of Mathematics Emeritus Professor of Mathematics

Geographical Laboratory, Carnegie Inst. of Washington Metallurgical Laboratory, University of Chicago Mathematics Deptartment, University of Chicago Ramo-Wooldridge Corporation System Development Corporation

1956–57 1958–62 1962–67 1967–87 1987–97

Concurrent Positions 1949–52 Mathematician

System Development Corporation California State University, Northridge California State University, Northridge

1953

Member, Technical Staff

1954–55

Member, Technical Staff

Inst. Numerical Anal., Nat. Bur. Standards, UCLA Research & Development Labs., Hughes Aircraft Ramo-Wooldridge Corporation

1955–56

Ford Faculty Fellow

University of California, Los Angeles

bombs that the United States used on Hiroshima and Nagasaki, Japan. As it happens, though, William Karush was one of 155 scientists of the Manhattan Project of Oak Ridge, Tennessee who in 1945 put their names to the so-called Szil´ ard Petition which was drafted by physicist L´eo Szil´ard “and asked President Harry S. Truman to consider an observed demonstration of the power of the atomic bomb first, before using it against people” [41]. The petition never reached Truman. In later years, Will Karush became an outspoken peace advocate [32]. The portrait of him presented here shows Karush wearing a “Beyond War” pin on his shirt collar. In general, William Karush listed his research interests as belonging to operations research, calculus of variations, and applied mathematics. His published works in operations research include papers in mathematical programming, queueing, and dynamic programming. He is also known for having edited two different dictionaries of mathematics [20], [22]. As is evident from the table above, Karush had a varied career: part of it in industry, and a somewhat larger part in academia. At the University of Chicago (1945–56) he rose from instructor to associate professor. He took a leave of absence in southern California and never returned to the University of Chicago. Eleven years later, he joined the faculty of California State University (at the time called “San Fernando Valley College”) as a full professor where his Documenta Mathematica

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duties involved the teaching of undergraduate-level mathematics. He taught there until 1987 at which time he retired and became an emeritus professor. Will Karush and his wife, Rebecca, were close friends of Richard E. Bellman of dynamic programming fame. For a number of years, Rebecca was a technical typist for Bellman. Will and Rebecca had two children, Larry and Barbara, both of whom live in California. Larry is a musician (see [17]). Barbara is a retired school teacher. In January 1991, Will and Rebecca took a short vacation in Palm Springs, California. One evening after dinner, Rebecca was struck by a car and fatally injured. Will Karush lived until February 22, 1997, one week before his 80th birthday. He died of complications from surgery. Acknowledgements Many people deserve credit for helping me to produce this view of William Karush and the Karush-Kuhn-Tucker Theorem. Accordingly, it is a great pleasure to acknowledge that I received various sorts of information and assistance from John R. Birge, Harold W. Kuhn, Tinne Hoff Kjeldsen, Mohamed A. ElHodiri, Larry Karush, Esther Diamond, Stephen D. Brady, Philip Wolfe, Saul I. Gass, Kenneth J. Arrow, Ilan Adler, Werner Horn, William Watkins, George Biriuk, Malcolm Soule, Joel L. Zeitlin, Efrem Ostrow, Ingram Olkin, Edwin Knihnicki, Margaret H. Wright, April E. Bacon, Joseph B. Keller, and library and departmental staff from coast to coast. As usual, the flaws are mine alone. References [1] American Men & Women of Science, 17th Edition, R.R. Bowker, New York, 1989. [2] A. Assad and S. I. Gass, Profiles in Operations Research, Springer, New York, 2011. [3] G. A. Bliss, The Problem of Lagrange in the Calculus of Variations, Lectures given by Professor G. A. Bliss at the University of Chicago in the Summer Quarter 1925. [Prepared by O. E. Brown, Northwestern University, Evanston, Ill.] [4] G. A. Bliss, The problem of Lagrange, American Journal of Mathematics 52 (1930), 673–744. [5] G. A. Bliss, Normality and abnormality in the calculus of variations, Transactions of the American Mathematical Society 43 (1938), 365–376. [6] G. A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, 1946. ¨ [7] O. Bolza, Uber den Abnormalen Fall beim Lagrangeschen und Mayerschen Problem mit gemischten Bedingungen und variabeln Endpunkten, Mathematische Annalen 74 (1913), 430–446. Documenta Mathematica

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¨ [8] O. Bolza, Uber Variationsprobleme mit Ungleichungen als Nebenbedingungen, Mathematische Abhandlungen (1914), 1–18. [9] R. Courant and D. Hilbert, Methoden der Mathematischen Physik I, Verlag von Julius Springer, Berlin, 1924. [10] M. A. El-Hodiri, Constrained Extrema of Functions of a Finite Number of Variables: Review and Generalizations, Krannert Institute Paper No. 141, Purdue University, 1966. [See also Constrained Extrema: Introduction to the Differentiable Case with Economic Appications. Springer-Verlag, Berlin, 1971. [11] M. A. El-Hodiri, The Karush Characterization of Constrained Extrema of Functions of a Finite Number of Variables. Ministry of Treasury UAR, Research Memoranda. series A, no. 3, July 1967. [12] M. A. El-Hodiri, private correspondence to Richard W. Cottle, March 3, 2012. [13] A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley & Sons, New York, 1968. [14] M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley & Sons, New York, 1966. [15] M. R. Hestenes, Optimization Theory: The Finite Dimnsional Case, Krieger Publishing Company, Huntington, N.Y., 1975 (reprinted 1981). [16] F. John, Extremum problems with inequalities as subsidiary conditions, in (K.O. Friedrichs, O.E. Neugebauer, and J.J. Stoker, eds.) Studies and Essays, Courant Anniversary Volume, Wiley-Interscience, New York, 1948, pp. 187–204. [17] larrykarush.com/ABOUT.html. [18] W. Karush, Minima of Functions of Several Variables with Inequalities as Side Conditions, Master’s Thesis, Department of Mathematics, University of Chicago, 1939. [19] W. Karush, Isoperimetric Problems and Index Theorems in the Calculus of Variations, Doctoral Dissertation, Department of Mathematics, University of Chicago, 1942. [20] W. Karush, The Crescent Dictionary of Mathematics, The Macmillan Company, New York, 1962. [21] W. Karush, Mathematical Programming, Man-Computer Search and System Control. Technical Report SP-828, System Development Corporation, Santa Monica, Calif., 1962. Documenta Mathematica

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[22] W. Karush, ed., Webster’s New World Dictionary of Mathematics. MacMillan, New York, 1989. [23] T. H. Kjeldsen, The Kuhn–Tucker Theorem in Nonlinear Programming: A Multiple Discovery? TEKST NR 377, IMFUFA, Roskilde Universitetscenter, Roskilde, Denmark. [24] T. H. Kjeldsen, A contextualized historical analysis of the Kuhn–Tucker Theorem in nonlinear programming: The impact of World War II, Historia Mathematica 27 (2000), 331–361. [25] T. H. Kjeldsen, New mathematical disciplines and research in the wake of World War II, in (B. Booß-Bavnbek and J. Høyrup, eds.) Mathematics and War, Birkh¨ auser, Basel, 2003, pp. 126–152. [26] T. H. Kjeldsen, The development of nonlinear programming in post war USA: Origin, motivation , and expansion, in (H.B. Andersen et al., eds.) The Way Through Science and Philosophy: Essays in Honour of Stig Andur Pederson, College Publications, London, 2006, pp. 31–50. [27] H. W. Kuhn, Nonlinear programming: A historical view, in (R. W. Cottle and C. E. Lemke, eds.) Nonlinear Programming [SIAM-AMS Proceedings, Volume IX]. American Mathematical Society, Providence, R.I., 1976. [28] H. W. Kuhn, Nonlinear programming: A historical note, in (J. K. Lenstra, A. H. G. Rinnooy Kan, and A. Schrijver, eds.) History of Mathematical Programming: A Collection of Personal Reminiscences, CWI and NorthHolland Publishing Company, Amsterdam, 1991. [29] H. W. Kuhn, Being in the right place at the right time, Operations Research 50 (2002), 132–134. [30] H. W. Kuhn and A. W. Tucker, Nonlinear programming, in (J. Neyman, ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951, pp. 481–492. [31] Los Angeles Times, William Karush: Wrote ‘Webster’s Dictionary of Mathematics’. [Online at http://articles.latimes.com/1997-02-28/ news/mn-33402_1_william-karush.] [32] H. Marquez Estrada, From A-bomb to drive for peace, Santa Barbara New-Press, May, 1987, pages B-1 and B-4. [Precise date unknown.] [33] Mathematics Genealogy Project www.genealogy.ams.org. [34] L. A. Pars, An Introduction to the Calculus of Variations, John Wiley & Sons, New York, 1962. Documenta Mathematica

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[35] L. L. Pennisi, An indirect sufficiency proof for the problem of Lagrange with differential inequalities as added side conditions, Transactions of the American Mathematical Society 74 (1953), 177–198. [36] A. Takayama, Mathematical Economics., Dryden Press, Hinsdale, Ill.: 1974. [37] A. Takayama, Mathematical Economics, 2nd Edition, Cambridge University Press, Cambridge, 1985. [38] R. Tichatschke, “Auf den Schultern von Giganten” Zur Geschichte der Mathematischen Optimierung, Forschungsbericht Nr. 08-4, Mathematik/Informatik, Universit¨at Trier (Germany). [39] United States census data. [40] F. A. Valentine, The Problem of Lagrange with Differential Inequalities as Added Side Conditions, Doctoral Dissertation, Department of Mathematics, University of Chicago, 1937. [41] http://en.wikipedia.org/wiki/Szilard_petition.

Richard W. Cottle Department of Management Science and Engineering Stanford University Stanford, California 94305-4121 USA [emailprotected]

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Nelder, Mead, and the Other Simplex Method Margaret H. Wright

2010 Mathematics Subject Classification: 49M30, 65K10, 90C56 Keywords and Phrases: Nelder-Mead, direct search simplex method, derivative-free optimization, non-derivative optimization In the mid-1960s, two English statisticians working at the National Vegetable Research Station invented the Nelder–Mead “simplex” direct search method. The method emerged at a propitious time, when there was great and growing interest in computer solution of complex nonlinear real-world optimization problems. Because obtaining first derivatives of the function f to be optimized was frequently impossible, the strong preference of most practitioners was for a “direct search” method that required only the values of f ; the new Nelder– Mead method fit the bill perfectly. Since then, the Nelder–Mead method has consistently been one of the most used and cited methods for unconstrained optimization. We are fortunate indeed that the late John Nelder1 has left us a detailed picture of the method’s inspiration and development [11, 14]. For Nelder, the starting point was a 1963 conference talk by William Spendley of Imperial Chemical Industries about a “simplex” method recently proposed by Spendley, Hext, and Himsworth for response surface exploration [15]. Despite its name, this method is not related to George Dantzig’s simplex method for linear programming, which dates from 1947. Nonetheless, the name is entirely appropriate because the Spendley, Hext, and Himsworth method is defined by a simplex; the method constructs a pattern of n + 1 points in dimension n, which moves across the surface to be explored, sometimes changing size, but always retaining the same shape. Inspired by Spendley’s talk, Nelder had what he describes as “one useful new idea”: while defining each iteration via a simplex, add the crucial ingredient that the shape of the simplex should “adapt itself to the local landscape” [12]. During a sequence of lively discussions with his colleague Roger Mead, where “each of us [was] able to try out the ideas of the previous evening on the other the following morning”, they developed a method in which the simplex could “elongate itself to move down long gentle slopes”, or “contract itself on to the final minimum” [11]. And, as they say, the rest is history. 18

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p3

p3

p3

p3

pin p¯

p¯

p¯

p¯

p1

p2

pout pr

pe Reflection

Outside contraction

Expansion

Inside contraction

Shrink

The 1965 Nelder–Mead paper [12] appeared in the Computer Journal, a prestigious publication of the British Computer Society. Implementations and numerical testing followed almost immediately in which the Nelder–Mead method performed well compared to existing algorithms. In addition, one should not underestimate the degree to which the Nelder–Mead method appealed to practitioners because its moves are easy to describe. The Nelder–Mead simplex can change in five different ways during an iteration, as illustrated here in two dimensions. Except in the case of a shrink, the worst vertex of the simplex at iteration k (the point p3 in the figure) is replaced at iteration k + 1 by one of the reflection, expansion, or contraction points. Based on this picture, users felt (and feel) that they understand what the method is doing. As Nelder said while trying to explain the method’s popularity [11], “. . . the underlying ideas are extremely simple – you do not have to know what a Hessian matrix is to understand them”. Nelder’s recollection of events [11] following publication of the Nelder–Mead paper is that some “professional optimizers” were “surprised” because they “had convinced themselves that direct search methods . . . were basically unpromising”. Nelder notes with relish that “our address (National Vegetable Research Station) also caused surprise in one famous US laboratory,2 whose staff clearly doubted if turnipbashers could be numerate”. The Nelder–Mead paper has been cited thousands of times, and qualified by the late 1970s as a “Science Citation Classic”. The Nelder–Mead method soon became so much more popular than other simplex-based methods that it began to be called “the” simplex method, in the context of unconstrained optimization.3 The story of the subsequent position of the Nelder–Mead method in mainstream optimization clearly illustrates a sea change, sometimes called “math2 To

the present author’s knowledge, this laboratory has never been identified. the LP simplex method is much better known, the Nelder–Mead method is sometimes lightheartedly called “the other simplex method”. 3 Because

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ematization”, that has taken place since the 1960s and early 1970s. A 1972 survey paper by Swann [16, page 28] concludes by saying Although the methods described above have been developed heuristically and no proofs of convergence have been derived for them, in practice they have generally proved to be robust and reliable . . . The lack of theoretical foundations and motivation would almost certainly be regarded as unacceptable in an optimization journal today. As optimization became more mathematical, by the late 1970s textbooks tended to dismiss the Nelder–Mead method (and other direct search methods) as “ad hoc” or “heuristic”. Of course there were a small number of scholarly works about the Nelder–Mead method (see the references in [20, 6]). Among these, the analysis of [4] is of particular interest. Of equal or (to some) greater concern, the Nelder–Mead method was well known to experience practical difficulties ranging from stagnation to failure. As a result, even in its early years papers were published that described how the Nelder–Mead method could be modified so that it would work well on a particular problem. Although not center stage in mainstream optimization, direct search methods other than Nelder–Mead were being studied and implemented, especially in China and the Soviet Union, but the associated work was not well known in the West. (Several references to these papers are given in [20, 6].) This situation changed significantly in 1989, when Virginia Torczon, a PhD student at Rice University advised by John Dennis, published a thesis [17] that not only proposed a direct search method (“multidirectional search”), but also provided a proof that, under various conditions, lim inf k∇f k → 0, where f is the function to be optimized. Once rigorous convergence results had been established for one method, the floodgates opened, and since 1989 there has been a subsequent (and still ongoing) renaissance of interest in derivative-free methods. The level of intensity has been especially high for research on model-based derivative-free methods, which (unlike Nelder–Mead and other direct search methods) create evolving simple models of f . A nice discussion of the different classes of derivative-free methods can be found in [2]. How does the Nelder–Mead method fit into today’s landscape of derivativefree methods? It is fair to describe Nelder–Mead as a far outlier, even a singularity, in the emerging families of mathematically grounded direct search methods such as generalized pattern search and generating set search [2]. Hence the position of the Nelder–Mead method in mainstream nonlinear optimization is anomalous at best, and is subject to a wide range of attitudes. From the positive end, several researchers have created modified Nelder– Mead methods with the goal of retaining the favorable properties of the original while avoiding its known deficiencies. See, for example, [19, 5, 18, 10, 13, 1]. Strategies for remedying the defects of the original Nelder–Mead include using a “sufficient decrease” condition for acceptance of a new vertex (rather than Documenta Mathematica

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simple decrease) and restarting when the current simplex becomes excessively ill-conditioned. Taking a negative view, some researchers believe that Nelder–Mead is pass´e because modern derivative-free methods are consistently better: The Nelder-Mead algorithm, however, can work very well and it is expected to survive a very long time. Nevertheless, it is seriously defective: it is almost never the best method and indeed it has no general convergence results . . . we believe that ultimately more sophisticated and successful methods will earn their rightful place in practical implementations . . . [2, page 7]. Whichever view prevails in the long run, as of 2012 the Nelder–Mead method is not fading away. As in its early days, it remains remarkably popular with practitioners in a wide variety of applications. In late May 2012, Google Scholar displayed more than 2,000 papers published in 2012 that referred to the Nelder– Mead method, sometimes when combining Nelder–Mead with other algorithms. In addition, certain theoretical questions remain open about the original Nelder–Mead method. Why is it sometimes so effective (compared to other direct search methods) in obtaining a rapid improvement in f ? One failure mode is known because Ken McKinnon produced a fascinating family of strictly convex functions in two dimensions for which Nelder–Mead executes an infinite sequence of repeated inside contractions and thereby fails to converge to the minimizer from a specified starting configuration [9] – but are there other failure modes? An initial exploration of the effects of dimensionality [3] provides some insights, but there is more to be learned. Why, despite its apparent simplicity, should the Nelder–Mead method be difficult to analyze mathematically? (See [7, 8].) One can argue that, before the original method is retired, we should achieve the maximum possible mathematical understanding of how and why it works. In an interview conducted in 2000, John Nelder said about the Nelder–Mead method: There are occasions where it has been spectacularly good . . . Mathematicians hate it because you can’t prove convergence; engineers seem to love it because it often works. And he is still right. We end with a picture of John Nelder and George Dantzig, fathers of two different simplex methods, together at the 1997 SIAM annual meeting at Stanford University:

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John Nelder and George Dantzig, Stanford University, 1997, photographed by Margaret Wright References [1] Burmen, A., Puhan, J., and Tuma, T., Grid restrained Nelder–Mead algorithm, Computational Optimization and Applications 34 (2006), 359–375. [2] Conn, A. R., Scheinberg, K., and Vicente, L. N., Introduction to Derivative-Free Optimization, SIAM, Philadelphia, 2009. [3] Han, L. and Neumann, M., Effect of dimensionality on the Nelder–Mead simplex method, Optimization Methods and Software 21 (2006), 1–16. [4] Hensley, D., Smith, P., and Woods, D., Simplex distortions in Nelder– Mead reflections, IMSL Technical Report Series No. 8801, IMSL, Inc., Houston, Texas (1988). [5] Kelley, C. T., Detection and remediation of stagnation in the Nelder– Mead algorithm using a sufficient decrease condition, SIAM Journal on Optimization 10 (1999), 43–55. [6] Kolda, T. G., Lewis, R. M., and Torczon, V., Optimization by direct search: new perspectives on some classical and modern methods, SIAM Review 45 (2003), 385–482. [7] Lagarias, J. C., Reeds, J. A., Wright, M. H., and Wright, P. E., Convergence properties of the Nelder–Mead simplex algorithm in low dimensions, SIAM Journal on Optimization 9 (1998), 112–147. [8] Lagarias, J. C., Poonen, B., and Wright, M. H., Convergence of the restricted Nelder–Mead method in two dimensions, SIAM Journal on Optimization 22 (2012), 501–532. Documenta Mathematica

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[9] McKinnon, K. I. M., Convergence of the Nelder–Mead simplex method to a non-stationary point, SIAM Journal on Optimization 9 (1998), 148–158. [10] Nazareth, L. and Tseng, P., Gilding the lily: A variant of the Nelder–Mead algorithm based on golden section search, Computational Optimization and Applications 22 (2002), 133–144. [11] Nelder, J. A., This week’s citation classic, Citation Classics Commentaries 15 (1979). [12] Nelder, J. A. and Mead, R., A simplex method for function minimization, Computer Journal 7 (1965), 308–313. [13] Price, C. J., Coope, I. D., and Byatt, D., A convergent variant of the Nelder–Mead algorithm, J. Optimization Theory and Applications 113 (2002), 5–19. [14] Senn, S., A conversation with John Nelder, Statistical Science 18 (2003), 118–131. [15] Spendley, W., Hext, G. R., and Himsworth, F. R., Sequential application of simplex designs in optimization and Evolutionary Operation, Technometrics 4 (1962), 441–461. [16] Swann, W. H., “Direct search methods”, in Numerical Methods for Unconstrained Optimization (P. E. Gill and W. Murray, eds.), Academic Press, London, 13–28 (1972). [17] Torczon, V., Multi-Directional Search: A Direct Search Algorithm for Parallel Machines, PhD thesis, Department of Mathematical Sciences, Rice University, Houston, Texas, 1989. [18] Tseng, P., Fortified-descent simplicial search method: A general approach, SIAM Journal on Optimization, 10 (1999), 269–288. [19] Woods, D. J., An Interactive Approach for Solving Multi-Objective Optimization Problems, PhD thesis, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, 1985. [20] Wright, M. H., Direct search methods: once scorned, now respectable. in Numerical Analysis 1995: Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis, D. F. Griffiths and G. A. Watson (eds.), 191–208, Addison Wesley Longman, Harlow, UK, 1996. Margaret H. Wright Courant Institute of Mathematical Sciences New York, New York 10012 USA [emailprotected] Documenta Mathematica

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Subgradient Optimization in Nonsmooth Optimization (including the Soviet Revolution) Jean-Louis Goffin

2010 Mathematics Subject Classification: 26A27, 46N10 Keywords and Phrases: Nondifferentiable optimization, nonsmooth optimization, subgradient optimization, relaxation method, Soviet revolution

1

Introduction

Convex nondifferentiable, also known as convex nonsmooth, optimization (NDO) looks at problems where the functions involved are not continuously differentiable. The gradient does not exist, implying that the function may have kinks or corner points, and thus cannot be approximated locally by a tangent hyperplane, or by a quadratic approximation. Directional derivatives still exist because of the convexity property. NDO problems are widespread, often resulting from reformulations of smooth, or linear problems, that are formulated in a space with much smaller number of variables than in the original problem. Examples of this are the reformulation implicit in Dantzig-Wolfe decomposition or column generation [4] and [5], which are equivalent by duality to Cheney’s cutting plane method [20]. These methods do not work well if an aggregated formulation is used. Shor’s subgradient method [35, 36] provided a superior alternative, leading to a true Soviet revolution. His work was expanded both in theory and in practice by numerous authors. Held and Karp [17], unaware of the work of Shor, developed a method for the traveling salesman problem that uses subgradient optimization to compute a bound in a Lagrangean relaxation scheme. This seminal contribution also led to a huge following; see for instance Fisher [11]. 2

Basic definitions

The basic nondifferentiable optimization problem takes the form [N DO] Documenta Mathematica

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where f is a real-valued, continuous, convex, and nondifferentiable function. Sometimes there is a restriction that x ∈ X, a closed convex set, for which a projection map is available: x∗ (x) = ΠX (x) = x ¯ : k¯ x − xk ≤ ky − xk, ∀y ∈ X ; and the problem becomes:

[N DOc]

min f (x).

x∈X

The convexity of f implies that it has at least one supporting hyperplane at every point of Rn . The subdifferential is the set of such slopes, i.e., ∂f (x) = ξ : f (x) + hξ, (y − x)i ≤ f (y), ∀y ∈ Rn .

At differentiable points there is a unique supporting hyperplane whose slope is the gradient. At nondifferentiable points, there is an infinite set of subgradients and, hence, an infinite set of supporting hyperplanes. The derivative in the direction d is given by: f ′ (x; d) = sup ξ T d : ξ ∈ ∂f (x) and the direction of steepest descent is given by d∗ : min f ′ (x; d) = f ′ (x; d∗ );

kdk=1

it can be shown that if 0 ∈ / ∂f (x) and dˆ is the element of minimum norm in the subdifferential ∂f (x), then d∗ = −

dˆ . ˆ kdk

The use of the steepest descent method with exact line searches is not recommended as: 1. The steepest descent method with exact line searches may converge to a nonoptimum point, see Wolfe [43]; 2. In the frequent case where f (x) = maxi∈I {hai , xi + bi }, and the set I is computed by an oracle or subroutine, an LP or an IP, the cardinality of I may be exponential, and the subdifferential is given by: n X ∂f (x) = αi ai : i∈I(x)

X

i∈I(x)

o αi = 1, αi ≥ 0 ,

I(x) = i : hai , xi + bi = f (x) ;

so that it is unrealistic to expect that the full subdifferential will be available. In NDO, one assumes that the function f is given by an oracle which for every value of x returns the value of f , i.e., f (x), and one arbitrary subgradient ξ(x) ∈ ∂f (x). Documenta Mathematica

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Subgradient methods: The Soviet revolution

Subgradient methods were developed by Shor [35] and [36] in the 1960’s. To quote from a paper by B. T. Polyak [33] delivered at the Task Force on nondifferentiable optimization organized at IIASA by Lemar´echal and Mifflin, (this paper also includes an excellent bibliography of work done in the USSR before 1977): The subgradient method was developed in 1962 by N.Z. Shor and used by him for solving large-scale transportation problems of linear programming [35]. Altough published in a low-circulation publication, this pioneering work became widely known to experts in the optimization area in the USSR. Also of great importance for the propagation of nondifferentiable concepts were the reports by the same author presented in a number of conferences in 1962–1966. Publication of papers by Ermoliev [9], Polyak [30] and Ermoliev and Shor [10] giving a precise statement of the method and its convergence theorems may be regarded as the culmination of the first stage in developing subgradient techniques. All of their massive contributions to the field are well reported in their two books Shor[40] and Polyak[32], as well as in the second book by Shor[41]; see also the book by Nesterov [27]. So subgradient optimization simply moves the current iterate in the direction of a scaled subgradient by a stepsize that is decided a prori: ξk xk+1 = ΠX xk − tk , kξk k where xk is the current point, ξk ∈ ∂f (xk ) is an arbitrary subgradient of f at xk , tk is a stepsize and ΠX is the projection map on the constraint set X. It is assumed that the projection map is easily computed, such as if X is a sphere, a box or a simplex. A subgradient is not a direction of descent for the function f but it is one for the distance to the optimal set. Shor [35] states that a constant stepsize tk = t does not converge, as the example of f (x) = |x| clearly shows. He also shows that the iterates eventually reach an O(t) neighborhood of the optimum. This follows from an equivalent proof, extended to the case of a constraint set: Theorem 3.1 (Nesterov [27]). Let f be Lipschitz continuous on B2 (x∗ , R) with constant M and x0 ∈ B2 (x∗ , R). Then fk∗

Pk R2 + i=0 h2i . −f ≤M Pk 2 i=0 hi ∗

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In this statement fk∗ = minki=0 f (xk ) and f ∗ = minx∈X f (x). It follows that if the sequence tk is chosen as tk = Rǫ, ∀k = 1, . . . , N , and ∗ N = ⌈ ǫ12 ⌉ then: fN − f ∗ ≤ M Rǫ; see also Shor [40] pp. 23–24. This means that subgradient optimization is an optimal algorithm, uniformly in the dimension of the problem, see Nemirovski and Yudin [25]. Almost quoting from Polyak again [33]: Reference [35] has described the following way of stepsize regulation resting upon this result, although it is not entirely formalized. A certain ǫ is chosen and the computation proceeds with tk = Rǫ until the values of f (xk ) start to oscillate about a certain limit. After this ǫ is halved and the process is repeated. This leads readily to the divergent series of stepsizes, suggested by Polyak [30] and Ermoliev[9], and studied in Shor and Ermoliev [10]: ∞ X

tk = ∞,

tk → 0

tk > 0.

k=0

Theorem 3.2. Theorem 3.1 shows that fk∗ converges to f ∗ . R , which guarantees convergence in An often used stepsize is tk = √k+1 ∗ steps [27], where O means the term of higher order, ignoring lower order terms; the proof of this can be improved, see Nemirovski [26], who shows √ , where εN = f ∗ − f ∗ . that εN ≤ O(1) RM N N Unfortunately, the divergent stepsize rule can and is extremely slow. So the question arose, as to whether geometric convergence can be obtained. The answer is given in the following theorem, proved only in the unconstrained case: 1 O∗ ( √k+1 )

Theorem 3.3 (Shor [40] pp. 30–31). Let f be a convex function defined on Rn . Assume that for some ϕ satisfying 0 ≤ ϕ < π/2, and for all x ∈ Rn the following inequality holds:

ξ(x), x − x∗ (x) ≥ cos ϕkξ(x)k kx − x∗ (x)k, (2) where ξ(x) ∈ ∂f (x), and x∗ (x) is the point in the set of minima that is nearest to x. If for a given x0 we choose a stepsize t1 satisfying: kx∗ (x0 ) − x0 k cos ϕ for π/4 ≤ ϕ < π/2 t1 ≥ kx∗ (x0 ) − x0 k/(2 cos ϕ) for 0 ≤ ϕ < π/4, define {tk }∞ k=1 by where r(ϕ) =

tk+1 = tk r(ϕ),

sin ϕ 1/(2 cos ϕ)

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and generate {xk }∞ k=0 according to the formula xk+1 = xk − tk+1

ξ(xk ) . kξ(xk )k

Then either ξ(x∗k ) = 0 for some k ∗ , i.e., x∗k is a minimum point, or for all k = 1, . . . , ∞ the following inequality holds ( tk+1 / cos ϕ for π/4 ≤ ϕ < π/2 kxk − x∗ (xk )k 2tk+1 cos ϕ for 0 ≤ ϕ < π/4 This theorem was first proved in this form by Shor and Gamburd [38] and by Shor [39]. An earlier version that used the asphericity σ of the level set of f instead of cos ϕ was proved by Shor [37]. This is a slightly weaker result as cos ϕ ≥ 1/σ. In practice, a most widely used stepsize is tk = λ(f (xk ) − f¯)/||ξk || where λ ∈ (0, 2) and f¯ is expected to be a good estimate of the optimal value f (x∗ ). It can be either the exact optimum f ∗ , an overestimate fˆ > f ∗ , or an underestimate fˇ < f ∗ . This was suggested and studied by Polyak, see for instance [32]. The most general theorem is due to Nemirovski [26], under the assumption that f¯ = f ∗ : εN ≤ M kx0 − x∗ kN −1/2 . Polyak [31], see also Shor [40] shows that if in addition to the Lipschitz condition on f one has a lower bound on the variation of f such as f (x) ≥ md(x, X ∗ )α where d(x, X ∗ ) is the distance to the optimal set X ∗ and α = 1 or 2 then: kxk − x∗ k ≤ q k kx0 − x∗ k, where q =

q m2 1 − λ(2 − λ) M 2.

The more practical case of f¯ < f ∗ , as an underestimate of f ∗ , can be computed by getting a feasible dual solution, was studied by Eremin [6, 7, 8] who studied the Chebyshev solution to an infeasible system of linear inequalities: P = x : hai , xi + bi ≤ 0, ∀i ∈ I .

This is equivalent to minimizing the function f (x) = maxi∈I {< ai , x > +bi }, where f ∗ > 0, and taking the stepsize tk = λk f (xk )/kξk k. He shows convergence of (xk )k=1,...,∞ to a point in X ∗ if (λk ) k=0,...,∞ > 0 is a divergent series that converges to 0. From a practical point of view subgradient optimization has solved quite successfully a wide range of problems. This means that many problems are quite surprisingly well conditioned. Subgradient optimization fails miserably on ill conditioned problems such as highly nonlinear multicommodity flow problems. Documenta Mathematica

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Nonsmooth problems are encountered in many disciplines. In some instances, they occur naturally and in others they result from mathematical transformations. The most complete reference on NDO problems is Chapter 5 of Shor’s book [40]. In Shor original work [35], he mentions solving the transportation problem using subgradient optimization. A standard transportation problem is a special case of an NDO that occurs when optimizing the Lagrangean dual of a constrained optimization problem: min s.t.

hc, yi Ay ≥ b By ≥ d

Dualizing the first set of constraints, with dual variables x, one gets the partial dual: f (x) = max(hb, xi + minhc − AT x, yi), y∈Y

x≥0

where Y = {y : By ≥ d} is a polyhedron, assumed to be compact, and with a set of extreme points given by{y i : i ∈ I}. One subgradient is thus any b−Ay i(x) where y i(x) is a minimizer of miny∈Y < c − AT x, y >. The formulation with an objective variable: min s.t.

hb, xi + w w ≤ hc − AT x, y i i∀i ∈ I

is the dual of the extended form of the Dantzig-Wolfe decomposition reformulation. 5

Other contributions

The seminal contribution by Held and Karp [17] on the traveling salesman problem introduced Lagrangean relaxation and the solution of the partial Lagrangean dual by subgradient optimization. They were not aware at that time of the Soviet revolution in this field, so they developed subgradient optimization from scratch. The symmetric traveling-salesman problem seeks to find a minimum cost tour in a complete undirected graph. A minimum tour k ∗ can be shown to be a 1-tour k with the added constraint that every node has degree 2. A 1-tree consists of a tree on the vertex set {2, 3, . . . , n}, together with two distinct edges at vertex 1. Therefore a formulation of the TSP is: mink s.t. : Documenta Mathematica

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and di,k is the degree of vertex i in the k th 1-tree, and ck is the cost of the 1-tree. Dualizing the degree constraints with multipliers πk leads to: ( ) n X f (π) = min ck + (di,k − 2)πi k

i=1

The cost of a minimum cost tour C ∗ is greater than or equal to maxπ f (π), which provides a lower bound on C ∗ . The computation of f (π) and a subgradient ξ involves the computation of a minimum cost 1-tree which can be done in O(n) steps. This formulation can be solved by the dual of Dantzig-Wolfe decomposition; this method shows the long tail typical of DW when no disaggregation is available, as seems the case here. Held and Karp [17] suggested the use of subgradient optimization, i.e., π m+1 = π m + tm ξ m , and proved a result analogous to Shor’s [35], with a constant tm = t¯ and convergence to within O(t¯) of the optimum is achieved. The solution of the TSP by branch and bound, using the bound computed here, was extremely successful, and led the authors to claim that: In fact, this experience with the traveling-salesman problem indicates that some form of the relaxation method may be superior to the simplex method for linear programs including a very large number of inequalities. The authors sought the wisdom of Alan Hoffman, who advised them that the method they just developed was closely related to the relaxation method for linear inequalities due to Agmon [1], and Motzkin and Schoengerg [23]. The relaxation method attempts to solve a system of linear inequalities {x : hai , xi+ bi ≤ 0 : i ∈ I} by projecting, in the case of Agmon, or reflecting in the case of Motzkin and Schoenberg on the most distant inequality. This amounts to minimizing the convex function < ai , x > +bi f (x) = max 0, max , i∈I kai k by using what became known as subgradient optimization with a stepsize that uses the information that f ∗ = 0. The algorithm is thus xk+1 = xk + λk ξk , where a¯ ξk = i , ka¯i k ,xi+bi with ¯i one of the indices that satisfies haika = f (x). ik Agmon [1] showed that for λ = 1 the convergence to a feasible point x∗ ∈ p 2 ∗ P = {x : f (x) = 0} is geometric at a rate 1 − µ , unless finite convergence occurs. Motzkin and Schoenberg [23] showed that if P is full-dimensional, finite

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convergence occurs if λ = 2. It was shown by the author [14] that Agmon’s definition of µ∗ can be written as µ∗ = inf x6∈P f (x)/d(x, P ), where d(x, P ) is the distance from x to P . It can also be shown [14] that µ∗ = cos ϕ as defined by Shor and Gamburd in Theorem 3.3. The works by Agmon and Motzkin and Schoenberg may be viewed as a precursors to the Soviet revolution. The successful solution of the traveling-salesman problem by computing bounds using subgradient optimization led to a true explosion of works in Lagrangean relaxation in the West; for example Fisher [11] and the many references therein. Karp, who was my thesis adviser, asked me to read the Held and Karp [17] paper as well as the ones by Agmon [1] and Motzkin and Schoenberg [23], and apply subgradient optimization to the transportation problem, and see if something could be done to explain the success of subgradient optimization. He also mentioned that the simplex method when applied to a “normallly” formulated system of equalities converges in a number of iterations which is a small multiple of the number of constraints, but that in the case where the number of variables is exponential, as in Dantzig-Wolfe decomposition, this estimate does not hold, thus requiring another solution technique. I engaged in a thorough review of the Soviet literature, and found the works of Eremin and Polyak, but missed the huge contributions by Shor. My 1971 thesis, published later as Goffin [12], has the following result, extending Motzkin and Schoenberg: the relaxation method converges finitely to a point x∗ ∈ P , where P is assumed to be full dimensional, if λ ∈ [1, 2] if P is obtuse h i √ 2 p λ∈ , 2 , if ν(P ) < 2/2, 1 + 2ν(P ) 1 − ν 2 (P )

where the condition number ν(P ) equals the minimum over all tangent cones to P of the sine of the half aperture of the largest spherical cone included in a tangent cone. It is easy to show that µ∗ ≥ ν(P ), and that if the constraints defining every tangent cone are linearly independent then µ∗ = ν(P ). Unfortunately, both ν(P ) and µ∗ are not polynomial, showing that the relaxation method is not a polynomial algorithm; see, for instance, Todd [42]. An unpublished result by the author shows that if {ai : i ∈ I} forms a totally unimodular matrix, then ν(P ) ≥ 1/n. The author then extended this convergence theory to subgradient optimization [13], and at the IIASA meeting in 1977, B. T. Polyak mentioned the work by Shor and Gamburd [38], and helped translate it, showing that this author’s results were essentially identical to that work. A very nice extension of the geometric convergence to the case of functional constraints has been published by Rosenberg [34], extending also results by Polyak [30]. A thorough study of subgradient optimization and its applications was performed by Held, Wolfe and Crowder [18]. They cite Polyak [30, 31] and Documenta Mathematica

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Shor [36]. As stepsize they use an underestimate f¯ of the function minimum f ∗ = minx∈X f (x) and use the Agmon relaxation step for an infeasible system: f (xk ) − f¯ xk+1 = ΠX xk − λk ξk kξk k2

(3)

where ξk ∈ ∂f (xk ). Paraphrasing from the Held et al. [18] paper on the “Validation of Subgradient Optimization”: We observed that the results did not seem to depend critically on the exact value of f¯. Of course it is necessary that the stepsize converges to 0, which we will not accomplish, with an underestimate f¯, unless we choose a sequence λk which tends to zero. Generally (but not always) a good rule is to set λ = 2 for 2n iterations (where n is a measure of the problem size), and then successively halve both the value of λ and the number of iterations until the number of iterations reaches some threshold z. λ is then halved every z iterations until the resulting λk is sufficiently small. It is thus possible to converge to a point not in the optimal set, altough in our work that almost never happened. We would particularly point out choice of stepsize as an area which is imperfectly understood. The answers provided to that question did not appear in the works of Shor [40] or Polyak [31], who prove rather weak results. The following result which extends [12] for Part 1 and Eremin [6, 7] for Part 2 appears in Allen et al. [2]: Theorem 5.1. In algorithm (3), 1. given δ > 0 and 0 < λk = λ < 2, there is some K such that f (xK ) ≤ f ∗ + λ/(2 − λ) (f ∗ − f¯) + δ; 2. if

∞ P

K

k=1

∗ λk = ∞, and λk → 0, then fK = min f (xk ) converges to f ∗ . k=1

This shows that the strategy of using λk → 0 is the correct one. The stepsize chosen by Held et al. [18] was, towards the end of the sequence, a halving of λ at each five iterations. This is equivalent to r(ϕ) = ( 12 )1/5 ∼ = .85, where r(ϕ) is defined in Shor’s theorem (3.3), assuming that Shor’s result of (3.3) applies in this case, which nobody has proven, but which seems quite likely to be provable. Held et al. [18] experimented with great success on a variety of problems, including the assignment problem, the multicommodity flow problems and the TSP, concluding: Briefly, we think that subgradient optimization holds promise for alleviating some of the computational difficulties of large-scale optimization. It is no panacea, though, and needs careful work to make it effective, but its basic simplicity and its wide range of applicability indicate that it deserves to be more widely studied. Further developments include: Documenta Mathematica

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1. An updating procedure for the target f¯ which can be either an overestimate f¯ > f ∗ or an underestimate f¯ < f ∗ , which now becomes a variable f¯k to be adjusted depending on the behaviour of the sequence f (xk ). Both Ahn et al. [21] and [15] show an updating rule for f¯k that guarantees that f∞ = inf k f (xk ) = f ∗ . 2. The computation of the primal variables y in section 4 can be done in the limit. This was shown by Shor [40] pp. 117–118 and improved by Anstreicher and Wolsey [3] and Nesterov [28]. Define the subgradient optimization by the recursive relation: xk+1 = ΠX (xk − tk ξk ), and the convex combination ti t¯ki = Pk

.

j=1 tj

Then the sequence defined by

y¯k =

k X

t¯ki y i

i=1

has the following properties Theorem 5.2. Let the sequence xk in the problem of section 4 be generated according to the formulae above, and ti → 0,

∞ X i=1

ti = ∞, and

∞ X

t2i < ∞.

i=1

Then xk → x∗ ∈ X ∗ , and any accumulation point of y¯k is in the optimal set Y ∗. 3. Nedic and Berstsekas [24] showed how to use the disaggregation structure, often available in problems obtained from Dantzig-Wolfe decomposition, by introducing an incremental subgradient method that cycles between the subgradients of the individual functions. 4. A recent paper by Nesterov [29] shows how to use subgradient optimization successfully on huge-scale problems, by using sparse updates of the subgradient, leading to excellent computational results. 6

Conclusions From my doctoral thesis: “To simplex, to relax: This thesis’ question Whether ’tis faster on P to iterate On the narrowing edge slung between vertices Documenta Mathematica

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Or to take the normal against a sea of planes And by opposing it, to leap to end today.”1 Silly and somewhat arrogantly optimistic. But as we have seen in this journey, subgradient optimization outperforms the simplex method in many instances. When it is good it’s very good, but when it is bad it is very bad, as is the case of ill-conditioned problems, or in the terminolgy of Shor, gully shaped functions. This has given rise to a set of more complex methods that deal well with ill conditioned problems. Among them are: 1. The r-algorithm due to Shor [40], which introduces a variable metric on top of the subgradient; it worked quite well with a heuristic choice of pararmeters, until a theoretically selected choice of the parameters by Yudin and Nemirovksi [25] led to the ellipsoid method and its deep theoretical significance 2. The mirror descent method of Yudin and Nemirovski [25] 3. The bundle method developed by Lemar´echal and Kiwiel and many others, about which a chapter appears in this book by Mifflin and Sagastizabal [22] 4. The analytic center cutting plane method by Goffin and Vial [16] Acknowledgments. The author’s research has been funded by the Natural Research Council in Science and Engineering of Canada for 39 years. I sincerely apologize to the many friends whose work I could not cite. References [1] S. Agmon, “The Relaxation Method for Linear Inequalities”, Canadian Journal of Mathematics, 6, 1954, 382–392. [2] E. Allen, R. Helgason and J. Kennigton, “ A Generalization of Polyak’s Convergence Result for Subgradient Optimization”, Mathematical Programming, 37, 1987, 309–317. [3] K.M Anstreicher and L.A. Wolsey, “Two ‘Well-Known’ properties of Subgradient Optimization”, Mathematical Programming, Ser. B 2009 120:213– 220. [4] G. B. Dantzig and P. Wolfe, “The Decomposition Algorithm for Linear Programming”, Econometrica 29 (4), (1961), 767–778. [5] G. B. Dantzig and P. Wolfe, “Decomposition Principle for Llinear Programs”, Operations Research, 8, (1960) 101–111. [6] I.I. Eremin, “Incompatible Solutions of Linear Inequalities”, Soviet Mathematics Doklady, 2, 1961, 821–824. 1 The

simplex method referred here is the one applied to a problem with an exponential number of hyperplanes. On normally formulated linear programs, A. Hoffman et al. [19] showed that the simplex method is vastly superior to the relaxation method. Documenta Mathematica

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[7] I.I. Eremin, “An Iterative Method for Cebysev Approximation of Incompatible Solutions of Linear Inequalities”, Soviet Mathematics Doklady, 3, 1962, 570–572. [8] I.I. Eremin, “A Generalization of the Motzkin-Agmon Relaxation Method”, Uspekhi Mathematicheski Nauk, 20, 1965, 183–187. [9] Yu.M Ermoliev: M. “Methods of Solutions of Nonlinear Extremal Problems”, Cybernetics 2,4, 1–16. [10] Yu.M. Ermoliev and N.Z. Shor, “On the Minimization of Nondifferentable Functions”, Cybernetics, 3, 1, 72. [11] M. L. Fisher, “The Lagrangian relaxation method for solving integer programming problems”, Management Science 27 (1981) 1–18. [12] J.L. Goffin: “The Relaxation Method for Solving Systems of Linear Inequalities”, Mathematics of Operations Research, 5,3 1980, 388–414. [13] J.L. Goffin, “On Convergence Rates of Subgradient Optimization Methods”, Mathematical Programming, 13, 1977, 329–347. [14] J.L. Goffin, “Nondifferentiable Optimization and the Relaxation Method”, Nonsmooth optimization: Proceedings of the IIASA workshop March 28– April 8, 1977 C. Lemar´echal and R. Mifflin eds. Pergamon Press 1978, 31–50. [15] J.L. Goffin and K.C. Kiwiel, “ Convergence of a Simple Subgradient method”, Mathematical Programming, 85, 1999, 207–211. [16] J.L. Goffin and J.P. Vial, “Convex Nondifferentiable Optimization: a Survey Focused on the Analytic Center cutting Plane Method”, Optimization Methods and Software, 17, 2002, 805–867. [17] M. Held and R.M. Karp, “The Traveling-Salesman Problem and Minimum Spanning Trees:Part II”, Mathematical Programming 1, 1971, 6–25. [18] M. Held, P. Wolfe and H.P. Crowder, “Validation of Subgradient Optimization”, Mathematical Programming, 6, 1974, 62–88. [19] A. Hoffman, M. Mannos, D. Sokolovsky and N. Wiegmann, “Computational Experience in Solving Linear Programs”, Journal of the SIAM, Vol. 1, No. 1 Sep., 1953. [20] J. E. Kelley, “The cutting plane method for solving convex programs”, Journal of the SIAM 8 (1960), 703–712. [21] S. Kim, H. Ahn and S-C. Cho, “ Variable Target Value Subgradient Method”, Mathematical Programming, 49, 1991, 359–369 Documenta Mathematica

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[22] R. Mifflin and C. Sagastizabal, “A Scince Fiction Story in Nonsmooth Optimization Originating at IIASA”, this volume. [23] T. Motzkin and I.J. Schoenberg, “The Relaxation Method for Linear Inequalities”, Canadian Journal of Mathematics, 6, 1954, 393–404. [24] A. Nedic and D.P. Bertsekas, “Incremental Subgradient Methods for Nondifferentiable Optimization”, SIAM J. OPTIM., Vol. 12, No. 1., 2001. [25] A. S. Nemirovskii and D. B. Yudin, Problem complexity and method efficiency in optimization, John Wiley, Chichester (1983). [26] A.S. Nemirovski, “Efficient Methods in Convex Programming”. Lecture Notes, Technion-Faculty of Industrial Engineering & Management, Fall Semester 1994/1995. [27] Yu. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer Academic Publishers, Boston, Doordrecht, London, 2004 [28] Yu. Nesterov, “Primal-Dual Subgradient Methods for Convex Problems”, Mathematical Programming, Ser B. 2009, 120:221–259. [29] Yu. Nesterov, “Subgradient Methods for Huge-Scale Optimizations Problems”, CORE Discusion Papers, 2012/2. [30] B.T. Polyak, “ A General Method of Solving Extremal Problems”, Soviet Math. Doklady, 8, 593–597 , 1967. [31] B.T. Polyak, “ Minimization of Unsmooth Functionals”, U.S.S.R. Computational Mathematics and Mathematical Physics, 9, 509–521, 1969. [32] B.T. Polyak, Introduction to Optimization, Optimization Software, Inc., Publications Division, New York, 1987. [33] B.T. Polyak, “Subgradient Methods: A Survey of Soviet Research” Nonsmooth optimization: Proceedings of the IIASA workshop March 28–April 8, 1977 C. Lemar´echal and R. Mifflin eds. Pergamon Press 1978, 5–30. [34] E. Rosenberg, “A Geometrically Convergent Subgradient Optimization Method for Nonlnearly Conctrained Convex Programs”, Mathematics of Operations Research, 13, 3, 1988. [35] N.Z Shor: “An application of the method of gradient descent to the solution of the network transportation problem”. In: Materialy Naucnovo Seminara po Teoret i Priklad. Voprosam Kibernet. i Issted. Operacii, Nucnyi Sov. po Kibernet, Akad. Nauk Ukrain. SSSR, vyp. 1, pp. 9–17, Kiev 1962. [36] N.Z Shor: “On the structure of algorithms for numerical solution of problems of optimal planning and design”. Diss. Doctor Philos. Kiev 1964 Documenta Mathematica

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[37] N.Z. Shor, “On the Rate of Convergence of the Generalized Gradient Method”, Kibernetika, 4, 3, 1968. [38] N.Z. Shor and P.R. Gamburd, “Some Questions Concerning the Convergence of the Generalized Gradient Method”, Kibernetika, 7,6, 1971. [39] N.Z. Shor, “Generalizations of Gradient Methods for Nonsmooth Functions and their Applications to Mathematical Programming”, Economic and Mathematical Methods, Vo. 12, No. 2 pp. 337–356 (in Russian) 1976. [40] N. Z. Shor, Minimization Methods for Non-differentiable Functions (in Russian), Naukova Dumka, Kiev, 1979. [English translation: Springer, Berlin, 1985]. [41] N. Z. Shor, Nondifferentiable Optimization and Polynomial Problems, Kluwer Academic Publishers, Boston, Doordrecht, London 1998. [42] M.J. Todd, “Some Remarks on the Relaxation Method for Linear Inequalities”, Technical Report No. 468, SORIE, Cornell University, Ithaca, New York, 1980. [43] P. Wolfe. “A method of conjugate subgradients for minimizing nondifferentiable functions,” Mathematical programming study, 3 (1975) 145–173.

Jean-Louis Goffin Professor emeritus in Management Science Desautels Faculty of Management McGill University Montreal, Quebec Canada [emailprotected]

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A Science Fiction Story in Nonsmooth Optimization Originating at IIASA ´ bal Robert Mifflin and Claudia Sagastiza

2010 Mathematics Subject Classification: 65K05, 49J52, 49M05, 90C30 Keywords and Phrases: Nonsmooth optimization, bundle methods, superlinear convergence Warning to the reader: despite its title, this story has no otherworldly planets, robots or galactic monsters; just a collection of fading memories confirming that optimization research is a perfect example of human synergy and persistence. As in a fairy tale, this story starts in a castle, Schloss Laxenburg, one of the residences of the imperial Habsburg family located south of Vienna. In fact, it was one of Maria Theresa’s summer houses. Many long years ago (forty plus) there once was a meeting of representatives from the Eastern and Western blocks which begat an international research organization to be located in Laxenburg, Austria. The International Institute for Applied Systems Analysis (IIASA) was thus created, with the purpose of building bridges across the Iron Curtain by means of scientific cooperation. This global, rather than nationalistic, goal was very bold and innovative. Since its creation, IIASA has pursued the above goal and today it is focused on issues such as energy use and climate change, food and water supplies, poverty and equity, population aging, and sustainable development. The institute’s research is independent of political or national interests; and the motto “Science for global insight” appears in its logo. But this is another story; here, we will rather look back, all the way to the IIASA beginnings and somewhat before to 1959, in order to give an answer to the question of whether or not, superlinear convergence for nonsmooth optimization is science fiction, as nicely phrased by Claude Lemar´echal in the 1970s. The Founding Fathers Before 1975 Claude Lemar´echal and Philip Wolfe independently created bundle methods that minimize a convex function f for which only one subgradient at a point is computable. The work of both authors appears in a 1975 Mathematical Programming Study. Documenta Mathematica

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Bundle methods are based on and improve on cutting-plane methods due to E. W. Cheney and A. A. Goldstein (1959) and to J. E. Kelley (1960). But this primal interpretation came much later. At first, a dual view was predominant: algorithms were designed to approximate a subdifferential set in such a way as to asymptotically satisfy (the nondifferentiable version of) Fermat’s condition, 0 ∈ ∂f (¯ x) where x ¯ is a minimizer. Since the new methods seemed to resemble conjugate gradient ones, they were called conjugate subgradient methods by Wolfe. The same algorithms were named extended Davidon methods by Lemar´echal, possibly with the hope for rapid convergence in mind. Indeed, after W. Davidon (1959) and R. Fletcher and M. Powell (1963) developed superlinearly convergent quasi-Newton methods for smooth minimization, rapid convergence was on everyone’s mind. For nonsmooth functions, however, this goal was seen as a wondrous grail, the object of an extended and difficult quest, which would take more than 30 years to achieve. When Robert Mifflin heard about the new methods, he gave up on an algorithm that moved and shrank an n-dimensional simplex, because bundle methods use previously generated subgradient information in a more efficient manner. He then defined a large class of nonconvex functions, called semismooth, and a dual-type bundle algorithm that achieved convergence to stationary points for such functions. All of the above research provided a way to solve dual formulations of large-scale optimization problems where underlying special structure could be exploited through the future use of parallel computing. In view of the new advances in the area, Wolfe influenced IIASA to form a nonsmooth optimization (NSO) task-force, including Lemar´echal, Mifflin, and certain Russians and Ukrainians. Among the latter, E. A. Nurminskii was expected at the beginning, but, probably due to the actions of Soviet authorities, could not make it to Laxenburg until one year after the departure of Lemar´echal and Mifflin. With the support of Michel Balinski (Chairman of the System and Decision Sciences Area at IIASA), the task-force organized at Laxenburg in 1977 a two week long participant-named “First World Conference on Nonsmooth Optimization”. From the Soviet side, there were B. T. Polyak and B. N. Pshenichnyi, while the West was represented by R. Fletcher, J. Gauvin, J.-L. Goffin, A. Goldstein, C. Lemar´echal, R. Marsten, R. Mifflin and P. Wolfe. Most of the participants wrote articles published in a 1978 IIASA Proceedings Series book. At those times when politics mixed with science, researchers were warned that their phones might be tapped and looked for hidden microphones in their table lamps. So this first international workshop was viewed as going beyond mathematics and, in his opening speech, Lemar´echal, feeling the importance of the moment, welcomed the participants with the words, To begin, let us break the glass. His emotion made his French (glace) supersede his English (ice)!1 1 At

a later Cambridge meeting Claude topped that slip of the tongue with the line “Now, I am only speaking in words” rather than the English equivalent “roughly speaking”, meaning here, “without mathematical precision”. Documenta Mathematica

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At the meeting, each participant presented his work during a three hour period in the morning, and the afternoon was devoted to brainstorming. These exchanges increased the participants’ awareness of the strong connections between nonlinear programming and nonsmooth optimization. In particular, Roy Marsten explained boxstep methods, and Boris Pshenichnyi’s talk suggested a link with Sequential Quadratic Programming, hinting at the possibility of superlinear convergence. The new conjugate-subgradient-like methods were the subject of many discussions during this first workshop. Their novelty was in that, unlike most subgradient methods that could be thought of as being forgetful and also different from smooth algorithms, the new methods kept past basic information in memory. Indeed, for progressing from the current iterate to the next one, a direction is defined by solving a quadratic program with data consisting of function and subgradient values from several past points. It is precisely this collection of information generated at previous iterations that is referred to as “the bundle”. Actually, the terminology was born during a workshop lunch: • bundle in English; • faisceau in French, a word that raised some concerns among English speaking participants, who wondered if it would connote fascism (it does not); and • Schar in German. As noted by Wolfe (while chewing Wiener Schnitzel mit Sp¨atzle), the German word sounds close to Shor. In those times, the r-algorithm of N. Z. Shor was the bˆete noire of NSO researchers, because of its reported success in many practical applications. This is, in spite of the method (a combination of steepest descent and conjugate gradients) lacking a general convergence proof. When there is convergence little is known about its rate, except for a recent (2008) work by Jim Burke, Adrian Lewis and Michael Overton, interpreting the ralgorithm as a variable metric method that does not satisfy the secant equation (a partial convergence rate result is given, for a convex quadratic function of two variables). This interpretation could help in unveiling today’s mystery behind the excellent performance of the r-algorithm. The r-algorithm is a space-dilation method, a family of (not so amnesic!) subgradient algorithms using information from both a current and a previous iterate, and usually having excellent numerical behavior. This family includes a variant related to the symmetric rank-one quasi-Newton method. It was this type of recurrent finding that kept alive the quest for rapid convergence. The ε-subdifferential and the road to implementation A second international workshop took place at IIASA in 1980, with contributions from Y. M. Ermoliev, J.-L. Goffin, C. Lemar´echal, R. Mifflin, E. A. Documenta Mathematica

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Nurminskii, R. T. Rockafellar, A. Ruszczynski, and A. P. Wierzbicki. In the conference book, Terry Rockafellar wrote about the important class of lower C2 functions, calling them particularly amenable to computation;2 ten years before he had introduced the concept of approximate subgradients, which was extended to nonconvex functions by Al Goldstein in 1977. In 1991, after many years of joint climbing trips in the Dolomites with discussions on this subject, C. Lemar´echal and Jochem Zowe came up with the eclipsing concept, aimed at defining a first-order approximation of a multi-valued mapping. The idea of an approximate subdifferential turned out to be fundamental for nonsmooth optimization. In particular, it is crucial for the effectiveness of bundle methods for large problems, but this is not its only important property. Indeed, on the theoretical side, the incorporation of an “expansion” parameter ε makes the multifunction ∂ε f (x) both inner and outer semicontinuous in the variables ε and x. For the exact subdifferential, the latter semicontinuity property holds (the subdifferential has a closed graph). Inner semicontinuity is of paramount importance, since it guarantees that having sequences xk → x ¯ and εk → 0, and a zero subgradient, 0 ∈ ∂f (¯ x), there exists an approximate subgradient sequence g k converging to zero: ∂εk f (xk ) ∋ g k → 0. Since the goal of any sound optimization method is to asymptotically satisfy Fermat’s condition, without inner continuity there is no hope. Now, this essential property holds only for approximate subgradients, but the available information is from exact subgradients. What to do? Here arises an important algorithmic consequence of the concept, known in the area as a transportation ´ formula, introduced by Lemar´echal in his Th`ese d’Etat from 1980. This simple, yet powerful, formula for convex functions relates exact subgradients (at one point) to inexact ones (at another point), as follows:

g i ∈ ∂f (xi ) =⇒ g i ∈ ∂εˆf (ˆ x) for εˆ = f (ˆ x) − f (xi ) − g i , x ˆ − xi . By means of this relation, bundle methods relate past exact subgradient information to a special ε-subgradient at a so-called serious point x ˆ, a point which gives significant progress towards the goal of minimizing the objective function (in bundle jargon, non-serious points are called null ). This special subgradient and its corresponding εˆ are called the aggregate subgradient and error, respectively. Together with a serious subsequence of iterates, these aggregate objects ensure limiting satisfaction of Fermat’s condition. The notion of an approximate subdifferential was algorithmically exploited for the first time by Dimitri Bertsekas and S. Mitter, early on in 1971. In 1974 Rockafellar visited Kiev and gave a talk on the subject which was translated into Russian by Pshenichnyi. This made it possible for Evgenii Nurminskii to learn about the subject. He then started to study the semicontinuity properties of this new set-valued operator and, after some joint work with 2 These

functions had been introduced in 1974 by Robert Janin in his University of Paris IX PhD dissertation Sur la dualit´ e et la sensibilit´ e dans les probl` emes de programmation math´ ematique. Documenta Mathematica

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Lemar´echal, eventually established its continuity. A comprehensive set of useful ε-subdifferential calculus rules was developed by Jean-Baptiste Hiriart-Urruty. An interesting application of the ε-subdifferential, significant for numerical performance, is that past bundle information can be “compressed” into the aggregate subgradients and errors, without loss of global convergence. The compression mechanism allows for discarding bundle information, keeping only enough to construct the last bundle subproblem solution, for example, only the solution itself. This makes the next direction defining subproblem easier to solve, a feature that is not present in the original cutting-plane method, which has to keep all of the past information for the sake of convergence. For this reason cutting-plane methods often suffer from a slow tailing-off convergence effect. Thanks to their potential for practical implementation, bundle methods were considered in several variants in the early 1990s. Trust region bundle methods and zig-zag searches were developed for convex and nonconvex functions by Zowe and his PhD student H. Schramm. Level variants were brought from Moscow to Paris by Arkadi Nemirovski and Yuri Nesterov, who wrote a paper with Lemar´echal on this subject. The development of technical tools for showing convergence of bundle methods and incorporating a compression mechanism in the algorithmic process is due to Krzysztof Kiwiel. He also developed a very efficient quadratic programming solver for the bundle direction subproblems, and systematically extended the methodology to different cases such as nonconvex and constrained ones. The first VU and the primal view The issue of increasing convergence speed of NSO methods was a recurrent obsession. For single variable problems, a superlinearly convergent method was devised by Lemar´echal and Mifflin in 1982. It has a very simple rule for deciding if, near a serious point, the function’s graph looks V-shaped (nonsmooth piecewise linear), or U-shaped (smooth quadratic). In the former case, a V-model, made from two cutting planes, is used to approximate the function. In the latter case, the difference of two “serious-side” derivatives is used to give second-order information for creating a quadratic U-model. Since cutting-plane methods are known to have finite termination for piecewise affine functions, these cases are solved efficiently with V-model minimizers. The same holds for smooth cases, because they are handled well via quasi-Newton moves. Nevertheless, this fast algorithm had the handicap of not extending directly to functions of several variables. The difficulty with extending VU-concepts to multidimensional problems was eventually solved, but it took almost 20 years to find the right objects, after a detour involving work descending from that of J.-J. Moreau and K. Yosida. The challenge was to find a generalization for the notion of a Hessian which is adequate for a black-box setting, that is, one that could be constructed from Documenta Mathematica

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bundle information consisting of function and single subgradient values at each computed point. At this stage, the primal interpretation of bundle methods became handy, since when considered as a stabilized cutting-plane method, there is a direct link between certain bundle iterates and the proximal point theory initiated by B. Martinet in 1970. After the seminal work on this subject by Terry Rockafellar in 1976, theoretical proximal results blossomed during the 1980s and 90s. An important step towards practical implementation was taken by Masao f*ckushima and Alfred Auslender, who independently showed that by not stopping bundling with a serious point one produced a sequence converging to a proximal point. Ending null steps with a serious step leads to an approximation of a proximal point. In 1993 Claude Lemar´echal and Claudia Sagastiz´abal interpreted the bundle direction as coming from a preconditioned gradient direction for minimizing the Moreau-Yosida regularization function associated with the proximal points. This interpretation led to a BFGS proximal approach opening the way to variable prox-metric bundle methods, which made quasi-Newton updates for a Moreau-Yosida regularization that was not fixed (the proximal parameter varies with the iterations). So the approach looked, in fact, like a dog chasing its tail. The smoothing effect of the Moreau-Yosida operator led to the belief that the key to defining an appropriate Hessian was to find proper proximal parameters (as in the BFGS proximal approach). This was a false track; in 1997 Lemar´echal and Sagastiz´ abal showed that for the Moreau-Yosida regularization to have a Hessian everywhere, the (nonsmooth!) function f needed to be sufficiently smooth and have a Hessian itself . . . once again, the elusive rapid convergence seemed out of reach. Moving fast is possible, if in the right subspace In their negative results from 1997, when studying the Moreau-Yosida Hessian, Lemar´echal and Sagastiz´ abal noticed that a nonsmooth function f exhibits some kind of second order behavior when restricted to a special subspace. More precisely, the function has kinks on (a translation of) the tangent cone to ∂f (¯ x) at the zero subgradient and appears smooth or “U-shaped” on (a translation of) the normal cone. Under reasonable assumptions related to the minimizer x ¯ being nondegenerate, the cones above are in fact complementary subspaces, called V and U , because they concentrate, respectively, all of the nonsmoothness and smoothness of f near x ¯. In the same work it was noticed that a Newton step based on the Hessian of the Moreau-Yosida regularization has no V-subspace component. The seed of just dropping off the regularization began to germinate. In the period 1984–96 Mifflin came up with similar concepts and conclusions in a different manner based on the bundle algorithm itself. The algebra associated Documenta Mathematica

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with the bundle method subproblem solution naturally breaks it into local V and U components with all the active subgradients having the same Ucomponent, which suggests that U is the space of differentiability. Associated with this he also developed the idea of an algorithm step being the sum of a bundle serious step and a U-Newton step. The U-Lagrangian from 2000, defined by Lemar´echal, Fran¸cois Oustry, and Sagastiz´ abal, proved useful as a theoretical tool to extract implicitly second order information from a nonsmooth function without resorting to the MoreauYosida regularization. Its associated U-Hessian turns out to be the correct second order object for NSO, akin to the projected Hessian in smooth nonlinear programming. In some favorable cases (involving strong minimizers) a conceptual VU-Newton step, constructed from the sum of a V-step and a U-step depending on the result of the V-step, can produce a superlinearly convergent sequence of iterates. Paraphrasing Lemar´echal words: with the U-Lagrangian came the realization that, when moving along a V-shaped valley of nondifferentiability which is tangent to the U-subspace at the minimizer, a Newton-like method could drive the algorithm convergence with the desired speed. The jackpot had been finally hit! Or not yet? In a manner similar to the proximal point algorithm, the ULagrangian superlinear scheme was highly conceptual, as it depended on information at the minimizer being sought, i.e. assuming the dog had already caught its tail. It would take some more years of hard work to produce implementable VUversions. The process was started by Oustry, who produced a rapidly convergent VU-algorithm with dual steps for the special case of a max-eigenvalue function. Two quadratic programming problems needed to be solved per iteration, instead of only one, as in classical bundle algorithms. Unfortunately, the method, tailored for eigenvalue optimization, used rich black-boxes that computed more than one subgradient at each point. Mifflin and Sagastiz´ abal developed VU-theory further, defining a class of functions structured enough to generate certain primal and dual tracks (the class includes the max-eigenvalue case). In the meantime, the importance of structure producing nonsmoothness was noticed by Lewis, whose partly smooth functions formalize, in a general nonconvex setting, VU structure. This was followed by works by Aris Daniilidis, Warren Hare, Jerˆome Malick and others. A nice connection between U-Lagrangian methods and Sequential Quadratic Programming was given by Scott Miller and J. Malick. By relating primal and dual tracks to U-Lagrangians and proximal points, Mifflin and Sagastiz´ abal succeeded in creating a superlinearly convergent VU algorithm for very general convex functions. The method also sequentially solves pairs of quadratic programs, corresponding to finding approximations in both the primal and dual tracks. This culminated over 30 years of effort by many researchers, not limited to the ones mentioned here, and brought us to Documenta Mathematica

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Figure 1: Sublinear, linear, and supernatural convergence

our current realization of science fiction: Figure 1 shows rapid convergence of a quasi-Newton version of the VU-algorithm. √ The half-and-half function f (x) = xT Ax + xT Bx was created by Lewis and Overton to analyze BFGS behavior when minimizing a nonsmooth function. The 8-variable example in the figure has a matrix A with all elements zero, except for ones on the diagonal at odd numbered locations (A(i, i) = 1 for i = 1, 3, 5, 7). The matrix B is diagonal with elements B(i, i) = 1/i2 for i = 1, . . . , 8. The minimizer of this partly smooth convex function is at x ¯ = 0, where the V and U subspaces both have dimension 4; hence, the name halfand-half. Each graph in the figure shows function values from all points generated by its corresponding algorithm starting from the point having all components equal to 20.08. The top curve was obtained with a proximal bundle method, implemented in the code N1CV2 by Lemar´echal and Sagastiz´abal. The middle curve corresponds to the BFGS implementation by Overton, who adapted the method for nonsmooth functions via a suitable line search developed with Lewis. They argue that the linear convergence of “vanilla BFGS” as exhibited by this example is surprisingly typical for nonsmooth optimization. However, so far this has been proved only for a two variable example with the use of exact line searches, i.e., by exploiting nonsmoothness. It pays to exploit nonsmoothness, even in more than one dimension, and it can be done implicitly as shown by the (supernatural) curve at the bottom of the figure. This one results from the quasi-Newton VU algorithm that uses a BFGS update formula to approximate U-Hessians. Only its serious point subsequence has proven QDocumenta Mathematica

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superlinear convergence.3 The tops of the ending “humps” in this graph are due to “clumps” of null steps. In the bundle business, null steps remain the hard cookies to digest. Null points can be thought of as intermediate unavoidable steps, needed to make the bundle “sufficiently rich”, until enough progress is achieved and an iterate can be declared serious. This fact was also commented on by Stephen Robinson, who in 1999 proved R-linear convergence of ε-subgradient descent methods (including the serious subsequence of proximal bundle algorithms), for functions satisfying a certain inverse growth condition. The feature of eliminating unnecessary null steps is yet to be found in NSO, because it is not known what unnecessary means. An empirical observation of how the algorithmic process drives the aggregate gradient and error to zero shows that, in general, the aggregate error goes to zero fast, while it takes long time (including many null steps) for the aggregate gradient to attain a small norm. This phenomenon suggests there is a minimal threshold, which cannot be avoided, for the number of null steps between two serious iterates. But except for complexity results (referring to a worst case that is rare in practice), there is not yet a clear understanding of how to determine a realistic value for the threshold. Maybe in another 30 or 40 years the answer will be spoken in words in a future ISMP Optimization History book. In the meantime the quest continues with the search for rapid convergence to local minimizers for nonconvex functions. Concluding remarks The astute reader probably noticed that IIASA was not directly involved in VU theory and algorithm developments. The reason is that the institution discontinued support for nonsmooth optimization when its last man standing, Vladimir Demyanov, left IIASA in 1985. He had organized the last IIASA Workshop on Nondifferential Optimization, held in Sopron, Hungary in 1984, and was a very early contributor to the field with a minimax paper in 1968. The same reader of this article will notice a lack of references as the authors are “only speaking in words” to minimize the level of technicality. This choice was made to avoid the embarrassment of missed citations.

3 However, one can envision a smooth outer envelope function, starting at about evaluation number 37, which touches some points, is strictly concave and has an ending slope looking very close to minus infinity. It empirically shows R-superlinear convergence of the qNVU algorithm.

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Acknowledgements. The authors are grateful to C. Lemar´echal and E. A. Nurminskii for sharing various NSO memories, from IIASA and elsewhere. They also thank AFOSR, CNPq, Faperj, INRIA and NSF for many years of research support.

Robert Mifflin Neill 103 Washington State University Pullman WA 99164-3113 USA

Claudia Sagastiz´abal IMPA Estrada Dona Castorina 110 22460-320 Jardim Botanico Rio de Janeiro – RJ Brazil

[emailprotected]

[emailprotected]

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Documenta Math.

Broyden Updating, the Good and the Bad! Andreas Griewank

Abstract. 2010 Mathematics Subject Classification: 65H10, 49M99, 65F30 Keywords and Phrases: Quasi-Newton, secant condition, least change, bounded deterioration, superlinear convergence So far so good! We had an updating procedure (the ’full’ secant method) that seemed to work provided that certain conditions of linear independence were satisfied, but the problem was that it did not work very well. In fact it proved to be quite numerically unstable. Charles Broyden in On the discovery of the ‘good Broyden’ method [6]. The idea of secant updating As Joanna Maria Papakonstantinou recounted in her comprehensive historical survey [29], regula falsi and other variants of the secant method for solving one equation in one variable go back to the Babylonian and Egyptian civilizations nearly 4000 years ago. They may be viewed just as a poor man’s version of what is now known as Newton’s method, though we should also credit Al Tusi [20]. During antiquity the very concept of derivatives was in all likelihood unknown, and in modern times the evaluation (and in the multivariate case also factorization) of Jacobian matrices is frequently considered too tedious and computationally expensive. The latter difficulty was certainly the concern of Charles Broyden in the sixties, when he tried to solve nonlinear systems that arose from the discretization of nonlinear reactor models for the English Electric Company in Leicester [6]. Now we know that, due to diffusion, the resulting system of ODEs must have been rather stiff, but that property was only identified and analyzed a few years later by Dahlquist. Nevertheless, Broyden and his colleagues already used some implicit time integration schemes, which required solving sequences of slightly perturbed nonlinear algebraic systems F (x) = 0 for F : Rn 7→ Rn . Documenta Mathematica

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Broyden noted that one could avoid the effort of repeatedly evaluating and factoring the system Jacobian by exploiting secant information, i.e., function value differences with

yi ≡ Fi − Fi−1

Fj ≡ F (xj )

for

j≤i

Here, xi ∈ Rn denotes the current iterate and xj , for j < i, disctinct points at which F has been evaluated previously. With si ≡ xi − xi−1 the new approximation Bi to the Jacobian F ′ (xi ) ∈ Rn×n Bi si = yi = F ′ (xi )si + o(ksi k)

(1)

The first order Taylor expansion on the right is valid if F has a Jacobian F ′ (x) ∈ Rn×n that varies continuously in x. We will tacitly make this assumption throughout so that F ∈ C 1 (D) on some open convex domain D ⊂ Rn containing all evaluation points of interest. In the univariate case of n = 1, one can divide by si to obtain Bi = yi /si ≈ F ′ (xi ) uniquely. In the multivariate case, the secant condition merely imposes n conditions on the n2 degrees of freedom in the new approximating Jacobian Bi . A natural idea is to remove the indeterminacy by simultaneously imposing earlier secant conditions Bi sj = yj , for j = i − n + 1 . . . i. The resulting matrix equation for Bi has a unique solution provided the n + 1 points xi−n+j , for j = 0 . . . n, are in general position, i.e., do not belong to a proper affine subspace of Rn . Theoretically, that happens with probability 1, but in practice the step vectors sj , for j = i − n + 1 . . . i, are quite likely to be nearly linearly dependent, which leads to the observation of instability by Broyden cited above. Rather than recomputing Bi from scratch, Broyden reasoned that the previous approximation Bi−1 should be updated such that the current secant condition is satisfied, but Bi v = Bi−1 v in all directions v ∈ Rn orthogonal to si . As he found out ‘after a little bit of scratching around’, these conditions have the unique solution [2] ⊤ Bi = Bi−1 + ri s⊤ si si , with ri ≡ yi − Bi−1 si (2) i ⊤ Here the outer product Ci ≡ ri s⊤ i /si si of the column vector ri and the row ⊤ vector si represent a rank one matrix. This formula became known as the good Broyden update, because it seemed to yield better numerical performance than the so-called bad formula (6) discussed below. For a recent review of quasi-Newton methods see the survey by J. M. Martinez [25]. Broyden stated that the fact that Ci = Bi − Bi−1 turned out to be of rank one was pure serendipity. Even though he claimed ’When I was at University they did not teach matrices to physicists’, he realized right away that the low rank property could be used to reduce the linear algebra effort for computing the next quasi-Newton step

si+1 = −Bi−1 Fi Documenta Mathematica

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to O(n2 ). That compares very favourably with the n3 /3 arithmetic operations needed for a dense LU factorization of the new Jacobian F ′ (xi ) to compute the −1 Newton step −F ′ (xi )−1 Fi . If the previous step is given by si = −Bi−1 Fi−1 , one can easily check that the secant error vector ri defined in (2) is identical to the new residual, i.e., ri = Fi , which we will use below. Tacking on a sequence of rank one corrections to an initial guess B0 , and reducing the linear algebra effort in the process looks more like an engineering trick than an algorithmic device of mathematical interest. Yet after a few years and in close collaboration with his coauthors John Dennis and Jorge Mor´e, a beautiful theory of superlinear convergence theory emerged [7], which was later built upon by other researchers and extended to many update formulas. For a much larger class of methods named after Charles Broyden and his coauthors Abbaffy and Spedicato, see [1]. Least change interpretation John Dennis credits Jorge Mor´e with a short argument showing that the good Broyden formula is a least change update. Specifically, if we endow the real space of n × n matrices A with the inner product hA, Bi ≡ T r(A⊤ B) = T r(B ⊤ A) then the corresponding norm kAkF ≡

p hA, Ai ≥ kAk

(3)

is exactly the one introduced by Frobenius. It is bounded below by the consistent matrix norm kAk induced by the Euclidean vector norm kvk on Rn . The affine variety [yi /si ] ≡ B ∈ Rn×n : Bsi = yi has the n(n − 1) dimensional tangent space [0/si ] and the n dimensional orthogonal complement n×n [0/si ]⊥ ≡ vs⊤ : v ∈ Rn i ∈R

Hence, the smallest correction of Bi−1 to obtain an element of [yi /si ] is given by the correction ⊤ ⊥ Ci = ri s⊤ i /si si ∈ [ri /si ] ∩ [0/si ]

For formal consistency we will set Ci = 0 if si = 0 = yi , which may happen for all i ≥ j if we have finite termination, i.e., reach an iterate xj with Fj = 0. The geometry is displayed below and yields for any other element Ai ∈ [yi /si ] by Pythagoras kBi−1 − Ai k2F − kBi − Ai k2F = kCi k2F Documenta Mathematica

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In particular, we have the nondeterioration property kBi − Ai kF ≤ kBi−1 − Ai kF This to hold for all Ai ∈ [yi /si ] is in fact equivalent to the least change property of the update. Broyden stated this property apparently for the first time in his survey paper [4], which he rarely cited afterwards. Moreover, nondeterioration can be equivalently stated in the operator norm as kBi − Ai k ≤ kBi−1 − Ai k

(4)

which makes sense even on an infinite dimensional Hilbert space where k · kF is undefined. Sequential properties in the affine case ⊤ So far we have described the single least change update Ci = ri s⊤ i /si si , but the key question is of course how a sequence of them compound with each other. One can easily check that Bi+1 = Bi + Ci+1 = Bi−1 + Ci + Ci+1 satisfies the previous secant condition Bi+1 si = yi only if si and si+1 are orthogonal so that Ci+1 si = 0. In fact, exactly satisfying all n previous secant conditions is not even desirable, because that would lead back to the classical multivariate secant method, which was found to be rather unstable by Broyden and others. However, successive updates do not completely undo each other and thus eventually lead to good predictions Bi−1 si ≈ yi . Now we will briskly walk through the principal arguments for the case when F is affine on a finite dimensional Euclidean space. Later we will discuss

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whether and how the resulting relations extend to nonlinear systems and infinite dimensional Hilbert spaces. Suppose for a moment that our equation is in fact affine so that F (x) = A x + b

A ∈ Rn×n

with

and

b ∈ Rn .

−1 Then the secant conditions over all possible steps si = −Bi−1 Fi−1 are satisfied by the exact Jacobian A ∈ [yi /si ] since yi = Ai si by definition of F . Moreover, let us assume that A and all matrices B with kB −Ak ≤ kB0 −Ak have inverses with a uniform bound kB −1 k ≤ γ. This holds by the Banach Perturbation Lemma [27] for all B0 that are sufficiently close to a nonsingular A. Then we can conclude, as Broyden did in [3], that all Bi are nonsingular −1 and, consequently, all steps si = −Bi−1 Fi−1 are well defined and bounded by ksi k ≤ γkFi−1 k. Repeatedly applying Pythagoras’ identity we obtain for any i the telescoping result that i X

kCj k2F = kB0 − Ak2F − kBi − Ak2F ≤ kB0 − Ak2F .

j=1

Hence, we derive from Cj sj = rj and the fact that the Frobenius norm is stronger than the operator norm that lim kCj kF → 0 j

and

lim krj k/ksj k ≤ lim kCj k = 0. j

j

(5)

Whereas these limits remain valid in the nonlinear case considered below, they hold in a trivial way in the affine case considered so far. This follows from the amazing result of Burmeister and Gay [12] who proved that Broyden’s good method reaches the roots of affine equations exactly in at most 2 n steps. The proof appears a little like an algebraic fluke and there is nothing monotonic about the approach to the solution. Moreover, the restriction that the ball with radius kB0 − Ak contains no singular matrix can be removed by some special updating steps or line-searches as, for example, suggested in [26], [17], and [23], also for the nonlinear case. The glory: Q-superlinear convergence The property krj k/ksj k → 0 was introduced in [8] and is now generally known as the Dennis and Mor´e characterization of Q-superlinear convergence. The reason is that it implies, with our bound on the stepsize, that krj k/kFj−1 k ≤ γ −1 krj k/ksj k → 0 and thus kFi+1 k →0 kFi k

⇐⇒

kxi+1 − x∗ k → 0 kxi − x∗ k

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Charles Broyden and his fellow quasi-Newton musketeers, J. Dennis and J. Mor´e

kx − x∗ k and vice versa. Correspondingly, the central concept of Q-superlinear convergence is completely invariant with respect to the choice of norms, a highly desirable property that is not shared by the weaker property of Q-linear convergence, where the ratio of successive residual norms kF (xj )k or solution distances kxi − x∗ k is merely bounded away from 1. Under certain initial assumptions Q-superlinear convergence is also achieved in the nonlinear case, and under a compactness condition even in infinite dimensional space. All this without any exact derivative information or condition that the sequence of steps be in some sense linearly independent. Originally, it was widely believed that to ensure superlinear convergence one had to establish the consistency condition that the Bi converge to the true Jacobian F ′ (x∗ ). In fact, these matrices need not converge at all, but, theoretically, may wander around F ′ (x∗ ) in a spiral, with the correction norms kCj k square summable but not summable. This means that the predicted increments Bi−1 si /ksi k in the normalized directions si /ksi k cannot keep being substantially different from the actual increments yi /ksi k because the si /ksi k belong to the unit sphere, which is compact in finite dimensions. The seemingly counterintuitive nature of the superlinear convergence proof caused some consternation in the refereeing process for the seminal paper by Broyden, Dennis and Mor´e [7]. It eventually appeared in the IMA Journal of Applied Mathematics under the editorship of Mike Powell. Broyden had analyzed the affine case, John Dennis contributed the concept of bounded deterioration on nonlinear problems and Jorge Mor´e contributed the least change characterization w.r.t. the Frobenius norm leading to the proof of superlinear convergence. All this is not just for good Broyden, but for a large variety of unsymmetric and symmetric updates like BFGS, where the Frobenius norms must be weighted, which somewhat localizes and complicates the analysis. More specifically, suppose one starts at x0 in the vicinity of a root x∗ ∈ Documenta Mathematica

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F −1 (0) near which the Jacobian is nonsingular and Lipschitz continuous. Then the nondeterioration condition (4) becomes a bounded deterioration condition with Ai replaced by F ′ (x∗ ) and a multiplicative factor 1 + O(kxi − x∗ k) as well as an additive term O(kxi − x∗ k) on the right-hand side. From that one can derive Q-linear convergence provided B0 is close enough to F ′ (x∗ ), which, in turn, implies Q-superlinear convergence by the perturbed telescoping argument. More generally, we have the chain of implications Bounded deterioration =⇒ Linear Convergence =⇒ Q-superlinear Convergence. Actually, R-linear convergence is enough for the second implication. This modularization of the analysis is a very strong point of the Broyden-Dennis-Mor´e framework [7] and has allowed many other researchers to communicate and contribute in an economical fashion. Bad Broyden by inverse least change The BDM mechanism also applies to so-called inverse updates, especially Broyden’s second unsymmetric formula. It can be derived by applying the least change criterion to the approximating inverse Jacobian Hi = Bi−1

with

Hi y i = s i

The equation on the right is called the inverse secant condition, which must be satisfied by Hi if Bi = Hi−1 is to satisfy the direct secant condition (1). After exchanging si and yi and applying the good Broyden formula to Hi one obtains the inverse update on the left, which corresponds to the direct update of Bi on the right Hi = Hi−1 +

(si − Hi−1 yi )yi⊤ yi⊤ yi

⇐⇒

Bi = Bi−1 +

ri yi⊤ yi⊤ si

(6)

The correspondence between the two representations can be derived from the so-called Sherman–Morrison–Woodbury formula [13] for inverses of matrices subject to low rank perturbations. Broyden suggested this formula as well, but apparently he and others had less favourable numerical experience, which lead to the moniker Bad Broyden update. It is not clear whether this judgement is justified, since the formula has at least two nice features. First, the inverse is always well defined, whereas the inverse of the good Broyden update can be seen to blow up if yi⊤ Bi−1 si = 0. Second, the bad Broyden update is invariant with respect to linear variable transformations in that applying it to the system F˜ (˜ x) ≡ F (T x ˜) = 0 with det(T ) 6= 0 leads to a sequence of iterates x ˜i related to the original ones by ˜0 = B0 T . The good xi = T x ˜i , provided one initializes x ˜0 = T −1 x0 and B Documenta Mathematica

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Broyden formula, on the other hand, is dependent on the scaling of the variables via the Euclidean norm, but is independent of the scaling of the residuals, which strongly influences the bad Broyden formula. However, even for quasiNewton methods based on the good Broyden update, the squared residual norm often enters through the back door, namely as merit function during a line-search. The resulting stabilized nonlinear equation solver is strongly affected by linear transformations on domain or range. In this brief survey we have only considered full step iterations and their local convergence properties. Whether or not one should implement quasi-Newton methods by storing and manipulating the inverses Hi is a matter for debate. Originally, Broyden and his colleagues had apparently no qualms about this, but later it was widely recommended, e.g., by the Stanford school [14], that one should maintain a triangular factorization of the Bi for reasons of numerical stability. Now it transpires that the required numerical linear algebra games, e.g., chasing subdiagonal entries, are rather slow on modern computer architectures. In any case, the trend is to limited memory implementations for large scale applications, in view of which we will first try to study the influence of the variable number n on Broyden updating. Estimating the R-order and efficiency index One might fault the property of Q-superlinear convergence for being not sufficiently discriminating, because it can be established for all halfway sensible updating methods. In view of the limiting case of operator equations on Hilbert spaces to be considered later, one may wonder how the convergence rate of quasi-Newton methods depends on the dimension n. A finer measure of how fast a certain sequence xi → x∗ convergences is the so-called R-order ρ ≡ lim inf |log kxi − x∗ k|

1/i

i

The limit inferior on the right reduces to a proper limit when the sequence xi → x∗ satisfies kxi − x∗ k ∼ kxi−1 − x∗ kρ . This is well known to hold with ρ = 2 for all iterations generated by Newton’s method from an x0 close to a regular root x∗ . Generally, the R-order [27] of a method is the infimum over ρ for all locally convergent sequences (xi )i=1...∞ . The result of Burmeister and Gay implies 2 n step quadratic convergence of Broyden’s good √ method on smooth nonlinear equations. That corresponds to an R-order of 2n 2 = 1 + 1/(2n) + O(1/n2 ). We may actually hope for just a little more by the following argument adapted from a rather early paper of Janina Jankowska [21]. Whenever a Jacobian approximation Bi is based solely on the function values Fi−j = F (xi−j ) , for j = 0 . . . n, its discrepancy to the Jacobian F ′ (x∗ ) is likely to be of order O(kxj−n − x∗ k). Here we have assumed that things are going well in that the distances kxi − x∗ k decrease monotonically towards 0, so that the function value at the oldest iterate xi−n contaminates Bi most. Then the usual analysis of Newton-like iterations [9] Documenta Mathematica

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yields the proportionality relation kxi+1 − x∗ k ∼ kxi−n − x∗ k kxi − x∗ k The first term on the right represents the error in the approximating Jacobian Bi multiplied by the current residual Fi of order kxi − x∗ k. Substituting the i ansatz kxi − x∗ k ∼ cρ for some c ∈ (0, 1) into the recurrence and then taking the log base c one obtains immediately the relations ρi+1 ∼ ρi−n + ρi

=⇒

0 = Pn (ρ) ≡ ρn+1 − 1 − ρn

Hence, we can conclude that the best R-order we may expect from Broyden updating is the unique positive root ρn of the polynomial Pn (ρ). For n = 1, both Broyden updating methods reduce to the classical √ secant scheme, which is well known [27] to have the convergence order ρ1 = (1+ 5)/2. The larger n, the smaller ρn , and it was shown in [19] that asymptotically Pn−1 (0) ∋ ρn ≈ 1 + ln(n)/n ≈

√ n n

Here an ≈ bn means that the ratio an /bn tends to 1 as n goes to infinity. The second approximation means that we may hope for n step convergence of order n rather than just 2 n step convergence of order 2 as suggested by the result of Burmeister and Gay. The first approximation implies that the efficiency index [28] in the sense of Ostrowski (namely the logarithm of the R-order divided by the evaluation cost and linear algebra effort per step) satisfies asymptotically ln(n)/n ln(2) ln(ρn ) ≈ ≥ OP S(F ) + O(n2 ) OP S(F ) + O(n2 ) n OP S(F ) + O(n3 ) The lower bound on the right-hand side represents Newton’s method with divided difference approximation of the Jacobian, and dense refactorization at each iteration. As we can see there is a chance for Broyden updating to yield an efficiency index that is ln(n)/ ln(2) = log2 n times larger than for Newton’s method under similar conditions. This hope may not be in vain since it was shown in [19] that the R-order ρn is definitely achieved when the Jacobian is updated by the adjoint Broyden formula Bi = Bi−1 + ri ri⊤ (F ′ (xi ) − Bi−1 ) ri⊤ ri However, this rank-one-update is at least twice as expensive to implement since it involves the transposed product F ′ (xi )⊤ ri , which can be evaluated in the reverse mode of Algorithmic Differentiation. The latter may be three times as expensive as pure function evaluation, so that the efficiency gain on Newton’s method can be bounded below by (log2 n)/4 = log16 n. Documenta Mathematica

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Whether or not simple Broyden updating itself achieves the optimal R-order ρn has apparently not yet been investigated carefully. To be fair, it should be noted that taking roughly n/ log(n) simplified Newton steps before reevaluating and refactorizing the Jacobian in the style of Shamanski˘ı [22], yields the convergence order near 1 + n/ log(n) for any such cycle and the corresponding effort is approximately [n OP S(F ) + O(n3 )][1 + 1/ log(n)]. The resulting efficiency index is asymptotically identical to the optimistic estimate for Broyden updating derived above. Pushing n to infinity While Broyden updating is well established in codes for small and medium scale problems, its usefulness for large dimensional problems is generally in doubt. The first author who applied and analyzed Broyden’s method to a control problem in Hilbert space was Ragnar Winther [31]. Formally, it is easy to extend the Broyden method to an operator equation y = F (x) = 0 between a pair of Hilbert spaces X and Y . One simply has to interpret transposition as taking the adjoint so that v ⊤ represents a linear function in X = X ∗ such that v ⊤ w ≡ hv, wi yields the inner product. The Good Broyden Update is still uniquely characterized by the nondeterioration condition (4) in terms of the operator norm k · k. This implies bounded nondeterioration in the nonlinear case and everything needed to derive local and linear convergence goes through. However, the least change characterization and its consequences cannot be extended, because there is no generalization of the Frobenius norm (3) and the underlying inner product to the space B(X, Y ) of bounded linear operators. To see this, we simply have to note that, in n dimensions, the Frobenius norm of the identity operator is n, the sum of its eigenvalues. That sum would be infinite for the identity on l2 , the space of square summable sequences to which all separable Hilbert spaces are isomorphic. There is apparently also no other inner product norm on B(X, Y ) that is at least as strong as the operator norm so that the implication (5) would work. These are not just technical problems in extending the superlinear result, since X is infinite dimensional exactly when the unit ball and, equivalently, its boundary, the unit sphere, are not compact. That means one can keep generating unit directions s¯i ≡ si /ksi k along which the current approximation Bi is quite wrong. Such an example with an orthogonal sequence of si was given by Griewank [18]. There, on an affine bicontinuous problem, Broyden’s method with full steps converges only linearly or not at all. To derive the basic properties of Broyden’s method in Hilbert space we consider an affine equation 0 = F (x) ≡ A x − b with a bounded invertible operator A ∈ B(Y, X). Then we have the discrepancies Di = A−1 Bi − I ∈ B(X, Y )

and Ei ≡ Di⊤ Di ∈ B(X)

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inite. Now the Broyden good update can be rewritten as Di+1 = Di I − s¯i s¯⊤ =⇒ Ei+1 ≡ Ei − r¯i r¯i i

with r¯i ≡ A−1 ri /ksi k. In the finite dimensional case one could show that the Frobenius norm of the Di decreases monotonically. Now we see that the operators Ei are obtained from the E0 = D0⊤ D0 by the consistent subtraction of rank-one terms. Hence, they have a selfadjoint semidefinite limit E∗ . This implies, by a generalization of the interlacing eigenvalue theorem, that the eigenvalues (λj (Ei ))j=1...∞ of Ei are monotonically declining towards their limits p(λj (E∗ ))j=1...∞ . Correspondingly, we find for the singular values σj (Di ) = λj (Ei ) of the Di that q σj (Di+1 ) ≤ σj (Di ) and σj (Di ) → λj (E∗ ) for i → ∞ Similarly, it was proven by Fletcher that the BFGS update monotonically moves −1/2 −1/2 all eigenvalues of the symmetric discrepancy B∗ Bi B∗ − I between the Hessian B∗ and its approximations Bi towards zero. With regards to convergence speed it was shown in [18] for C 1,1 operator equations that Broyden’s method yields locally lim sup kA−1 Fi+1 k kA−1 Fi k ≤ σ∞ (D0 ) ≡ lim σj (D0 ) j→∞

i→∞

In other words, the Q-factor is bounded by the essential spectrum σ∞ (D0 ) of the initial relative discrepancy D0 = A−1 B0 − I. Hence, we must have Q-superlinear convergence if D0 or, equivalently, just B0 − A is compact, an assumption that is of course trivial in finite dimensions. Equivalently, we can require the preconditioned discrepancy D0 to be compact or at least to have a small essential norm. Thus we can conclude that Broyden updating will yield reasonable convergence speed in Hilbert space if D0 is compact or has at least a small essential norm σ∞ (D0 ) = σ∞ (Dj ). It is well known that the essential norm is unaffected by modifications of finite rank. On the other hand, all singular values σj (D0 ) > σ∞ (D0 ) are effectively taken out as far as the final rate of convergence is concerned. Limited memory and data sparse For symmetric problems the idea of limited memory approximations to the Hessian of the objective [24] has been a roaring success. In the unsymmetric case things are not so clear. Whereas in the unconstrained, quadratic optimization case conjugate gradients generates the same iterates as BFGS in an almost memoryless way, there is, according to a result of Faber and Manteuffel [11], no short recurrence for unsymmetric real problems. Correspondingly, the more or less generic iterative solver GMRES for linear problems requires 2 i vectors of storage for its first i iterations. The same appeared be true of Broyden’s Documenta Mathematica

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method, where starting from a usually diagonal B0 , one could store the secant pairs (sj , yj ) for j = 1 . . . i. The same appeared be true for Broyden’s method in inverse form, where starting from an usually diagonal H0 = B0−1 one could store the secant pairs (sj , zj ) with zj ≡ Hj−1 yj for j = 1 . . . i. Then the inverse Hessian approximations have the product representation " # Y i (sj − zj )s⊤ (si − zi )s⊤ j i Hi−1 Hi = Hi−1 + I+ = H0 s⊤ s⊤ i zi j zj j=1 Deuflhard et al. noticed in [10] that for the fullstep iteration successive sj and sj+1 = −Hj Fj satisfy the relation sj+1 = (sj − zj )ksj k2 /s⊤ i zj . Hence, one only needs to store the sj and one can then cheaply reconstruct the zj for applying the inverse in product form to any vector v usually the current residual Fi . Hence the storage requirement is only i + O(1) vectors of length n up to the i-th iteration. In contrast the storage requirement for i iterations of Bad Broyden appears to be twice as large [10], so at least in that sense the derogatory naming convention is justified. In either case, one normally wishes to limit the number of vectors to be stored a priori and thus one has to develop strategies for identifying and discarding old information. This issue has been extensively studied for the limited memory BFGS method and for Broyden updating it has been the focus of a recent PhD thesis [30]. Usually one wishes to get rid of information from earlier iterates because nonlinearity may render it irrelevant or even misleading near the current iterates. On discretizations of infinite dimensional problems, one may wish to discard all corrections of a size close to the essential norm σ∞ (D0 ), since no amount of updating can reduce that threshhold. In good Broyden updating the correction made to any row of the approximating Jacobian is completely independent of what goes on in the other rows. In other words we are really updating the gradients ∇Fk of the component functions Fk independently. That shows immediately that one can easily use the method for approximating rectangular Jacobians F ′ (x) for F : Rn 7→ Rm with m independent of n. Also in updating the k−th row one can disregard all variables that have no impact on Fk so that the corresponding Jacobian entries are zero. The resulting sparse update is known as Schubert’s method [5]. The least change characterization now applies in the linear subspace of matrices with the appropriate sparsity pattern, and the whole BDM locally linear and Q-superlinear convergence goes through without any modification. However, since the update matrices Cj are now of high rank, there is no longer any advantage compared to Newton’s method with regards to the linear algebra effort per step. On the other hand, large sparse Jacobians can often be evaluated exactly, possibly using algorithmic differentiation [16], at an entirely reasonable cost. In particular it was found that none of the constraint Jacobians in the optimization test collection CUTEr takes more than 18 times the effort of evaluating the Documenta Mathematica

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vector functions of constraints themselves. Since the sparsity patterns also tend to be quite regular, no methods based on Broyden type updating [15] can here compete with methods based on exact derivatives values. Whether or not that situation is really representative for problems from applications is not entirely clear. In any case we have to count the inability to effectively exploit sparsity as part of the Bad about Broyden updating. Still, there is a lot of Good as well, for which we have to thank primarily Charles Broyden, who passed away last year at the age of 78 after an eventful life with various professional roles and countries of residence. Acknowledgement. The author is indebted to Jorge Mor´e, Trond Steihaug, and other colleagues for discussions on the historical record. References [1] Joszef Abaffy, Charles Broyden, and Emilio Spedicato. A class of direct methods for linear systems. Numer. Math., 45(3):361–376, 1984. [2] C. G. Broyden. A class of methods for solving nonlinear simultaneous equations. Math. Comp., 19:577–593, 1965. [3] C. G. Broyden. The convergence of single-rank quasi-Newton methods. Math. Comp., 24:365–382, 1970. [4] C. G. Broyden. Recent developments in solving nonlinear algebraic systems. In Numerical methods for nonlinear algebraic equations (Proc. Conf., Univ. Essex, Colchester, 1969), pages 61–73. Gordon and Breach, London, 1970. [5] C. G. Broyden. The convergence of an algorithm for solving sparse nonlinear systems. Math. Comp., 25:285–294, 1971. [6] C. G. Broyden. On the discovery of the “good Broyden” method. Math. Program., 87(2, Ser. B):209–213, 2000. Studies in algorithmic optimization. [7] C. G. Broyden, J. E. Jr. Dennis, and J. J. Mor´e. On the local and superlinear convergence of quasi-Newton methods. JIMA, 12:223–246, 1973. [8] J. E. Dennis, Jr. and Jorge J. Mor´e. A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comp., 28:549–560, 1974. [9] J. E. Jr. Dennis and R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, 1996. Documenta Mathematica

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[10] Peter Deuflhard, Roland Freund, and Artur Walter. Fast secant methods for the iterative solution of large nonsymmetric linear systems. In IMPACT of Computing in Science and Engineering, pages 244–276, 1990. [11] Vance Faber and Thomas Manteuffel. Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Numer. Anal., 21(2):352–362, 1984. [12] D. M. Gay. Some convergence properties of Broyden’s method. SIAM J. Numer. Anal., 16:623–630, 1979. [13] D. M. Gay and R. B. Schnabel. Solving systems of nonlinear equations by Broyden’s method with projected updates. In Nonlinear Programming 3, O. Mangasarian, R. Meyer and S. Robinson, eds., Academic Press, NY, pages 245–281, 1978. [14] Philip E. Gill, Walter Murray, and Margaret H. Wright. Numerical linear algebra and optimization. Vol. 1. Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1991. [15] A. Griewank and A. Walther. On constrained optimization by adjoint based quasi-Newton methods. Opt. Meth. and Soft., 17:869–889, 2002. [16] A. Griewank and A. Walther. Principles and Techniques of Algorithmic Differentiation, Second Edition. SIAM, 2008. [17] Andreas Griewank. The “global” convergence of Broyden-like methods with a suitable line search. J. Austral. Math. Soc. Ser. B, 28(1):75–92, 1986. [18] Andreas Griewank. The local convergence of Broyden-like methods on Lipschitzian problems in Hilbert spaces. SIAM J. Numer. Anal., 24(3):684– 705, 1987. [19] Andreas Griewank, Sebastian Schlenkrich, and Andrea Walther. Optimal r-order of an adjoint Broyden method without the assumption of linearly independent steps. Optim. Methods Softw., 23(2):215–225, 2008. [20] Hermann Hammer and Kerstin Dambach. Sharaf al-tusi, ein vorl¨aufer von newton und leibnitz. Der mathematische und naturwissenschaftliche Unterricht, 55(8):485–489, 2002. [21] Janina Jankowska. Theory of multivariate secant methods. SIAM J. Numer. Anal., 16(4):547–562, 1979. [22] C. T. Kelley. A Shamanski˘ı-like acceleration scheme for nonlinear equations at singular roots. Math. Comp., 47(176):609–623, 1986. Documenta Mathematica

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[23] Dong-Hui Li and Masao f*ckushima. A derivative-free line search and global convergence of Broyden-like method for nonlinear equations. Optim. Methods Softw., 13(3):181–201, 2000. [24] Dong C. Liu and Jorge Nocedal. On the limited memory BFGS method for large scale optimization. Math. Programming, 45(3, (Ser. B)):503–528, 1989. [25] Jos´e Mario Mart´ınez. Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math., 124(1–2):97–121, 2000. Numerical analysis 2000, Vol. IV, Optimization and nonlinear equations. [26] J. J. Mor´e and J. A. Trangenstein. On the global convergence of Broyden’s method. Math. Comp., 30(135):523–540, 1976. [27] J. M. Ortega and W. C. Reinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, 2000. [28] A. Ostrowski. Solution of Equations and Systems of Equations. Academic Press, New York, 1966. [29] Joanna Maria Papakonstantinou. Historical Development of the BFGS Secant Method and Its Characterization Properties. PhD thesis, Rice University, Houston, 2009. [30] Bart van de Rotten. A limited memory Broyden method to solve highdimensional systems of nonlinear equations. PhD thesis, Mathematisch Instituut, Universiteit Leiden, The Netherlands, 2003. [31] Ragnar Winther. A numerical Galerkin method for a parabolic control problem. PhD thesis, Cornell University, 1977.

Andreas Griewank Institut f¨ ur Mathematik Humboldt Universit¨at zu Berlin Unter den Linden 6 10099 Berlin Germany [emailprotected]

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Documenta Math.

´odory on the Road to the Maximum Principle Carathe Hans Josef Pesch

Abstract. On his Royal Road of the Calculus of Variations 1 the genious Constantin Carath´eodory found several exits – and missed at least one – from the classical calculus of variations to modern optimal control theory, at this time, not really knowing what this term means and how important it later became for a wide range of applications. How far Carath´eodory drove into these exits will be highlighted in this article. These exits are concerned with some of the most prominent results in optimal control theory, the distinction between state and control variables, the principle of optimality known as Bellman’s equation, and the maximum principle. These acchievements either can be found in Carath´eodory’s work or are immediate consequences of it and were published about two decades before optimal control theory saw the light of day with the invention of the maximum principle by the group around the famous Russian mathematician Pontryagin. 2010 Mathematics Subject Classification: 01A60, 49-03, 49K15 Keywords and Phrases: History of calculus of variations, history of optimal control, maximum principle of optimal control, calculus of variations, optimal control

1

On the road

Carath´eodory’s striking idea was to head directly for a new sufficient condition ignoring the historical way how the necessary and sufficient conditions of the calculus of variations, known at that time, had been obtained. This article contains material from the author’s paper: Carath´ eodory’s Royal Road of the Calculus of Variations: Missed Exits to the Maximum Principle of Optimal Control Theory, to appear in Numerical Algebra, Control and Optimization (NACO). 1 Hermann Boerner coined the term “K¨ onigsweg der Variationsrechnung” in 1953; see H. Boerner: Carath´ eodorys Eingang zur Variationsrechnung, Jahresbericht der Deutschen Mathematiker Vereinigung, 56 (1953), 31–58. He habilitated 1934 under Carath´ eodory. Documenta Mathematica

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Figure 1: Constantin Carath´eodory – ΚωνσταντÐνος Καραθεοδορ¨ (1938) (Born: 13 Sept. 1873 in Berlin, Died: 2 Feb. 1950 in Munich, Germany) and Constantin Carath´eodory and Thales from Milet on a Greek postage stamp (Photograph courtesy of Mrs. Despina Carath´eodory-Rodopoulou, daughter of Carath´eodory. See: ∆. Καραθεοδορ-ΡοδοπÔλου, ∆. ΒλαχοστεργÐουΒασβατèκη: ΚωνσταντÐνος Καραθεοδορ: Ο σοφìς ÇΕλλην του Μονχου, Εκδìσεις Κακτος, Athens, 2001.) We follow, with slight modifications of the notation,2 Carath´eodory’s book of 1935, Chapter 12 “Simple Variational Problems in the Small” and Chapter 18 “The Problem of Lagrange”.3 We begin with the description of Carath´eodory’s Royal Road of the Calculus of Variations directly for Lagrange problems that can be regarded as precursors of optimal control problems. We will proceed only partly on his road, in particular we are aiming to Carath´eodory’s form of Weierstrass’ necessary condition in terms of the Hamilton function. For the complete road, see Carath´eodory’s original works already cited. Short compendia can be found in Pesch and Bulirsch (1994) and Pesch (to appear), too. ⊤ Let us first introduce a C 1 -curve x = x(t) = (x1 (t), . . . , xn (t)) , t′ ≤ t ≤ t′′ , in an (n + 1)-dimensional Euclidian space Rn+1 . The line elements (t, x, x) ˙ of the curve are regarded as elements of a (2n + 1)-dimensional Euclidian space, say S2n+1 . Minimize Z t2

I(x) =

L(t, x, x) ˙ dt

(1)

t1

2 We generally use the same symbols as Carath´ eodory, but use vector notation instead of his component notation. 3 The book was later translated into English in two parts (1965–67). The German edition was last reprinted in 1994.

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subject to, for the sake of simplicity, fixed terminal conditions x(t1 ) = a and x(t2 ) = b, t′ < t1 < t2 < t′′ , and subject to the implicit ordinary differential equation G(t, x, x) ˙ =0 (2) with a real-valued C 2 -function L = L(t, x, x) ˙ 4 and a p-vector-valued C 2 function G = G(t, x, x) ˙ with p < n, both defined on an open domain A ⊂ S2n+1 . It is assumed that the Jacobian of G has full rank, ∂Gk rank = p. (3) ∂ x˙ j k=1,...,p j=1,...,n

1st Stage: Definition of extremals. Carath´eodory firstly coins the term extremal in a different way than today. According to him, an extremal is a weak extremum of the problem (1), (2).5 Hence, it might be either a so-called minimal or maximal. 2nd Stage: Legendre-Clebsch condition. Legendre-Clebsch necessary condition Lx˙ x˙ (t, x, x) ˙

Carath´eodory then shows the

must not be indefinite.

Herewith, positive (negative) regular, resp. singular line elements (t, x0 , x˙ 0 ) ∈ A can be characterized by Lx˙ x˙ (t, x0 , x˙ 0 ) being positive (negative) definite, resp. positive (negative) semi-definite. Below we assume that all line elements are positive regular. In today’s terminology: for fixed (t, x) the map v 7→ L(t, x, v) has a positive definite Hessian Lvv (t, x, v). ´odory’s sufficient 3rd Stage: Existence of extremals and Carathe condition. We consider a family of curves which is assumed to cover simply a certain open domain of R ⊂ Rn+1 and to be defined, because of (3), by the differential equation x˙ = ψ(t, x) with a C 1 -function ψ so that the constraint (2) is satisfied. Carath´eodory’s sufficient condition then reads as follows. Theorem 1 (Sufficient condition). If a C 1 -function ψ and a C 2 -function S(t, x) can be determined such that L(t, x, ψ) − Sx (t, x) ψ(t, x) ≡ St (t, x),

(4)

L(t, x, x′ ) − Sx (t, x) x′ > St (t, x)

(5)

4 The twice continuous differentiability of L w. r. t. all variables will not be necessary right from the start. 5 In Carath´ eodory’s terminology, any two competing curves x(t) and x ¯(t) must lie in a close neighborhood, i.e., |¯ x(t) − x(t)| < ǫ and |x ¯˙ (t) − x(t)| ˙ < η for positive constants ǫ and η. The comparison curve x ¯(t) is allowed to be continuous with only a piecewise continuous derivative; in today’s terminology x ¯ ∈ P C 1 ([t1 , t2 ], Rn ). All results can then be extended to analytical comparison curves, if necessary, by the well-known Lemma of Smoothing Corners.

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Figure 2: Constantin Carath´eodory as a boy (1883), as ´el`eve ´etranger of the ´ Ecole Militaire de Belgique (1891), a type of military cadet institute, and together with his father Stephanos who belonged to those Ottoman Greeks who served the Sublime Porte as diplomats (1900) (Photographs courtesy of Mrs. Despina Carath´eodory-Rodopoulou, daughter of Carath´eodory. See: ∆. Καραθεοδορ-ΡοδοπÔλου, ∆. ΒλαχοστεργÐου-Βασβατèκη: ΚωνσταντÐνος Καραθεοδορ: Ο σοφìς ÇΕλλην του Μονχου, Εκδìσεις Κακτος, Athens, 2001.) for all x′ , which satisfy the boundary conditions x′ (t1 ) = a and x′ (t2 ) = b and the differential constraint G(t, x, x′ ) = 0, where |x′ −ψ(t, x)| is sufficiently small with |x′ −ψ(t, x)| 6= 0 for the associated line elements (t, x, x′ ), t ∈ (t1 , t2 ), then the solutions of the boundary value problem x˙ = ψ(t, x), x(t1 ) = a, x(t2 ) = b are minimals of the variational problem (1), (2). 2

Exit to Bellman’s Equation

Carath´eodory stated verbatim (translated by the author from the German edition of 1935, p. 201 [for the unconstrained variational problem (1)]: According to this last result, we must, in particular, try to determine the functions ψ(t, x) and S(t, x) so that the expression L∗ (t, x, x′ ) := L(t, x, x′ ) − St (t, x) − Sx (t, x) x′ ,

(6)

considered as a function of x′ , possesses a minimum for x′ = ψ(t, x), which, moreover, has the value zero. In today’s terminology: St = min {L(t, x, x′ ) − Sx x′ } ; ′ x

(7)

see also the English edition of 1965, Part 2) or the reprint of 1994, p. 201. This equation became later known as Bellman’s equation and laid the foundation of his Dynamic Programming Principle; see the 1954 paper of Bellman.6 6 In Breitner: The Genesis of Differential Games in Light of Isaacs’ Contributions, J. of Optimization Theory and Applications, 124 (2005), p. 540, there is an interesting comment

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Actually, the principle of optimality traces back to the founding years of the Calculus of Variations,7 to Jacob Bernoulli. In his reply to the famous brachistochrone problem8 by which his brother Johann founded this field in 16969 , Jacob Bernoulli wrote: Si curva ACEDB talis sit, quae requiritur, h.e. per quam descendendo grave brevissimo tempore ex A ad B perveniat, atque in illa assumantur duo puncta quantumlibet propinqua C & D: Dico, proportionem Curvae CED omnium aliarum punctis C & D terminatarum Curvarum illam esse, quam grave post lapsum ex A brevissimo quoque tempore emetiatur. Si dicas enim, breviori tempore emetiri aliam CFD, breviori ergo emetietur ACF DB, quam ACEDB, contra hypoth. (See Fig. 3.) If ACEDB is the required curve, along which a heavy particle descends under the action of the downward directing gravity from A to B in shortest time, and if C and D are two arbitrarily close points of the curve, the part CED of the curve is, among all other parts having endpoints C and D, that part which a particle falling from A under the action of gravity traverses in shortest time. Viz., if a different part CF D of the curve would be traversed in a shorter time, the particle would traverse ACF DB in a shorter time as ACEDB, in contrast to the hypothesis. Jacob Bernoulli’s result was later formulated by Euler10 (Carath´eodory: in one of the most wonderful books that has ever been written about a mathematical subject) as a theorem. Indeed, Jacob Bernoulli’s methods were so powerful and general that they have inspired all his illustrious successors in the field of the calculus of variations, and he himself was conscious of his outstanding results which is testified in one of his most important papers (1701)11 (Carath´eodory: by W. H. Flemming: Concerning the matter of priority between Isaacs’ tenet of transition and Bellman’s principle of optimality, my guess is that these were discovered independently, even though Isaacs and Bellman were both at RAND at the same time . . . In the context of calculus of variations, both dynamic programming and a principle of optimality are implicit in Carath´ eodory’s earlier work, which Bellman overlooked. For more on Bellmann and his role in the invention of the Maximum Principle, see Plail (1998) and Pesch and Plail (2009, 2012) 7 For roots of the Calculus of Variations tracing back to antiquity, see Pesch (2012). 8 Bernoulli, Jacob, Solutio Problematum Fraternorum, una cum Propositione reciproca aliorum, Acta Eruditorum, pp. 211–217, 1697; see also Jacobi Bernoulli Basileensis Opera, Cramer & Philibert, Geneva, Switzerland, Jac. Op. LXXV, pp. 768–778, 1744. 9 Bernoulli, Johann, Problema novum ad cujus solutionem Mathematici invitantur, Acta Eruditorum, pp. 269, 1696; see also Johannis Bernoulli Basileensis Opera Omnia, Bousquet, Lausanne and Geneva, Switzerland, Joh. Op. XXX (pars), t. I, p. 161, 1742. 10 Euler, L., Methodus inveniendi Lineas Curvas maximi minimive proprietate gaudentes, sive Solutio Problematis Isoperimetrici latissimo sensu accepti, Bousquet, Lausanne and Geneva, Switzerland, 1744; see also Leonhardi Euleri Opera Omnia, Ser. Prima, XXIV (ed. by C. Carath´ eodory), Orell Fuessli, Turici, Switzerland, 1952. 11 Bernoulli, Jacob, Analysis magni Problematis Isoperimetrici, Acta Eruditorum, pp. 213– Documenta Mathematica

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Figure 3: Jacob Bernoulli’s figure for the proof of his principle of optimality eine Leistung allerersten Ranges) not only by the dedication to the four mathematical heroes Marquis de l’Hˆospital, Leibniz, Newton, and Fatio de Duillier, but also by the very unusual and dignified closing of this paper: Deo autem immortali, qui imperscrutabilem inexhaustae suae sapientiae abyssum leviusculis radiis introspicere, & aliquousque rimari concessit mortalibus, pro praestita nobis gratia sit laus, honos & gloria in sempiterna secula. Trans.: Verily be everlasting praise, honor and glory to eternal God for the grace accorded man in granting mortals the goal of introspection, by faint (or vain) lines, into the mysterious depths of His Boundless knowledge and of discovery of it up to a certain point. – This prayer contains a nice play upon words: radius means ray or line as well as drawing pencil or also the slat by which the antique mathematicians have drawn their figures into the green powdered glass on the plates of their drawing tables. For the Lagrange problem (1), (2), Eq. (7) reads as St =

min

x′ such that G(t,x,x′ )=0

{L(t, x, x′ ) − Sx x′ } ;

(8)

compare Carath´eodory’s book of 1935, p. 349. Carath´eodory considered only unprescribed boundary conditions there. Carath´eodory’s elegant proof relys on so-called equivalent variational problems and is ommitted here; cf. Pesch (to appear). 3

On the road again

4th Stage: Fundamental equations of the calculus of variations. This immediately leads to Carath´eodory’s fundamental equations of the calculus of variations, here directly written for Lagrangian problems: Introducing 228, 1701; see also Jacobi Bernoulli Basileensis Opera, Cramer & Philibert, Geneva, Switzerland, Jac. Op. XCVI, pp. 895–920, 1744. Documenta Mathematica

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the Lagrange function M (t, x, x, ˙ µ) := L(t, x, x) ˙ + µ⊤ G(t, x, x) ˙ with the p-dimensional Lagrange multiplier µ, the fundamental equations are Sx = Mx˙ (t, x, ψ, µ) , St = M (t, x, ψ, µ) − Mx˙ (t, x, ψ, µ) ψ, G(t, x, ψ) = 0 .

(9) (10) (11)

These equations can already be found in Carath´eodory’s paper of 1926, almost 30 years prior to Bellman’s version of these equations. They constitute necessary conditions for an extremal of (1), (2). 5th Stage: Necessary condition of Weierstrass. Replacing ψ by x˙ in the right hand sides of (9)–(11), Weierstrass’ Excess Function for the Lagrange problem (1), (2) is obtained as E(t, x, x, ˙ x′ , µ) = M (t, x, x′ , µ) − M (t, x, x, ˙ µ) − Mx˙ (t, x, x, ˙ µ) (x′ − x) ˙

(12)

with line elements (t, x, x) ˙ and (t, x, x′ ) both satisfying the constraint (2). By a Taylor expansion, it can be easily seen that the validity of the Legendre-Clebsch condition in a certain neighborhood of the line element (t, x, x) ˙ is a sufficient condition for the necessary condition of Weierstrass, E(t, x, x, ˙ x′ , µ) ≥ 0 .

(13)

The Legendre–Clebsch condition can then be formulated as follows: The minimum of the quadratic form Q = ξ ⊤ Mx˙ x˙ (t, x, x, ˙ µ) ξ , subject to the constraint ∂G ξ=0 ∂ x˙ on the sphere kξk2 = 1, must be positive. This immediately implies Mx˙ x˙ G⊤ x˙ must be positive semi-definite . Gx˙ 0

(14)

This result will play an important role when canonical coordinates are now introduced. 6th Stage: Canonical coordinates and Hamilton function. New variables are introduced by means of y := Mx⊤ ˙ µ) , ˙ (t, x, x,

(15)

z := G(t, x, x) ˙ = Mµ⊤ (t, x, x, ˙ µ) .

(16)

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Figure 4: Constantin Carath´eodory in G¨ottingen (1904), his office in his home in Munich-Bogenhausen, Rauchstraße 8, and in Munich (1932) in his home office (Photographs courtesy of Mrs. Despina Carath´eodory-Rodopoulou, daughter of Carath´eodory. See: ∆. Καραθεοδορ-ΡοδοπÔλου, ∆. ΒλαχοστεργÐουΒασβατèκη: ΚωνσταντÐνος Καραθεοδορ: Ο σοφìς ÇΕλλην του Μονχου, Εκδìσεις Κακτος, Athens, 2001.) Because of (14), these equations can be solved for x˙ and µ in a neighborhood of a “minimal element” (t, x, x, ˙ µ),12 x˙ = Φ(t, x, y, z) ,

(17)

µ = X(t, x, y, z) .

(18)

Defining the Hamiltonian in canonical coordinates (t, x, y, z) by H(t, x, y, z) = −M (t, x, Φ, X) + y ⊤ Φ + z ⊤ X ,

(19)

the function H is at least twice continuously differentiable and there holds Ht = −Mt ,

Hx = −Mx ,

Hy = Φ⊤ ,

Hz = X ⊤ .

(20)

Letting H(t, x, y) = H(t, x, y, 0), the first three equations of (20) remain valid for H instead of H. Alternatively, H can be obtained directly from y = Mx⊤ ˙ µ) and 0 = G(t, x, x) ˙ because of (14) via the relations ˙ (t, x, x, x˙ = φ(t, x, y) and µ = χ(t, x, y), H(t, x, y) = −L(t, x, φ(t, x, y)) + y ⊤ φ(t, x, y) .

(21)

12 Carath´ eodory has used only the term extremal element (t, x, x, ˙ µ) depending whether the matrix (14) is positive or negative semi-definite. For, there exists a p-parametric family of extremals that touches oneself at a line element (t, x, x). ˙ However, there is only one extremal through a regular line element (t, x, x). ˙

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Note that φ is at least of class C 1 because L ∈ C 2 , hence H is at least C 1 , too. The first derivatives of H are, by means of the identity y = L⊤ ˙ ⊤, x˙ (t, x, x) Ht (t, x, y) = −Lt (x, y, φ) ,

Hx (t, x, y) = −Lx (t, x, φ),

Hy (t, x, y) = φ(t, x, y)⊤ . Therefore, H is even at least of class C 2 . This Hamilton function can also serve to characterize the variational problem completely. 4

Missed exit to optimal control

´odory’s closest approach to optimal control. 7th Stage: Carathe In Carath´eodory’s book of 1935, p. 352, results are presented that can be interpreted as introducing the distinction between state and control variables in the implicit system of differential equations (2). Using an appropriate numeration and partition x = (x(1) , x(2) ), x(1) := (x1 , . . . , xp ), x(2) := (xp+1 , . . . , xn ), Eq. (2) can be rewritten due to the rank condition (3):13 G(t, x, x) ˙ = x˙ (1) − Ψ(t, x, x˙ (2) ) = 0 . By the above equation, the Hamiltonian (21) can be easily rewritten as ¯ x, φ(2) ) + y (1) φ(1) + y (2) φ(2) H(t, x, y) = −L(t, ¯ x, φ(2) ) := L(t, x, Ψ, φ(2) ) with L(t, ⊤

⊤

(22)

and x˙ (1) = Ψ(t, x, φ(2) ) = φ(1) (t, x, y) and x˙ (2) = φ(2) (t, x, y). This is exactly the type of Hamiltonian known from optimal control theory. The canonical variable y stands for the costate and x˙ (2) for the remaining freedom of the optimization problem (1), (2) later denoted by the control. Nevertheless, the first formulation of a problem of the calculus of variations as an optimal control problem, which can be designated justifiably so, can be found in Hestenes’ RAND Memorandum of 1950. For more on Hestenes and his contribution to the invention of the Maximum Principle, see Plail (1998) and Pesch and Plail (2009, 2012). 8th Stage: Weierstrass’ necessary condition in terms of the Hamiltonian. From Eqs. (13), (15), (16), (19), and (20) there follows Carath´eodory’s formulation of Weierstrass’ necessary condition which can be interpreted as a precursor of the maximum principle E = H(t, x, y) − H(t, x, y ′ ) − Hy (t, x, y ′ ) (y − y ′ ) ≥ 0 , 13 The

(23)

original version is Γk′ (t, xj , x˙ j ) = x˙ k′ − Ψk′ (t, xj , x˙ j ′′ ) = 0, where k′ = 1, . . . , p, j = 1, . . . , n, j ′′ = p + 1, . . . , n. Note that Carath´ eodory used Γ in his book of 1935 instead of G which he used in his paper of 1926 and which we have inherit here. Documenta Mathematica

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where (t, x, y) and (t, x, y ′ ) are the canonical coordinates of two line elements passing through the same point. This formula can already be found in Carath´eodory’s paper of 1926. From here, there is only a short trip to the maximum principle, however under the strong assumptions of the calculus of variations as have been also posed by Hestenes (1950). For the general maximum principle see Bolyanskii, Gamkrelidze, and Pontryagin (1956). 5

Side road to a maximum principle of Optimal Control Theory

In Pesch, Bulirsch (1994), a proof for the maximum principle was given for an optimal control problem of type Z t2 ! L(t, z, u) dt = min subject to z˙ = g(t, z, u) t1

starting with Carath´eodory’s representation of Weierstrass’ necessary conditions (23) in terms of a Hamiltonian. In the following we pursue a different way leading to the maximum principle more directly, still under the too strong assumptions of the calculus of variations as in Hestenes (1950). Herewith, we continue the tongue-in-cheek story on 300 years of Optimal Control by Sussmann and Willems (1997) by adding a little new aspect. Picking up the fact that x˙ = v(t, x) minimizes v 7→ L∗v (t, x, v), we are led by (6) to the costate p = L⊤ ˙ [as in (15), now using the traditional v (t, x, x) notation] and the Hamiltonian H, H(t, x, p) = min{L(t, x, x) ˙ + p⊤ x} ˙ . x˙

Then Carath´eodory’s fundamental equations read as follows p = −Sx⊤ (t, x) ,

St = H(t, x, Sx⊤ ) .

This is the standard form of the Hamiltonian in the context of the calculus of variations leading to the Hamilton–Jacobi equation. Following Sussmann and Willems (1997) we are led to the now maximizing Hamiltonian (since we are aiming to a maximum principle), also denoted by H, H(t, x, u, p) = −L(t, x, u) + p⊤ u with p = L⊤ u (t, x, u) defined accordingly and the traditional notation for the degree of freedom, the control x˙ = u, when we restrict ourselves, for the sake of simplicity, to the most simplest case of differential constraints. It is then obvious that Hp⊤ = u as long as the curve x satisfies ˙ p(t) . x(t) ˙ = Hp⊤ t, x(t), x(t), Documenta Mathematica

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By means of the Euler-Lagrange equation d Lu (t, x, x) ˙ − Lx (t, x, x) ˙ =0 dt and because of Hx = −Lx , we obtain p(t) ˙ = −Hx⊤ (t, x, x, ˙ p(t)) .

(25)

Furthermore, we see Hu⊤ = −L⊤ u + p = 0. Since the Hamiltonian H(t, x, u, p) is equal to −L(t, x, u) plus a linear function in u, the strong Legendre–Clebsch condition for now maximizing the functional (1) is equivalent to Huu < 0. Hence H must have a maximum with respect to u along a curve (t, x(t), p(t)) defined by the above canonical equations (24), (25). If L depends linearly on u, the maximization of H makes sense only in the case of a constraint on the control u in form of a closed convex set Uad of admissible controls, which would immediately yield the variational inequality Hu (t, x, u ¯, p) (u − u ¯) ≤ 0 ∀u ∈ Uad

(26)

along a candidate optimal trajectory x(t), p(t) satisfying the canonical equations (24), (25) with u ¯ denoting the maximizer. That is the maximum principle in its known modern form. A missed exit from the royal road of the calculus of variations to the maximum principle of optimal control? Not at all! However, it could have been at least a first indication of a new field of mathematics looming on the horizon. See also Pesch (to appear). 6

´sume ´ Re

With Carath´eodory’s own words: I will be glad if I have succeeded in impressing the idea that it is not only pleasant and entertaining to read at times the works of the old mathematical authors, but that this may occasionally be of use for the actual advancement of science. [. . . ] We have seen that even under conditions which seem most favorable very important results can be discarded for a long time and whirled away from the main stream which is carrying the vessel science. [. . . ] It may happen that the work of most celebrated men may be overlooked. If their ideas are too far in advance of their time, and if the general public is not prepared to accept them, these ideas may sleep for centuries on the shelves of our libraries. [. . . ] But I can imagine that the greater part of them is still sleeping and is awaiting the arrival of the prince charming who will take them home.14 14 On

Aug. 31, 1936, at the meeting of the Mathematical Association of America in CamDocumenta Mathematica

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Figure 5: Constantin Carath´eodory on a hike with his students at Pullach in 1935 (Photographs courtesy of Mrs. Despina Carath´eodoryRodopoulou, daughter of Carath´eodory. See: ∆. Καραθεοδορ-ΡοδοπÔλου, ∆. ΒλαχοστεργÐου-Βασβατèκη: ΚωνσταντÐνος Καραθεοδορ: Ο σοφìς ÇΕλλην του Μονχου, Εκδìσεις Κακτος, Athens, 2001.) References Bellman, R. E. (1954) The Theory of Dynamic Programming. Bull. Amer. Math. Soc. 60, 503–516. Boltyanskii, V. G., Gamkrelidze, R. V., and Pontryagin, L. S. (1956) On the Theory of Optimal Processes (in Russian). Doklady Akademii Nauk SSSR 110, 7–10. ¨ Carath´eodory, C. (1926) Die Methode der geod¨atischen Aquidistanten und das Problem von Lagrange. Acta Mathematica 47, 199–236; see also Gesammelte Mathematische Schriften 1 (Variationsrechnung). Edited by the Bayerische Akademie der Wissenschaften, C. H. Beck’sche Verlagsbuchhandlung, M¨ unchen, Germany, 1954, 212–248. Carath´eodory, C. (1935) Variationsrechnung und partielle Differentialgleichungen erster Ordnung. Teubner, Leipzig, Germany. Carath´eodory, C. (1965–67) Calculus of Variations and Partial Differential Equations of the First Order, Part 1, Part 2. Holden-Day, San Francisco, bridge, Mass., during the tercentenary celebration of Harvard University; see Carath´ eodory, The Beginning of Research in the Calculus of Variations, Osiris 3 (1937), 224–240; also in Gesammelte Mathematische Schriften 2; edited by the Bayerische Akademie der Wissenschaften, C. H. Beck’sche Verlagsbuchhandlung, M¨ unchen, Germany, (1955), 93–107. Documenta Mathematica

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California. Reprint: 2nd AMS printing, AMS Chelsea Publishing, Providence, RI, USA, 2001. Carath´eodory, C. (1994) Variationsrechnung und partielle Differentialgleichungen erster Ordnung. With Contributions of H. Boerner and E. H¨ older. Edited, commented and extended by R. Kl¨otzler. Teubner-Archiv der Mathematik 18, Teubner-Verlagsgesellschaft, Stuttgart, Leipzig, Germany. Hestenes, M. R. (1950) A General Problem in the Calculus of Variations with Applications to the Paths of Least Time. Research Memorandum No. 100, ASTIA Document No. AD 112382, RAND Corporation, Santa Monica. Pesch, H. J. (2012) The Princess and Infinite-dimensional Optimization In: M. Gr¨ otschel (ed.): Optimization Stories. Documenta Mathematica. Pesch, H. J. and Plail, M. (2009) The Maximum Principle of Optimal Control: A History of Ingenious Ideas and Missed Opportunities. Control and Cybernetics 38, No. 4A, 973-995. Pesch, H. J. (to appear) Carath´eodory’s Royal Road of the Calculus of Variations: Missed Exits to the Maximum Principle of Optimal Control Theory. To appear in Numerical Algebra, Control and Optimization (NACO). Pesch, H. J., and Bulirsch, R. (1994) The Maximum Principle, Bellman’s Equation and Carath´eodory’s Work, J. of Optimization Theory and Applications 80, No. 2, 203–229. Pesch, H. J. and Plail, M. (2012) The Cold War and the Maximum Principle of Optimal Control. In: M. Gr¨otschel (ed.): Optimization Stories. Documenta Mathematica. Plail, M. (1998) Die Entwicklung der optimalen Steuerungen. Vandenhoeck & Ruprecht, G¨ ottingen. Sussmann, H. J. and Willems, J. C. (1997) 300 Years of Optimal Control: From the Brachystrochrone to the The Maximum Principle. IEEE Control Systems Magazine 17, No. 3, 32–44. Hans Josef Pesch Chair of Mathematics in Engineering Sciences University of Bayreuth 95440 Bayreuth Germany [emailprotected]

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Documenta Math.

The Cold War and the Maximum Principle of Optimal Control Hans Josef Pesch and Michael Plail

Abstract. By the end of World War II, the next global confrontation emerged: the confrontation between the USA and the USSR and their allies, so between the West and the East with their antagonistic fundamental political values and their ideological contradiction. This development may be seen as a consequence of Marxism-Leninism and its claim for the world revolution or as a consequence of the political and economical structure of the USA with its permanent pursuit of new markets. All this had had also consequences for mathematicians, because the flow of information, though not completely cut, was not as easy as before. Looking positively at side effects, however, the isolated research may have not been burdened by traditional thinking and that may have been fruitful. Russian mathematicians around Pontryagin in the Steklov Institute got, with the maximum principle, new results beyond former frontiers while the Americans around Hestenes at the RAND corporation were captured by the tradition of the Chicago School of the Calculus of Variations. Nevertheless, both groups paved the way for a new field in mathematics called Optimal Control – and their protagonists fell out with each other inside their groups. 2010 Mathematics Subject Classification: 01A60, 01A72, 01A79, 4903, 49K15, 00A06 Keywords and Phrases: History of optimal control, maximum principle of optimal control, optimal control With the advent of the Cold War mathematicians were immediately involved in the new global confrontation. A mathematical challenge of those times with This article is an easy-to-read and considerably shortened version of the authors’ paper entitled The Maximum Principle of Optimal Control: A History of Ingenious Ideas and Missed Opportunities [see Pesch and Plail (2009)], enriched by some anectodes. The conclusions therein and also here are extracted from the second author’s monograph on the development of optimal control theory from its commencements until it became an independent discipline in mathematics; see Plail (1998). Documenta Mathematica

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Figure 1: Johann Bernoulli’s price question of 1696 and its solution which was realized in the Zernike Science Park of the University of Groningen. This monument was erected in 1996 to honor one of the most famous former members of its faculty, Johann Bernoulli, who had been a professor there from 1695 to 1705. which they were confronted was: What is the optimal trajectory of an aircraft that is to be steered from a given cruise position into a favorable position against an attacking hostile aircraft? This problem became later known as the minimum-time-to-climb problem. It is the problem of determining the minimum-time aircraft trajectory between two fixed points in the range-altitude space. On the first glance, the answer to this question seems be easy. Every mathematician would immediately recognize its similarity to the famous prize question of Johann Bernoulli from 1696: what is the curve of quickest descent between two given points in a vertical plane (Fig. 1).1 This problem is considered to be the foundation stone of the Calculus of Variations to which so many famous mathematicians have contributed as the Bernoulli brothers Jacob and Johann, Euler, Lagrange, Legendre, Jacobi, Weierstrass, Hilbert, and Carath´eodory to mention only a few. Hence the calculus of variations should help to find a solution. On the other hand there was something hidden in those problems which was new and could not be revealed by the calculus of variations. 1 Bernoulli,

Johann, Problema novum ad cujus solutionem Mathematici invitantur, Acta Eruditorum, pp. 269, 1696; see also Johannis Bernoulli Basileensis Opera Omnia, Bousquet, Lausanne and Geneva, Switzerland, Joh. Op. XXX (pars), t. I, p. 161, 1742. Documenta Mathematica

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The following historical development will show that it is sometimes better to know too little than too much. Unbelievable? In mathematics?

1

The protagonists

Who were the mathematicians in this competition? Well, there were Magnus R. Hestenes (1906–1991), Rufus P. Isaacs (1914–1981), and Richard E. Bellman (1920–1984) in the “blue corner” (see Fig. 2) and Lev Semyonovich Pontryagin (1908–1988), Vladimir Grigorevich Boltyanskii (born 1925), and Revaz Valerianovich Gamkrelidze (born 1927) in the “red corner” (see Fig. 3). All members of the blue corner later complained about their missed opportunities. In contrast, the names of all members of the red corner will for ever be connected with the maximum principle, since the proof of the maximum principle designated the birth of a new field in applied mathematics named optimal control, which has, and continues to have, a great impact on optimization theory and exciting applications in almost all fields of sciences.

2

How did it happen?

Initially, engineers attempted to tackle such minimum-time interception problems for fighter aircraft. Due to the increased speed of aircraft, nonlinear terms no longer could be neglected. However, linearisation was not the preferred method. The engineers confined themselves to simplified models and achieved improvements step by step. For example, Angelo Miele’s (born 1922) solution for a simplified flight path optimization problem from the 1950s (with the flight path angle as control variable) exhibits an early example what later became known as bang – singular – bang switching structure (in terms of aerospace engineering: vertical zoom climb – a climb along a singular subarc – vertical dive). As early as 1946, Dmitry Yevgenyevich Okhotsimsky (1921–2005) solved the specific problem of a vertically ascending rocket to achieve a given final altitude with a minimum initial mass.2 His solution consists of a motion with maximum admissible thrust, an ascent with an optimal relation between velocity and altitude, and finally a phase with thrust turned off.3 However, mathematicians like to have general solution methods, or at least solution methods for a large class of equivalent problems. 2 This problem was firstly posed by Georg Karl Wilhelm Hamel (1877–1954) in 1927. Hamel’s and Okhotsimsky’s problem has to be distinguished from Robert Goddard’s (1882– 1945) earlier problem of 1919. In his problem the maximum altitude was sought which a rocket can reach with a given initial mass. The rocket pioneer Goddard is the eponym of the Goddard Space Flight Center in Greenbelt, Maryland. 3 Okhotsimsky contributed to the planning of multiple space missions including launches to Moon, Mars and Venus – and the launch of the first Sputnik satellite in 1957.

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Figure 2: The mathematicians at RAND: Magnus R. Hestenes, Rufus P. Isaacs, and Richard E. Bellman (Credits: Magnus R. Hestenes: Thanks to Dr. Ronald F. Boisvert, Mathematical and Computational Science Division of the Information Technology Laboratory at the National Institute of Standards and Technology in Gaithersburg, Maryland, who got this photo as part of a collection of photographs owned by John Todd (1911–2007), a professor of mathematics and a pioneer in the field of numerical analysis. John Todd worked for the British Admiralty during World War II. One of Todd’s greatest achievements was the preservation of the Mathematical Research Institute of Oberwolfach in Germany at the end of the war. Rufus P. Isaacs: Painting by Esther Freeman. Thanks to Mrs. Rose Isaacs, Po-Lung Yu, and Michael Breitner; see P. L. Yu: An appreciation of professor Rufus Isaacs. Journal of Optimization Theory and Applications 27 (1), 1979, 1–6. Richard E. Bellman: http://www.usc.edu/ academe/faculty/research/ethical_conduct/index.html.) 3

The traditionalists

After the end of World War II, the RAND coorporation (Research ANd Development) was set up by the United States Army Air Force at Santa Monica, California, as a nonprofit think tank focussing on global policy issues to offer research and analysis to the United States armed forces. Around the turn of the decade in 1950 and thereafter, RAND employed three great mathematicians, partly at the same time. 3.1

Magnus R. Hestenes

Around 1950, Hestenes wrote his two famous RAND research memoranda No. 100 and 102; see Hestenes (1949, 1950). In these reports, Hestenes developed a guideline for the numerical computation of minimum-time aircraft trajectories. In particular, Hestenes’ memorandum RM-100 includes an early formulation of what later became known as the maximum principle: the optimal control vector (the angle of attack and the bank angle) has to be chosen in such a way that it maximizes the so-called Hamiltonian function along a minimizing trajectory. Documenta Mathematica

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In his report, we already find the clear formalism of optimal control problems with its separation into state and control variables. The state variables are determined by differential equations, here the equations of motion of an aircraft. The control variables represent the degrees of freedom which the pilot has in hand to steer the aircraft – and, if mathematicians are sitting behind him, to do this in an optimal way. In the language of mathematics, Hestenes’ problem reads as follows: d ~ +D ~ +W ~ , (m ~v ) = T~ + L dt dw ˙ (v, T, h) , =W dt ~ and the drag vector D ~ are known functions of the where the lift vector L angle of attack α and the bank angle β; engineers have to give mathematicians ~ has the length w, m is the vehicle’s this information. The weight vector W mass assumed to be constant due to the short maneuver time. The thrust vector T is represented as a function of velocity v = |~v | and altitude h. Then the trajectory is completely determined by the initial values of the position ~ as well as by the values vector ~r, the velocity vector ~v and the norm w of W of α(t) and β(t) along the path. The task now consists of determining the functions α(t) and β(t), t1 ≤ t ≤ t2 , in such a way that the flight time t2 is minimized with respect to all paths which fulfill the differential equations and have prescribed initial and terminal conditions for ~r(t1 ), ~v (t1 ), w(t1 ), ~r(t2 ), ~v (t2 ), and w(t2 ). 3.2

Richard E. Bellman and Rufus P. Isaacs

Also in the early 1950s, Richard Bellman worked at RAND on multi-stage decision problems. Extending Bellman’s principle of optimality,4 it is possible to derive a form of a maximum principle. Bellman in his autobiography: I should have seen the application of dynamic programming to control theory several years before. I should have, but I did not. One of Bellman’s achievements is his criticism of the calculus of variations because of the impossibility of solving the resulting two-point boundary-value problems for nonlinear differential equations at that time. Finally, Isaacs, the father of differential games, complained with respect to his “tenet of transition” from the early 1950s: Once I felt that here was the heart of the subject . . . Later I felt that it . . . was a mere truism. Thus in (my book) “Differential Games” it is mentioned only by title. This I regret. I had no idea, that Pontryagin’s principle and Bellman’s maximal principle (a special case 4 based on Bellman’s equation which can already be found in Carath´ eodory’s earlier work of 1926. See Pesch (2012) and the references cited therein.

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Hans Josef Pesch and Michael Plail of the tenet, appearing little later in the RAND seminars) would enjoy such widespread citation.

Indeed, Isaacs’ tenet represents an even more general minimaximum principle. However, he had the greatness to understand: The history of mathematics has often shown parallel evolution when the time was ripe. 3.3

Priority quarrel in the blue corner

Concerning the matter of priority between Isaacs’ tenet of transition and Bellman’s principle of optimality, there was some level of contention between Isaacs and Bellman, as the following personal remembrance of Isaacs’ colleague at RAND, Wendell H. Fleming, indicates: One day in the early 1950s, Bellman was giving a seminar at RAND in which he solved some optimization problems by dynamic programming. At the end of Bellman’s seminar lecture, Isaacs correctly stated that this problem could also be solved by his own methods. Bellman disagreed. After each of the two reiterated his own opinion a few times, Isaacs said: “If the Bellman says it three times, it must be true.” This quote refers to a line from Lewis Carroll’s nonsense tail in verse “The Hunting of the Snark”. One of the main (and other absurd) characters in this tale is called the Bellman.5 Last but not least, Hestenes also claimed in a letter to Saunders MacLane: It turns out that I had formulated what is now known as the general optimal control problem. I wrote it up as a RAND report and it was widely circulated among engineers. I had intended to rewrite the results for publication elsewhere and did so about 15 years later. As a reason for the delay, he mentioned his workload as chairman at the University of Southern California and his duties at the Institute for Numerical Analysis. 3.4

Sometimes it may be better to know less

Hestenes was a student of Gilbert Ames Bliss (1876–1951) and an academic grandchild of Oskar Bolza (1857–1942)6 from the famous Chicago School of 5 The Hunting of the Snark (An Agony in 8 Fits) is usually thought of as a nonsense poem written by Lewis Carroll, the author of Alice’s Adventures in Wonderland. This poem describes with infinite humour the impossible voyage of an improbable crew to find an inconceivable creature; cf. Martin Gardner: The Annotated Snark, Penguin Books, 1974. 6 Mathematicians like to track their academic relationships; cf. the Mathematics Genealogy Project: http://genealogy.math.ndsu.nodak.edu/.

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the Calculus of Variations. Bolza in turn was a student of Felix Christian Klein (1849–1925) and Karl Theodor Wilhelm Weierstrass (1815–1897). He had attended Weierstrass’ famous 1879 lecture course on the calculus of variations. This course might have had a lasting effect on the direction Bolza’s mathematical interests have taken and that he has passed on to his descendants. In this tradition, Hestenes’ derivation of his maximum principle fully relied on Weierstrass’ necessary condition (and the Euler-Lagrange equation), in which the control functions are assumed to be continuous and to have values in an open control domain. These assumptions were natural for Hestenes’ illustrative example of minimum time interception, but have obfuscated the potential of this principle. It may be that Hestenes’ deep knowledge of the calculus of variations, standing in the tradition of the Chicago School, was his drawback. This may have caused Hestenes not to find the hidden secrets behind those problems. Since certain optimal control problems such as Hestenes’ interception problem can be classified as problems of the calculus of variations, this may have prevented him from separating his solution from that environment and generalizing his idea to problems with bounded controls. A major concerns namely was that, in aerospace engineering, the admissible controls cannot be assumed to lie always in open sets. The optimal controls may also run partly along the boundaries of those sets. This kind of problems were solved with short delay in the USSR. Hence, it seems that sometimes it may be better to know less! 3.5

Merits

More important are Hestenes’ merits. Hestenes indeed expressed Weierstrass’ necessary condition as a maximum principle for the Hamiltonian. Herewith he had observed the importance of Weierstrass’ condition for the theory of optimal control. Six years before the work at the Steklov Institute in Moscow began, the achievement of introducing a formulation that later became known as the general control problem was adduced by Hestenes in his Report RM-100. Nevertheless, this often has been credited to Pontryagin.7 Hestenes’ report was considered to be hardly distributed outside RAND. However, there were many contacts between staff members of RAND engaged in optimal control and those “optimizers” outside RAND. Therefore, the content of RM-100 cannot be discounted as a flower that was hidden in the shade. The different circulation of Hestenes’ RM-100 compared to Isaacs’ RM-257, 1391, 1399, 1411, and 1486 may have been caused by the fact that Hestenes’ memorandum contains instructions for engineers while Isaacs’ memoranda were considered to be cryptic. To this Wendell H. Fleming meant:8 7 First attemps to distinguish between state and control variables although not named this way can be found in Carath´ eodory’s work; see Pesch (2012) and the references cited therein. For an extensive estimation of Hestenes’ work considerating his surroundings and preconditions see Plail (1998). 8 on the occasion of the bestowal of the Isaacs Award by the International Society of Dynamic Games in Sophia-Antipolis, France, in July 2006

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Figure 3: The mathematicians at Steklov: Lev Semyonovich Pontryagin, Vladimir Grigor’evich Boltyanskii, and Revaz Valerianovich Gamkrelidze (Credits: Lev Semyonovich Pontryagin: http://www-history.mcs. st-andrews.ac.uk/PictDisplay/Pontryagin.html. Vladimir Grigor’evich Boltyanskii: From Boltyanskii’s former homepage at the Centro de Investigaci´ on en Matem´ aticas, Guanajuato, Mexico. Revaz Valerianovich Gamkrelidze: Photo taken by the first author at the Banach Center Conference on 50 Years of Optimal Control in Bedlewo, Poland, September, 2008.) One criticism made of Isaacs’ work was that it was not mathematically rigorous. He worked in the spirit of such great applied mathematicians as Laplace, producing dramatically new ideas which are fundamentally correct without rigorous mathematical proofs. 4

The advant-gardists

4.1

Lev Semyonovich Pontryagin

Lev Semyonovich Pontryagin (1908–1988),9 already a leading mathematician on the field of topology, decided to change his research interests radically towards applied mathematics around 1952. He was additionally encouraged by the fact that new serendipities in topology by the French mathematicians Leray, Serre and Cartan came to the fore. In addition, he also was pressured by M. V. Keldysh, director of the department of applied mathematics of the Steklov Institute, and by the organisation of the Communist Party at the institute to change his research direction. Maybe they wanted these mathematicians eventually to work for something more meaningful for the workers’ and peasants’ state than topology. Contact was then made with Colonel Dobrohotov, a professor at the military academy of aviation. In 1955, Pontryagin’s group got together with members of the air force. As in the US, minimum time interception problems were discussed. 9 Pontryagin

lost his eyesight as the result of an explosion at the age of about 14. His mother wrote down his mathematical notes. Since she did not know the meaning or names of all these mathematical “hieroglyphs”, they used a kind of a secret language to name them. Documenta Mathematica

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Already prepared since 1952 by a seminar on oscillation theory and automatic control that was conducted by Pontryagin and M. A. Aizerman, a prominent specialist in automatic control, it was immediately clear that a time optimal control problem was at hand there. However, to strengthen the applications, engineers were also invited. In particular, A. A. Fel’dbaum and A. J. Lerner focussed the attention on the importance of optimal processes of linear systems for automatic control. Pontryagin quickly noticed that Fel’dbaum’s method had to be generalized in order to solve the problems posed by the military. The first important step towards a solution was done by Pontryagin “during three sleepless nights”. A little later already the first results could be published by Pontryagin and his co-workers Boltyanskii and Gamkrelidze in 1956. Their early form of the maximum principle (of 1956) presents itself in the following form: Given the equations of motion x˙ i = f i (x1 , . . . , xn , u1 , . . . , ur ) = f i (x, u) and two points ξ0 , ξ1 in the phase space x1 , . . . , xn , an admissible control vector u is to be chosen10 in such way that the phase point passes from the position ξ0 to ξ1 in minimum time. In 1956, Pontryagin and his co-workers wrote: Hence, we have obtained the special case of the following general principle, which we call maximum principle: the function H(x, ψ, u) = ψα f α (x, u) shall have a maximum with respect to u for arbitrary, fixed x and ψ, ¯ We denote the if the vector u changes in the closed domain Ω. maximum by M (x, ψ). If the 2 n-dimensional vector (x, ψ) is a solution of the Hamiltonian system ∂H , i = 1, . . . , n , ∂ψi ∂f α ∂H ψ˙ i = − i ψα = − i , ∂x ∂x x˙ i = f i (x, u) =

and if the piecewise continuous vector u(t) fulfills, at any time, the condition H x(t), ψ(t), u(t) = M x(t), ψ(t) > 0 , then u(t) is an optimal control and x(t) is the associated, in the small, optimal trajectory of the equations of motion. 10 The

letter u stands for the Russian word for control: upravlenie. Documenta Mathematica

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Hans Josef Pesch and Michael Plail x(t) ˙ 10 u = −1 5

-5

u = +1 -10 -10

-5

5

10

x(t)

Figure 4: Phase diagram: optimal solution of the minimum-time harmonic oszillator problem: minimize the terminal time tf subject to the differential equation x ¨ + x = u with boundary conditions x(0) = x0 , x(0) ˙ = x˙ 0 , x(tf ) = 0, and x(t ˙ f ) = 0, and control constraint |u| ≤ 1. The problem allows a complete analytical solution and, moreover, a synthesis, i.e., for any given initial point (x(0) = x0 , x(0) ˙ = x˙ 0 ) in the phase plane, the origine (x(tf ) = 0 , x(t ˙ f ) = 0) can be reached in minimum time tf by a finite number of switches of the control u being of bang-bang type, i.e., it switches when ever the trajectories cross the garland-like switching curve. Thereby, the optimal control law satisfies a feedback law: the optimal value of u is −1 above and +1 below the switching curve while the phase trajectories piecewise consist of circles with shrinking radii. This condition was immediately verified to be successful by means of problems of the Bushaw-Fel’dbaum type, e.g., x ¨ + x = u. Such dynamical systems have to be steered from any point of the phase plane x˙ vs. x to its origin in minimum time, where the set of admissible control values is bounded by |u| ≤ 1. Just take x to be the distance between the aircraft and the missile, you immediately get an abstract planar aircombat problem. Its solution is described by Fig. 4. 4.2

Vladimir Grigor’evich Boltyanskii and Revaz Valerianovich Gamkrelidze

Their first theorem on the Maximum Principle was not correct in general cases. It is a necessary and sufficient condition only for linear problems (as proved by Gamkrelidze, 1957, 1958). Later in 1958 Boltyanskii showed that the maximum principle is only a necessary condition in the general case. He published the proof first separately, later on together with Pontryagin and Gamkrelidze in Documenta Mathematica

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1960. Boltyanskii’s proof was very intricate and required substantial knowledge of different fields of mathematics. Indeed, Boltyanskii’s proof greatly influenced the later development of the modern theory of extremal problems.11 The research efforts at the Steklov Institute led to a series of publications and culminated in their famous book of 1961 which became a standard work of optimal control theory until today. In 1962, Pontryagin, Boltyanskii, Gamkrelidze, and the fourth author of that book, Evgenii Frolovich Mishchenko (1922–2010), received the Lenin prize for their work. Both Boltyanskii and Gamkrelidze concur in statements to the authors, that the somehow comparable conditions of the calculus of variations were not known during the development phase of the maximum principle, although Bliss’ monograph of 1946 existed in a Russian translation from 1950. Fortunately, the Pontryagin group did no know too much about the calculus of variations. 4.3

Priority quarrel in the red corner

Boltyanskii claimed the version of the maximum principle as a necessary condition to be his own contribution and described how Pontryagin hampered his publication. He said Pontryagin intended to publish the results under the name of four authors. After Boltyanskii refused to do so, he was allowed to publish his results in 1958 but said that he had to praise Pontryagin’s contribution disproportionally and had to call the principle Pontryagin’s maximum principle. According to Boltyanskii, Rozono´er, an engineer, was encouraged to publish a tripartite work on the maximum principle in Avtomatika i Telemekhanika in 1959, in order to disseminate the knowledge of the maximum principle in engineering circles and to contribute this way to the honour of Pontryagin as discoverer of the maximum principle. This priority argument may be based on the fact that Pontryagin wanted to aim for a globally sufficient condition after Gamkrelidze’s proof of a locally sufficient condition, and not to a necessary condition as it turned out to be after Boltyanskii’s proof. Boltyanskii may have felt very uncomfortable to write in his monograph: The maximum principle was articulated as hypothesis by Pontryagin. Herewith he gave the decisive impetus for the development of the theory of optimal processes. Therefore the theorem in question and the closely related statements are called Pontryagin’s maximum principle in the entire world – and rightly so. Boltyanskii felt suppressed and cheated of the international recognition of his achievements. After the break-up of the USSR, Boltyanskii was able to extend his fight for the deserved recognition of his work. 11 For

precursors of Boltyankii’s proof and their influences see Plail (1998). Documenta Mathematica

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Gamkrelidze held a different view:12 My life was a series of missed opportunities, but one opportunity, I have not missed, to have met Pontryagin. For traces of the Maximum Principle before the time covered here, see Plail (1998), Pesch and Plail (2009) as well as Pesch (2012) and the references cited therein. 4.4

Distinctions

Pontryagin received many honours for his work. He was elected a member of the Academy of Sciences in 1939, and became a full member in 1959. In 1941 he was one of the first recipients of the Stalin prize (later called the State Prize). He was honoured in 1970 by being elected Vice-President of the International Mathematical Union. 5

´sume ´ Re

Hestenes, Bellman, and Isaacs as well as Pontryagin and his co-workers Boltyanskii and Gamkrelidze have not exclusively contributed to the development of optimal control theory, but their works were milestones on the way to modern optimal control theory. Their works are examples for demanding mathematical achievements with a tremendous application potential, today no longer solely in the military sector or in aeronautics, but also for many industrial applications. Today the second step after the numerical simulation of complicated nonlinear processes often requires an optimization post-processing. Not seldom side conditions as differential equations and other constraints must be taken into account for real-life models. Optimal control definitely is the germ cell of all those new fields in continuous optimization that have recently developed such as optimal control with partial differential equations or shape, respectively topology optimization, which are continuously contributing to the accretive role of mathematics for the development of present and future key technologies. References Boltyanskii, V. G., Gamkrelidze, R. V., and Pontryagin, L. S. (1956) On the Theory of Optimal Processes (in Russian). Doklady Akademii Nauk SSSR 110, 7–10. Hestenes, M. R. (1949) Numerical Methods for Obtaining Solutions of Fixed End Point Problems in the Calculus of Variations. Research Memorandum No. 102, RAND Corporation, Santa Monica. 12 in the historical session at the Banach Center Conference on 50 Years of Optimal Control in Bedlewo, Poland, on Sept. 15, 2008

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Hestenes, M. R. (1950) A General Problem in the Calculus of Variations with Applications to the Paths of Least Time. Research Memorandum No. 100, ASTIA Document No. AD 112382, RAND Corporation, Santa Monica. Pesch, H. J. (2012) Caratheodory on the road to the maximum principle, this volume. Pesch, H. J. and Plail, M. (2009) The Maximum Principle of Optimal Control: A History of Ingenious Ideas and Missed Opportunities. Control and Cybernetics 38, No. 4A, 973-995. Plail, M. (1998) Die Entwicklung der optimalen Steuerungen (The development of optimal control). Vandenhoeck & Ruprecht, G¨ottingen. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. and Mishchenko, E. F. (1961) Matematicheskaya teoriya optimal´nykh prozessov. Fizmatgiz, Moscow. Translated into English. The Mathematical Theory of Optimal Processes. John Wiley and Sons (Interscience Publishers), New York, 1962. Translated into German. Mathematische Theorie optimaler Prozesse. Akademie-Verlag, Leipzig, 1964. Second revised German edition, Oldenbourg, M¨ unchen, 1967.

Hans Josef Pesch Chair of Mathematics in Engineering Sciences University of Bayreuth 95440 Bayreuth Germany

Michael Plail Head of BGMI Consulting in Mathematics and Computer Science, Munich 82237 W¨orthsee Germany

[emailprotected]

[emailprotected]

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Documenta Math.

The Princess and Infinite-Dimensional Optimization Hans Josef Pesch

Abstract. Traces of infinite-dimensional optimization can be sourced to ancient Greek mathematics. According to a legend the knowledge about the solution of such kind of problems helped on the foundation of Carthage, and today’s new subfields of infinitedimensional optimization such as optimal control, shape or topology optimization are indispensable in propelling present and future technological developments. 2010 Mathematics Subject Classification: 00A05, 01A20, 01A45, 4903, 49J27 Keywords and Phrases: Ancient Greek mathematics, infinitedimensional optimization, calculus of variations, optimal control, shape optimization The wish for optimization seems to be deeply grounded in mankind. How often somebody says proudly: “Now I have optimized it again!” [for example, the author’s spouse or colleagues from engineering departments, etc. The author will not comment here on the word “again”.] Hence there must be traces of optimization deep in human history. Like most mathematicians, the author likes to trace the roots of his own research area and to search for his scientific ancestors and funny stories around them. Therefore, this article tries to answer the question “What is the first infinite-dimensional constrained optimization problem?”. But the reader may be warned. The answer may come from a subjective viewpoint and may be affected by the “optimization of the attractiveness” of the stories behind these questions. For the non-experts, in infinite-dimensional optimization we want to find optimal solutions of problems where the optimization variables are elements of infinite-dimensional spaces or even more complicated objects such as functions, curves, sets, shapes, topologies etc. The search for extremal points of realvalued functions of real variables known from school is not meant. At a first glance, this may indicate that we cannot go back farther than to the invention of calculus by Leibniz and Newton at the end of the 17th century. However, this is not true as we will see. Documenta Mathematica

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Hans Josef Pesch Renaissance in mathematics

Johann Bernoulli’s price question (acutissimis qui toto Orbe florent mathematicis, for the most astucious mathematicians of the entire globe)1 may come into mind first: “What is the curve of quickest descent between two given fixed points in a vertical plane?” (1696) and Newton’s problem: “What is the shape of a body of minimum resistance?” (1687).2 The first problem was created by Johann Bernoulli to tease his older brother Jacob from whom he knew that he was working on those kind of problems and Johann hoped that his brother and teacher would not be able to find an answer. He erred; see, e.g., Goldstine (1980). Both problems are typical infinite-dimensional problems. Their analytical solution is even today not possible without a solid knowledge of calculus, a few years earlier invented by Leibniz (1684),3 resp. Newton (1736).4 Johann Bernoulli’s problem reads as follows, cp. Fig. 1: Z xB p 1 + (y ′ (x))2 1 √ p inf dx , y∈Yad 2g 0 −y(x)

where the set of admissible functions Yad is defined by Yad := y is continuous on [0, xB ] and continuously differentiable on (0, xB ) with prescribed boundary conditions y(0) = 0 and y(xB ) = yB .

Here g denotes the Earth’s gravitational acceleration. Sir Isaac Newton’s problem reads as follows: the total resistance of particles that hit the body (nose of the aircraft or its airfoils; see Fig. 9) exactly once and transfer their momentum to the body, is the sum over the body of these transfers of momentum: Z dx inf , y∈Yad Ω 1 + k∇y(x)k2 2 with Yad := {y : Ω → [0, M ] ⊂ R : Ω ⊂ R2 bounded and y concave} . 1 Bernoulli, Johann, Problema novum ad cujus solutionem Mathematici invitantur, Acta Eruditorum, pp. 269, 1696; see also Johannis Bernoulli Basileensis Opera Omnia, Bousquet, Lausanne and Geneva, Switzerland, Joh. Op. XXX (pars), t. I, p. 161, 1742. 2 Newton, Isaac: Philosophiae Naturalis Principia Mathematica, submitted 1686 to the Royal Society, published 1687, 2nd ed. 1713, 3rd ed. 1726, commented 3rd ed. by the Franciscans Thomas Le Seur and Fran¸cois Jacquier using Leibniz’ calculus (!), 1739–1742. 3 Leibniz, Gottfried Wilhelm: Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus, Acta Eruditorum, 1984. 4 Newton, Isaac: The method of Fluxions and Infinite Series with its Application to the Geometry of Curve-lines, 1736. Newton’s work was already existent and ready for press in 1671 (in Latin). The English translation, however, appeared not until 1736 after Newton’s death. This has contributed to the priority quarrel between Newton and Leibniz; see, e.g., Wußing (2009), p. 471ff, and the references cited therein.

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y xB

A(0, 0)

x Pythagoras: q p 2 2 ∆s = ∆x + ∆y = 1 +

time t

∆y 2 ∆x

∆x (∆x 6= 0)

Galilei: s

v=

∆x

∆s ∆t

=

√

2g

√ −y

Time of fall: ∆y ∆s

∆t =

yB

√1 √∆s −y 2g

=

√1 2g

q ∆y 2 1+( ∆x ) √ −y

∆x

B(xB , yB ) time t + ∆t

Figure 1: Bernoulli’s Brachistochrone Problem: Pythagoras’ theorem, Galilei’s law of free fall and a summation over all infinitesimal time intervals ∆t yields the minimization problem. For its solution, Johann Bernoulli applied the idea of discretization and associated the curve of quickest descent with a ray of light through layers of different media and the fall velocity with the speed of light. By Fermat’s principle of least time, resp. Snell’s law of refraction, 2 Johann Bernoulli derived the differential equation y(x) 1 + (y ′ (x)) = −2 r, r > 0, as necessary condition, the solutions of which were known to be cycloids: x(θ) = r (θ − sin θ), y(θ) = −r (1 − cos θ), 0 ≤ θ ≤ θB , with r and θB defined by the terminal conditions x(θB ) = xB and y(θB ) = yB .

Newton: I reckon that this proposition will be not without application in the building of ships.2 This old problem is still inspiring current research; see, e.g., Buttazzo et. al. (1993) and Lachand-Robert and Peletier (2001). In his famous reply5 to the problem of his younger brother Johann, Jacob Bernoulli posed the following even more difficult problem: “What is the shape of the planar closed curve, resp. of the associated bounded set surrounded by

5 Bernoulli,

Jacob, Solutio Problematum Fraternorum, una cum Propositione reciproca aliorum, Acta Eruditorum, pp. 211–217, 1697; see also Jacobi Bernoulli Basileensis Opera, Cramer & Philibert, Geneva, Switzerland, Jac. Op. LXXV, pp. 768–778, 1744. Documenta Mathematica

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this curve that contains the maximum area while its perimeter is restricted?”, Z b dy dx inf x −y dt , γ∈Γad a dt dt where the set Γad of admissible curves is given by s Z b 2 2 dy dy x(t) Γad := γ : [a, b] ∋ t 7→ ∈ R2 : + dt = L > 0 . y(t) dt dt a Or, more generally, in modern mathematical language Z sup dx Ω∈Oad

Ω

with the set Oad of all admissible sets given by Oad := {Ω ⊂ Rn : Ω bounded , n ∈ N , and

Z

ds = L > 0} . ∂Ω

Here, ∂Ω denotes the (sufficiently smooth) boundary of the set Ω, and L is a given positive constant determining the perimeter, resp. surface. In all these problem statements, we are searching for a minimizer or maximizer being an element of an infinite-dimensional (huge) “something” where the criterion which is to be optimized depends on those objects. In addition, restrictions must be obeyed. Using an appropriate interpretation, all these problems can be considered to be the mother problems of important fields of continuous optimization: the classical Calculus of Variations, a playground of such mathematical heros like Euler, Lagrange, Legendre, Jacobi, Weierstrass, Hilbert, and Carath´eodory, and the modern theories of optimal control (Fig. 2), an offspring of the Cold War [Pesch and Plail (2012)], and the rather current fields shape, resp. topology optimization. This first so-called isoperimetric problem of Jacob Bernoulli is known as Dido’s problem in the mathematical literature. This points to an antique origin even far before the turn from the 17th to the 18th century, far before the times of those mathematical pioneers Leibniz, Newton, and the Bernoulli brothers. Hence this problem, at least a simplified version of it, must be solvable by geometric means, too. 2

Florescence in mathematics in antiquity

Indeed, the first isoperimetric problem, more complicated than Euclid’s earlier theorem6 saying that the rectangle of maximum area with given perimeter is 6 Little is known about Euclid’s life, but we have more of his writings than of any other ancient mathematician. Euclid was living in Alexandria about 300 B.C.E. based on a passage in Proclus’ Commentary on the First Book of Euclid’s Elements; cp. http://aleph0.clarku. edu/~djoyce/java/elements/Euclid.html.

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x2 x1B

A(0, 0)

x1 Formulation as optimal control: Rt minu∈Uad 0 f 1 dt

M at time t vx1 (t)

local horizontal

u(t) vx2 (t)

subject to the constraints: √ p x˙ 1 (t) = 2 g −x2 (t) cos u(t) √ p x˙ 2 (t) = 2 g −x2 (t) sin u(t) x1 (0) = 0 , x2 (0) = 0

x1 (tf ) = x1B , x2 (tf ) = x2B x2B

v(t) B(x1B , x2B )

Figure 2: The Brachistochrone problem formulated as problem of optimal control with the class Uad of admissible controls (slopes) defined by Uad := {u : [0, tf ] → (0, 2 π) : u continuous}. The optimal control u∗ is determined by the minimum principle: u∗ (t) = arg minu∈Uad H(x(t), p(t), u) with the state ⊤ vector x := (x1 , x2 )⊤ and the adjoint state vector √ p√:= (p1 , p2 ) . Hereby, the Hamiltonian is defined by H(x, p, u) := 1 + 2 g −x2 (p1 cos u + p2 sin u) and the adjoint state vector p must satisfy the canonical equation p˙ = −Hx .

the square, came down to us in written form by Theon Alexandreus7 in his commentaries on Klaudios Ptolemaios’8 Mathematical Syntaxis, a handbook of astronomy called Almagest.9 In this syntaxis one can find a theorem which is

7 Theon Alexandreus: ∗ about 335 C.E. probably in Alexandria, † ca. 405 C.E.; see. e.g., http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Theon.html. He edited Euclid’s Elements, published around 364 C.E., authoritative into the 19th century. His daughter Hypatia (∗ about 351 C.E., about † ca. 370 C.E.; cf. Fig. 7) also won fame as the first historically noted women in mathematics. She was murdered by a Christian mob after being accused of witchcraft. For more see http://www-history.mcs.st-and.ac.uk/Biographies/Hypatia.html. 8 Klaudios Ptolemaios: ∗ about. 85–100 C.E. in Upper Egypt, † about 165– 180 C.E. in Alexandria; see, e.g., http://www-groups.dcs.st-andrews.ac.uk/~history/ Mathematicians/Ptolemy.html. In contrast to Aristarchos of Samos and Seleukos of Seleukia, who both already pleaded the heliocentric system, Ptolemaios held on the geocentric system. 9 See http://en.wikipedia.org/wiki/Almagest.

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Figure 3: Zenodoros’ theorem in a fourteenth century manuscript of the city library of Nuremberg (Cod. N¨ ur. Cent. V App. 8, detail of p. 58r ). accredited to Zenodoros, but may be even older.10 See also Heath (1981). It is cited there from Zenodoros’ treatise “ΠερÈ Êσοµèτρων σχηµτων” (On isometric figures) of about the 2nd century B.C.E.: In the middle of the antepenultimate line of Fig. 3, we can read: ÃΩσαÔτως δ' îτι τÀν Òσην περÐµετρον âχìντων σχηµτων διαφìρων, âπειδ µεÐζον âστι τ πολυγωνι¸τερα, τÀν µàν âπιπèδων å κÔκλος e [ligature ⊙] γÐνεται µεÐζων, τÀν δà στερεÀν σφαØρα [ligature ⊕]. Ποιησìµεθα δ τν τοÔτων πìδειξιν âν âπιτοµ¨ù âκ τÀν Ζηνοδ¸ρωú δεδειγµèνων âν τÀú `ΠερÈ Êσοµèτρων σχηµτων'. Just as well, since those of different figures which have the same contour are larger which have more angles, the circle is larger than the (other) plane figures and the sphere than the (other) solids. We are going to present the proof for this in an extract of the arguments as has been given by Zenodoros in his work ‘On isometric figures’. Figure 4 shows the entire page No. 58r with Zenodoros’ theorem in a fourteenth century manuscript of the city library of Nuremberg. The reverse side shows his proof whose elegance was praised by Carath´eodory.11 For Zenodoros’ proof in modern mathematical language and other proofs of his theorem, it is referred to Bl˚ asj¨ o (2005). This ancient problem also still inspires mathematicians until today; see, e.g., Almeida et. al. (2012) for a very recent contribution. This codex was in possession of the Lower-Franconian mathematician and astronomer Johannes M¨ uller better known as Regiomontanus,12 who received 10 Zenodoros: ∗ about 200 B.C.E. in Athen, † about 140 B. C. in Greece; see. e.g., http:// www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Zenodorus.html. 11 Carath´ eodory, C.: Basel und der Beginn der Variationsrechnung, publication in honor of the sixtieth birthday of Professor Dr. Andreas Speiser, Z¨ urich, Switzerland, 1945; see also Carath´ eodory, C.: Gesammelte Mathematische Schriften 2, C. H. Beck’sche Verlagsbuchhandlung, Munich, Germany, pp. 108–128, 1955. 12 Johannes M¨ uller (Regiomontanus): ∗ 1436 in K¨ onigsberg in Bavaria, † 1476 in

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Figure 4: Zenodoros’ theorem in a fourteenth century manuscript of the city library of Nuremberg (Cod. N¨ ur. Cent. V App. 8, p. 58r ), entire page. it as a gift from his patron Cardinal Johannes Bessarion, titular patriarch of Constantinople. The codex served as the original printing copy for the editio princeps of 1538 published in Basel. Already hundreds of years before Zenodoros’ theorem was proven, “engineering intuition” brought the Phoenician princess Elissa (Roman: Dido) of Tyros, today Sur, Lebanon, to take advantage of it. According to a legend,13 Dido was on the run from her power-hungry brother Pygmalion, who already had ordered the murder of her husband Acerbas and strived for her life and wealth. Dido with her abiders came, on a sail boat, to the shores of North Africa in the region of today’s Tunis, Tunesia at around 800 B.C.E. The local habitans were friendly, but did not want to have the armed strangers in their vicinity Rome; see, e.g., http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/ Regiomontanus.html. Regiomontanus is the Latin word for his native town K¨ onigsberg (it is not the famous K¨ onigsberg in Prussia, today’s Kaliningrad, Russia, which gave Euler’s Problem of the Seven Bridges of K¨ onigsberg its name. 13 The legend seems to be apocryphal and may be fictious, but very appropriately invented for the Ionian-Greek word βÜρσα meaning oxhide. Documenta Mathematica

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Figure 5: The Punic Carthage and Zenodoros’ optimal solution as realized by Dido. Surely Dido has chosen a piece of land by the coast so as to exploit the shore as part of the perimeter permanently. Therefore the resourceful princess asked the local king Iarbas for a small amount of grassland for their livestock, only so small that it can be covered by an oxhide. Iarbas laughed and could not refuse Dido’s modest request. Dido then cut the hide into thin strips (Fig. 6), encircled a large area (Fig. 5) on which their fellows erected the new city Qart-Hadaˇst (Phoenician for new city) with the citadel Byrsa, from which the ancient superpower Carthage later developed. So far the first part of the legend. We will omit here the tragic lovestory between Dido and the Trojan hero Aeneas, who came to Dido’s adopted home after his fly from Troja. He left her by command of Jupiter whereupon Dido threw herself into the flames of the fire by which she burned all things that Aeneas and his compagnions left behind. This curse is said to be the source for the later enmity between Rome and Carthage. The legend of the founding of Carthage was sung by Publius Vergilius Maro14 in his famous Aeneid (book one, verses 365–368): devenere locos, ubi nunc ingentia cernis moenia surgentemque novae Karthaginis arcem, mercatique solum, facti de nomine Byrsam, taurino quantum possent circumdare tergo. and in English verses, perpetuating the hexameter, from the translation of the famous English poet John Dryden, a contemporary of the Bernoulli Brothers: At last they landed, where from far your Eyes May view the Turrets of new Carthage rise: 14 Publius Vergilius Maro, Roman poet: ∗ 70 B.C.E. in Andes (Pietole?) near Mantua, † 19 B.C.E. in Brundisium (Brindisi)

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Figure 6: Dido purchases land for the foundation of Carthage, engraving by Mathias Merian the elder from Historische Chronica, Frankfurt a. M., 1630. There bought a space of Ground, which ‘Byrsa’ call’d From the Bull’s hide, they first inclos’d, and wall’d. or in an older translation by the sixteenth century authors Thomas Phaer and Thomas Twyne: Than past they forth and here they came, where now thou shalt espie The hugy walles of new Carthage that now they rere so hie. They bought the soile and Birsa it cald whan first they did begin, As much as with a bull hide cut they could inclose within.

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Florescence in mathematics today

Back to present and future: What is the optimal shape of a very fast aircraft, say which is able to fly at supersonic speed with minimal drag? Indeed, that is a modern version of Dido’s problem. Figure 8 shows effects of aerodynamic drag minimizing on airfoil and body of a supersonic cruise transporter due to Brezillon and Gauger (2004). More challenges are waiting such as fuel optimization of aircraft using laminar flow airfoils with blowing and sucking devices or using morphing shape airfoils with smart materials and adaptive structures built-in. Figure 9 shows the, in this sense, non-optimized flow around the Airbus A 380 computed by numerical simulation. Optimization with those respects may be next steps for which infinite-dimensional optimization in various specifications must be employed: optimal control of ordinary and partial differential equations as well as Documenta Mathematica

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Figure 7: Hypathia, detail of The Schooll of Athens’ by Raphael shape and topology optimization. Their roots can be traced, tongue-in-cheek, to the renaissance of mathematics with the invention of calculus and even as far as to the geometricians in antiquity.

Figure 8: Drag minimization for the EUROSUP SCT (supersonic cruise transporter) at Mach number 2: Optimized shape geometry (upper wing) versus initial design (lower wing) with local flow Mach number distribution. The strong shock on the wing could be reduced. [Brezillon, Gauger (2004)] (Copyright: Prof. Dr. Nicolas Gauger, Head of Computational Mathematics Group, Department of Mathematics and Center for Computational Engineering Science, RWTH Aachen University, Aachen, Germany) Documenta Mathematica

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Figure 9: Numerical flow simulation for the Airbus A380 (picture credit: Airbus. Copyright: Dr. Klaus Becker, Senior Manager Aerodynamic Strategies, EGAA, Airbus, Bremen, Germany) Mathematical optimization has become and will continue to be an important tool in modern high technology. Mathematics in total has even become a key technology by itself. References Almeida, R., Ferreira, R. A. C., and Torres, D. F. M. (2012) Isoperimetric Problems of the Calculus of Variations with Fractional Derivatives. Acta Mathematica Scientia 32B(2), 619–630. ¨ , V. (2005) The Isoperimetric Problem. Amer Math Monthly 112(6), Bl˚ asjo 526–566. Brezillon, J. and Gauger, N. (2004) 2D and 3D aerodynamic shape optimisation using adjoint approach. Aerospace Science and Technology 8 (8), 715–727, 2004. Buttazzo, G., Ferone, V., and Kawohl, B. (1993) Minimum Problems over Sets of Concave Functions and Related Questions. Mathematische Nachrichten 173, 71–89. Goldstine, H. H. (1980) A History of the Calculus of Variations from the 17th through the 19th Century. Studies in the History of Mathematics and Physical Sciences, Springer, New York, Heidelberg, Berlin. Documenta Mathematica

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Heath, T. L. (1981) A History of Greek Mathematics II. Dover Publications, Mineola, NY, USA. Lachand-Robert, T. and Peletier, M. A. (2001) Newton’s Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions. Mathematische Nachrichten 226, 153–176. Pesch, H. J. and Plail, M. (2012) The Cold War and the Maximum Principle of Optimal Control. In: M. Gr¨otschel (ed.): Optimization Stories. Documenta Mathematica. Wußing, H. (2009) 6000 Jahre Mathematik I. Springer, Berlin, Germany.

Hans Josef Pesch Chair of Mathematics in Engineering Sciences University of Bayreuth 95440 Bayreuth Germany [emailprotected]

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Computing Stories

Optimization theory has existed before computers were invented, but the expansion of optimization and its wide range of applications was only possible due to the enormous growth and accessibility of modern computing machinery. To address the importance of computing theory and practice for optimization I have asked four authors to cover some of these aspects. One article is on the history of NP-completeness where, for instance, some new insights into the prehistory of this important concept can be found. Another article is on the history of optimization modeling systems which are tools helping users to employ optimization algorithms efficiently. This is an area usually neglected by academic researchers but of high relevance for practitioners. A third article deals with the history of the reverse mode of differentiation, which is a methodology supporting, in particular, continuous optimization techniques by improving the information flow, memory management, sensitivity analysis, error estimation, conditioning, etc. Finally, the history of “Moore’s Law” is reviewed that describes/postulates the exponential growth of computing power. How long will it stay alive? The history of computing hardware is long and surveyed in many books and articles. One driving force of the computing machine development has always been the aim to reduce the effort necessary to carry out long calculations. Leibniz, for instance, stated: “It is unworthy of excellent men to lose hours like slaves in the labor of calculation which could safely be relegated to anyone else if machines were used.” Leibniz himself made significant contributions to the design of mechanical computing devices. Today, it is generally accepted that Konrad Zuse (1910–1995) built the first program-controlled computing machine in the world. Zuse studied civil engineering and earned his Diploma in 1935 at Technische Hochschule BerlinCharlottenburg (today TU Berlin). He was annoyed by the repetitive statics calculations and decided to automate these procedures. His first computer, the Z1 finished in 1938, was mechanical. His Z3 was operational in 1941; it had the same logic design as the Z1, but used electrical components. It was a fully digital, floating-point, programmable machine. There are various Internet archives that document Zuse’s achievements in detail. I recommend http://www.zib. de/zuse/home.php, maintained by Raul Rojas, and the Web page http:// www.zuse.org of Horst Zuse, Konrad’s son, that also provides numerous documents about his father and the computer technology he invented. Konrad Documenta Mathematica

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¨ tschel Martin Gro

Figure 1: Zuse memorial plate http://en.wikipedia.org/wiki/File: Gedenktafel_Methfesselstr_10_(Kreuzb)_Konrad_Zuse.JPG

Zuse did most of his work in the prewar time in the living room of his parents, see Fig. 1, in intellectual isolation, assisted and financially supported by his family and a few friends only. Zuse has been honored, e.g., by naming the Konrad-Zuse-Zentrum f¨ ur Informationstechnik Berlin after him. Martin Gr¨otschel

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A Brief History of NP-Completeness, 1954–2012 David S. Johnson

2010 Mathematics Subject Classification: 68-03, 68Q17, 68Q25, 68W25, 90C05, 90C22 Keywords and Phrases: NP-completeness, polynomial time, approximation algorithms, bin packing, unique games conjecture The year 2012 marks the 40th anniversary of the publication of the influential paper “Reducibility among combinatorial problems” by Richard Karp [37]. This paper was the first to demonstrate the wide applicability of the concept now known as NP-completeness, which had been introduced the previous year by Stephen Cook and Leonid Levin, independently. 2012 also marks the 100th anniversary of the birth of Alan Turing, whose invention of what is now known as the “Turing machine” underlay that concept. In this chapter, I shall briefly sketch the history and pre-history of NP-completeness (with pictures), and provide a brief personal survey of the developments in the theory over the last 40 years and their impact (or lack thereof) on the practice and theory of optimization. I assume the reader is familiar with the basic concepts of NPcompleteness, P, and NP, although I hope the story will still be interesting to those with only a fuzzy recollection of the definitions. The New Prehistory When the Garey & Johnson book Computers and Intractability: A Guide to the Theory of NP-Completeness [23] was written in the late 1970s, the sources of the theory were traced back only to 1965. In particular, we cited papers by Cobham [13] and Edmonds [18], which were the first to identify the class of problems solvable in polynomial time as relevant to the concept of efficient solvability and worthy of study. We also cited a second paper of Edmonds [17], which in a sense introduced what was later to be called the class NP, by proposing the notion of a problem having a “good characterization.” It turns out, however, that a pair of eminent mathematicians had touched on the issues involved in NP-completeness over a decade earlier, in handwritten private letters that took years to come to light. The first to be rediscovered (and the second to be written) was a letter from Kurt G¨odel to John von Neumann, both then at the Institute for Advanced Study in Princeton, New Jersey. G¨ odel is perhaps most famous for his 1931 “Incompleteness Theorems” about Documenta Mathematica

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mathematical logic. His letter, written in German and dated 20 March 1956, was not publicized until 1989, when Juris Hartmanis published a translation and commentary [27]. In this letter, G¨ odel considered first a problem of finding proofs in a given proof system: Given a first order formula F and an integer n, is there is a proof of F having length no more than n? Let A be a Turing machine that solves this problem, and, following G¨ odel, let ψA (F, n) denote the number of steps that A takes when applied to the instance consisting of formula F and bound n. Now let φA (n) be the worst-case value of ψA (F, n) over all formulas F of length n. Note that a Turing machine A performing exhaustive search would have a value for φA (n) that was no worse than exponential in n. G¨odel pointed out how wonderful it would be if there were an A with φA (n) = O(n) or even O(n2 ), observing that such a speedup had already been observed for the problem of computing the quadratic residue symbol. Finally, he asked “how strongly in general” one could improve over exhaustive search for combinatorial problems, in particular mentioning the problem of primality testing (a problem whose worst-case complexity remained open for almost 50 more years, until it was shown to be polynomial-time solvable by Agrawal, Kayal, and Saxena in 2002 [3]). Note that G¨ odel did not make the generalization from O(n) and O(n2 ) to polynomial time. He was more interested in algorithms that might plausibly be practical. He was also not measuring running time in terms of the modern concept of “input length”. For that he would have had to explicitly specify that n was written in unary notation. (If n were written in standard binary notation, then exhaustive search for his problem might have been doubly exponential in the input size.) On the other hand, he does seem to have assumed binary, or at least decimal, input size when he discussed primality testing. Moreover, he used the idea of worst-case running time analysis for algorithms and problems, something that was not all that common at the time, and which dominates algorithmic research today. And he does seem to have an idea of the class of problems solvable by exhaustive search, which can be viewed as a generalization of NP, and his final question hints at the question of P versus NP. At any rate, G¨ odel’s letter, once discovered, was immediately recognized as an important precursor to the theory of NP-completeness. When an annual prize for outstanding journal papers in theoretical computer science was established in 1992, it was only natural to name it the G¨odel Prize. More recently, the letter has even lent its name to a well-written and popular blog on algorithms and computational complexity (G¨ odel’s Lost Letter and P = NP, http://rjlipton.wordpress.com). The other famous mathematician whose letters foreshadowed the theory of NP-completeness was John Nash, Nobel Prize winner for Economics and subject of both the book and the movie A Beautiful Mind. In 1955, Nash sent several handwritten letters about encryption to the United States National Security Agency, which were not declassified and made publicly available until 2012 [1]. In them, he observes that for typical key-based encryption processes, Documenta Mathematica

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Figure 1: Stephen Cook, Richard Karp, and Leonid Levin, photographed in the 1980s

if the plain texts and encrypted versions of some small number of messages are given, then the key is determined. This is not technically correct, since in addition there must be sufficient entropy in the plain texts, but Nash’s arguments apply as well to the problem of finding some key consistent with the encryptions. His central observation was that even if the key is determined, it still may not be easy to find. If the key is a binary string of length r, exhaustive search will work (as it did for G¨ odel), but takes time exponential in r. For weak cryptosystems, such as substitution ciphers, there are faster techniques, taking time O(r2 ) or O(r3 ), but Nash conjectured that “for almost all sufficiently complex types of enciphering,” running time exponential in the key length is unavoidable. This conjecture would imply that P 6= NP, since the decryption problem he mentions is polynomial-time equivalent to a problem in NP: Given the data on plain and encrypted texts and a prefix x of a key, is there a key consistent with the encryptions which has x as a prefix? It is a stronger conjecture, however, since it would also rule out the possibility that all problems in NP can, for instance, be solved in time nO(log n) , which, although non-polynomial, is also not what one typically means by “exponential.” Nash is also making a subsidiary claim that is in essence about the NP-hardness of a whole collection of decryption problems. This latter claim appears to be false. Nash proposed an encryption scheme of the type he specified, but the NSA observed in private notes that it provided only limited security, and since the publication of the letters modern researchers have found it easy to break [2]. Also, like G¨odel, Nash did not make the leap from low-order polynomial time to polynomial time in general. He did however, correctly foresee the mathematical difficulty of the P versus NP problem. He admitted that he could not prove his conjecture, nor did he expect it to be proved, even if it were true. Documenta Mathematica

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Cook, Karp, and Levin The theory of NP-completeness is typically traced back to Steve Cook’s 1971 paper “The complexity of theorem-proving procedures” [14], which provided the first published NP-completeness results. However, Leonid Levin, then a student in Moscow, proved much the same results at roughly the same time, although his results were not published until 1973. Over the years, the contemporaneous and independent nature of Levin’s accomplishment have come to take precedence over publication dates, and what used to be called “Cook’s Theorem” is now generally referred to as the “Cook-Levin Theorem.” Let me say a bit about these two parallel developments. When Cook wrote his paper, he was an Associate Professor in the Computer Science Department of the University of Toronto, where he is now a University Professor. Earlier, he had received his PhD from Harvard in 1966, and spent four years as an Assistant Professor in the Mathematics Department of University of California, Berkeley, which foolishly denied him tenure. Cook’s paper appeared in the proceedings of the 1971 ACM Symposium on Theory of Computing (STOC), and there are apocryphal stories that it almost was not accepted. This seems unlikely, although it wouldn’t be the first time a major breakthrough was not recognized when it occurred. The paper’s significance was certainly recognized as soon as it appeared. Not only did the paper prove that satisfiability is NP-complete (in modern terminology), but it also proved the same for 3SAT, and hinted at the broader applicability of the concept by showing that the same also holds for subgraph isomorphism (more specifically, the special case now known as the clique problem). I was a grad student at MIT at the time, and Albert Meyer and Mike Fischer included these results in their Fall 1971 Algorithms course. Others had also been busy, as became clear at the March 1972 conference on “Complexity of Computer Computations” at the IBM T.J. Watson Research Center in Yorktown Heights, NY, where Richard Karp presented his famous paper. Karp was also a Harvard PhD recipient (1959), and after an 11-year stint at the same IBM Research Center that housed the conference, had moved to a professorship at UC Berkeley in 1968, where he remains today, after a brief sojourn to the University of Washington in Seattle. Karp’s paper showed that 19 additional problems were NP-complete, including such now-famous characters as vertex cover, chromatic number, the directed and undirected hamiltonian circuit problems, subset sum, and the knapsack problem. Most of the proofs were due to Karp himself, but a few were attributed to Gene Lawler, Bob Tarjan, and “the Algorithms Seminar at Cornell.” The paper appears to be the first to use the notations P and NP, although its term for “NP-complete” was “polynomial complete,” a locution used in several early papers before the modern terminology took hold. The paper also introduced the distinction between a polynomial transformation, where an instance of the first problem is transformed into one of the second that has the same yes-no answer, and a polynomial reduction, in which the first problem is solved using Documenta Mathematica

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one or more calls to a subroutine that solves the second. Cook had stated his results in terms of the latter notion, but his proofs had essentially relied only on the first. This was the first conference that I had attended, and I was suitably awed by all the famous participants whom I was meeting for the first time - including John Hopcroft, Michael Rabin, Bob Tarjan, Jeff Ullman, and Richard Karp himself. I even got to sit across the table from Dick at one lunch. I took the opportunity to mention to him that I had already proved one polynomial completeness result myself, that for bin packing, the problem that was to be the topic of my thesis. Albert Meyer had proposed I work on it just a month earlier, saying “This is perfect for you, Johnson. You don’t need to know anything – you just have to be clever.” Albert had learned about the problem from a preprint of a 1972 STOC paper by Garey, Graham, and Ullman [21]. In the problem, one is given a sequence of numbers a1 , a2 , . . . , an ∈ (0, 1] and a target k, and asked if the numbers be partitioned into k sets, each summing to no more than 1. Dick showed polite interest, but, as the words came out of my mouth, I was embarrassed to realize how trivial my proof was compared to the ones in hisP paper (subset sum is the special case of bin packing where n k = 2 and the i=1 ai = 2.) In addition to many other interesting papers, the conference included a lively panel discussion, a transcript of which is contained in the proceedings [45]. It covered issues raised by many of the preceding talks, but the discussion kept coming back to the P versus NP question. The most remembered (and prescient) comment from the panel was by John Hopcroft. He observed that, although a consensus seemed to be forming that the two classes were not equal, for all we currently knew, every problem in NP could be solved in linear time. He concluded that it would be “reasonably safe” to conjecture that, within the next five years, no one would prove that any of the polynomial complete problems even required more than quadratic time. It is now 40 years and counting, and we still have yet to see any such proofs. Meanwhile, in a much different world, Leonid Levin was thinking about the same issues, but not getting nearly the same publicity. In the Soviet Union at the time, many researchers were considering questions related to the P versus NP question. In particular, there was the notion of the class of problems that could only be solved by perebor, the Russian name for algorithms that were essentially based on exhaustive search [52]. Levin was a PhD student at the University of Moscow. In 1971, he completed a thesis on Kolmogorov complexity, but although it was approved by Kolmogorov (his advisor) and by his thesis committee, the authorities refused to grant the degree for political reasons. (Levin admits to having been a bit intractable himself when it came to toeing the Soviet line [51, 151–152].) Levin continued to work on other things, however, in particular perebor, coming up with his version of NP-completeness that same year, and talking about it at various seminars in Moscow and Leningrad [52]. He also wrote up his results, submitting them for publication in June 1972 [52], although the paper did not appear until the Documenta Mathematica

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second half of 1973. Its title, translated into English, was “Universal sequential search problems” [42] (“Sequential search” was a mistranslation of perebor ). The 2-page paper was brief and telegraphic, a trait shared by many of Levin’s subsequent papers (e.g., see [55, 43]), omitting proofs entirely. A corrected translation appears as an appendix in [52]. In his paper, Levin deals with the generalization of NP to search problems: Relations A(x, y) on strings, such that for all pairs (x, y) such that A(x, y) holds, the length of y is polynomially bounded in the length of x, and such that for all pairs (x, y), one can determine in polynomial time whether A(x, y) holds. Here x stands for an instance of the problem, and y a corresponding “solution.” The search problem for A is, given x, find a y such that A(x, y) holds. The corresponding problem in NP is, given x, does there exist a y such that A(x, y) holds. Levin mentions this version, calling it a “quasi-search” problem, but concentrates on the search problem version. He describes what we would now view as the standard notion of a polynomial reduction from one search problem A to another one, and calls a problem a “universal search problem” if there exist polynomial reductions to it from all the search problems in the above class. He then goes on to list six search problems that he can prove are universal search problems. These include the search versions of satisfiability, set cover, and subgraph isomorphism, along with others that were not on Karp’s list, such as the following tiling problem: Given a square grid whose boundary cells each contain an integer in the range from 1 to 100, together with rules constraining the contents of interior cells, given the contents of the four neighboring cells (to the left, right, top, and bottom), find a legal tiling that agrees with the given assignment to the boundary cells. Those who heard Levin speak about these results were immediately impressed. Trakhtenbrot [52] quotes Barzdin, who heard Levin speak in Novosibirsk in April, 1972, as saying “Just now Levin told me about his new results; it is a turning point in the topic of perebor !” Note that this is clear evidence that the work of Cook and Karp had not yet received wide attention in Russia. However, neither did the work of Levin. In 1973, when Russian theoreticians finally did take up NP-completeness, it was mainly through the Cook and Karp papers [25]. Levin’s impact appears not to have spread much beyond those who had heard him speak in person. In 1978, Levin emigrated to the US, where I first met him while visiting MIT. There he finally received an official PhD in 1979, after which he took up a position at Boston University, where he is now a Full Professor. He has made many additional contributions to complexity theory, including • A theory of average case completeness [43], using which he shows that a variant of his above-mentioned tiling problem, under a natural notion of a uniform distribution for it, cannot be solved in polynomial expected time unless every other combination of a problem in NP with a reasonably constrained probability distribution can be so solved. • A proof that the one-way functions needed for cryptography exist if and Documenta Mathematica

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only if pseudorandom number generators exist that cannot in polynomial time be distinguished from true random number generators [28]. • A proof that a 1965 precursor of the ellipsoid algorithm, in which simplices play the role of ellipses, also runs in polynomial time [55] (thus there is a simplex algorithm that runs in polynomial time . . . ). Cook and Karp also have made significant contributions to complexity theory since their original breakthroughs. Karp’s many contributions are well known in the mathematical programming community and too extensive to list here. Cook’s main work has been in the study of proof complexity, but he is responsible for introducing at least one additional complexity class, one that provides an interesting sidelight on NP-completeness. This is the class SC, the set of decision problems that can be solved by algorithms that run in polynomial time and require only polylogarithmic space, that is, use O(logk n) space for some fixed k. Here “SC” stands for “Steve’s Class,” the name having been suggested by Nick Pippenger in recognition of Steve’s surprising 1979 result that deterministic context-free languages are in this class [15], but also in retaliation for Steve’s having introduced the terminology “NC” (“Nick’s Class”) for the set of decision problems that can be solved in polylogarithmic time using only a polynomial number of parallel processors [26]. The significance of these two classes is that, although it is easy to see that each is contained in P, one might expect them both to be proper subclasses of P. That is, there are likely to be problems in P that cannot be solved in polynomial time if restricted to polylog space, and ones that cannot be solved in polylog time if restricted to a polynomial number of processors. By analogy with NP-completeness, one can identify candidates for such problems by identifying ones that are “complete for P” under appropriate reductions. One famous example, complete for P in both senses, is linear programming [16]. Both Cook and Karp have won multiple prizes. Cook won the 1982 ACM Turing Award (the top prize in computer science) and the 1999 CRM-Fields Institute Prize (the top Canadian award for research achievements in the mathematical sciences). Karp won the Lanchester Prize in 1977, the Fulkerson Prize in discrete mathematics in 1979, the ACM Turing Award in 1985, the ORSATIMS von Neumann Theory Prize in 1990, and many others. Levin is long overdue for his own big award, although I expect this will come soon. And, of course, the biggest prize related to NP-completeness is still unawarded: The question of whether P equals NP is one of the six remaining open problems for the resolution of which the Clay Mathematics Institute is offering a $1,000,000 Millenium Prize. Garey, Johnson, and Computers and Intractability My own most influential connection to the theory of NP-completeness is undoubtedly the book Computers and Intractability: A Guide to the Theory of NP-completeness, which I wrote with Mike Garey and which was published in Documenta Mathematica

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Figure 2: Michael Garey and David Johnson in 1977 1979. At the time, we optimistically promised the publishers that we would sell 5,000 copies, but it has now sold over 50,000, picking up some 40,000 citations along the way, according to Google Scholar. My early involvement with the theory, beyond the lunchtime conversation mentioned above, mainly concerned one of the methods for coping with NPcompleteness: Designing and analyzing approximation algorithms. While at MIT I wrote a PhD thesis on approximation algorithms for the bin packing problem [32] and a paper exploring how the same approach could be extended to other problems, such as graph coloring, set covering, and maximum satisfiability [33]. On the strength of this research, I was recruited to come to work at Bell Labs by Ron Graham and Mike Garey, whose initial paper on bin packing had introduced me to the topic. After receiving my PhD in June 1973, I moved to New Jersey and began my Bell Labs/AT&T career. One of my first collaborations with Mike was in producing a response to a letter Don Knuth had written in October to many of the experts in the field. The letter sought a better name than “polynomial complete” for the class of problems that Cook and Karp had identified. Knuth asked for a vote on three terms he was proposing (“Herculean,” “formidable,” and “arduous”). We did not particularly like any of Knuth’s alternatives, and proposed “NP-complete” as a write-in candidate. We were not the only ones, and when Knuth announced the results of his poll in January 1974 [41], he gave up on his original proposals, and declared “NP-complete” the winner, with “NP-hard” chosen to designate problems that were at least as hard as all the problems in NP, although possibly not in NP themselves. See Knuth’s article or [23] for an amusing summary of some of the other proposals he received. Mike and I also began an active research collaboration, covering both bin packing and scheduling algorithms and the proof of new NP-completeness results. When Karp wrote a journal article [38] derived from his original proceedings paper, his expanded list, now of 25 problems, included some of our new results. This set the stage for our book [23], with its much longer list, although the actual genesis of the book was more happenstance. In April 1976, Mike Documenta Mathematica

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and I attended a conference at Carnegie-Mellon University on “New Directions and Recent Results in Algorithms and Complexity,” where I gave a talk on the various types of approximation guarantees we had seen so far. Afterwards, at a coffee break, an editor for the Prentice-Hall publishing company came up to me and suggested that Mike and I write a book on approximation algorithms. In thinking about that proposal, we realized that what was needed, before any book on approximation algorithms, was a book on NP-completeness, and by the time we left the conference we were well on our way to deciding to write that book ourselves. One of my tasks was to collect NP-completeness results for our planned list, which in those days before personal computers meant writing the details by hand onto file cards, stored in plastic box. At that time, it was still possible to aim for complete coverage, and our eventual list of some 300 problems covered most of what had been published by the time we finished our first draft in mid1978, including many results we came up with ourselves when we identified interesting gaps in the literature, and for which we provided the unhelpful citation “[Garey and Johnson, unpublished].” We did keep notes on the proofs, however (in that same plastic box), and most can still be reconstructed . . . After detailed discussions about what we wanted to say, I wrote first drafts of the chapters, with Mike then clarifying and improving the writing. (A quick comparison of the writing in [23] with that in this memoir will probably lead most readers to wish Mike were still doing that.) We did resort to computers for the actual typesetting of the book, although I had to traipse up to the 5th floor UNIX room to do the typing, and put up with the invigorating smell of the chemicals in the primitive phototypesetter there. Because we were providing camera-ready copy, we had the final say on how everything looked, although our publisher did provide thorough and useful copy-editing comments, including teaching us once and for all the difference between “that” and “which.” There was only one last-minute glitch, fortunately caught before the book was finalized – the cover was supposed to depict the graph product of a triangle and a path of length two, and the initial artist’s rendering of this was missing several edges. Over the years, the book has remained unchanged, although later printings include a 2-page “Update” at the end, which lists corrigenda and reports on the status of the twelve open problems listed in Appendix A13 of the book. As of today only two remain unresolved: graph isomorphism and precedence constrained 3-processor scheduling. Of the remaining ten, five are now known to be polynomial-time solvable and five are NP-complete. For details, see [35, 46]. A second edition is perpetually planned but never started, although I have resumed my NP-completeness column, now appearing on a sporadic basis in ACM Transactions on Algorithms, as groundwork for such an undertaking. We never did write that book on approximation algorithms, and indeed no such book seems to have appeared until Dorit Hochbaum’s Approximation Algorithms for NP-Hard Problems [29] appeared in 1997. This was an edited collection, to which Mike, Ed Coffman, and I contributed a chapter. The first Documenta Mathematica

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textbook on approximation algorithms was Vijay Vazirani’s Approximation Algorithms [53], which did not appear until 2001. Although Mike and I never got around to writing a second book, there is a second “Garey and Johnson” book of a sort. In 1990, our wives, Jenene Garey and Dorothy Wilson, respectively a Professor of Nutrition at NYU and a school teacher, coauthored The Whole Kid’s Cookbook, copies of which were sold to raise funds for the Summit Child Care Center, a local institution where Dorothy had worked. The Last Forty Years: Hardness of Approximation It would be impossible, in the limited space left to me, to give a thorough history of the developments in the theory of NP-completeness since the 1970s, so in this section I shall restrict myself to just one thread: applying the theory to approximation algorithms. An approximation algorithm does not necessarily return an optimal solution, but settles for some feasible solution which one hopes will be near-optimal. A standard way to evaluate an approximation algorithm A is in terms of the “worst-case guarantee” it provides. Let us suppose for simplicity that the problem X for which A is designed is a minimization problem. Then A provides a worst-case guarantee equal to the maximum, over all instances I of the problem, of A(I)/OP T (I), where A(I) is the value of the solution that algorithm yields for instance I, and OP T (I) is the optimal solution value. For example, Christofides’ algorithm for the Traveling Salesman Problem (TSP) has a worst-case guarantee of 3/2 if we restrict attention to instances satisfying the triangle inequality [12]. We are of course most interested in approximation algorithms for NP-hard problems that run in polynomial time. Unfortunately, it turns out that sometimes designing such an approximation algorithm can be just as hard as finding an optimal solution. The first paper to make this observation appeared in 1974, written by Sahni and Gonzalez [49]. They showed, for example, that if one does not assume the triangle inequality, then for any constant k, the existence of a polynomial-time approximation algorithm for the TSP with worst-case guarantee k or better would imply P = NP. The proof involves a “gap” construction, by transforming instances of HAMILTON CIRCUIT to TSP instances whose optimal tours have length n if the Hamilton Circuit exists, and otherwise have length greater than kn (for example by letting the distance between u and v be 1 if {u, v} is an edge in the original graph, and kn otherwise). By the time our NP-completeness book appeared, there were a few more results of this type. Of particular interest were results ruling out “approximation schemes.” A polynomial-time approximation scheme (PTAS) for a problem is a collection of polynomial-time algorithms Aǫ , where Aǫ has a worst-case guarantee of 1 + ǫ or better. In 1975, Sahni [48] showed that the Knapsack Problem has such a scheme. His algorithms, and many like them, were seriously impractical, having running times exponential in 1/ǫ, although for any fixed ǫ they do run in polynomial time. Nevertheless, over the years much effort has been Documenta Mathematica

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devoted to finding such schemes for a wide variety of problems. Given how impractical PTASs tend to be, one could perhaps view this everpopular pastime of designing them as providing “negative-negative” results, rather than positive ones. One can rule out the existence of such a scheme (assuming P 6= NP) by proving that there exists an ǫ such that no polynomialtime approximation can have a worst-case guarantee of 1 + ǫ or better unless P = NP. This is trivially true for bin packing, since if an algorithm could guarantee a ratio less than 3/2, then one could use it to solve the subset sum problem. The existence of a PTAS for a problem thus merely shows that there is no ǫ such that one can prove a 1 + ǫ inapproximability result. There is one particular type of PTAS, however, that can perhaps be viewed more positively. Shortly after Sahni’s knapsack PTAS appeared, Ibarra and Kim [31] significantly improved on it, designing what we now call a fully polynomial-time approximation scheme (FPTAS): An algorithm A that takes as input both an instance I and an ǫ > 0, returns a solution that is no worse than (1 + ǫ)OP T (I), and runs in time bounded by a polynomial not just in the size of I, but also in 1/ǫ. Unfortunately, it was quickly realized that FPTASs were much less common than ordinary PTASs. In particular, the TSP with the triangle inequality could not have an FPTAS unless P 6= NP, something that could not then be ruled out for ordinary PTASs. This was because it was “NP-hard in the strong sense,” which means it was NP-hard even if we restrict all numbers in the input (in this case the inter-city distances) to integers that are bounded by some fixed polynomial in the input length, rather than the exponentially large values normally allowed by binary notation. It is an easy result [22] that no optimization problem that is strongly NP-hard can have an FPTAS unless P = NP (in which case none is needed). On the other end of the scale (problems for which no algorithms with a bounded performance guarantee could exist, or at least were known), there were fewer results, although the best performance guarantee then available for P∞ the SET COVER problem was H(n) = i=1 1/i ∼ ln n [33, 44], and no algorithms for clique were known with guarantees better than O(n/polylog(n)) [33]. Whether this was best possible (assuming P 6= NP) was unknown, and the field remained in this state of ignorance for more than a decade. Indeed, although there was the occasional interesting problem-specific result, approximation algorithms remained only a minor thread of algorithms research until 1991, when a seemingly unrelated result in NP-completeness theory suddenly gave them an explosive new life. This result was the discovery of a new characterization of NP, in terms of “probabilistically checkable proofs” (PCPs). A PCP is a proof whose validity can be estimated by looking at only a few, randomly chosen, bits. If the proof is valid, then any choice of those bits will support this fact. If it is defective, than a random choice of the bits to be examined will, with probability 1/2 or greater, confirm that the proof is not valid. This basic concept developed out of a series of papers, starting with the study of interactive proofs involving Documenta Mathematica

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multiple provers and one verifier. (These papers include one with Leonid Levin as a co-author [10].) If f (n) and g(n) are two functions from the natural numbers to themselves, let PCP(f, g) denote that class of all problems that have PCPs using O(f (n)) random bits and looking at O(g(n)) bits of the proof. In late 1991, Feige, Goldwasser, Lov´ asz, Safra, and Szegedy [20] showed that NP ⊆ PCP(log n log log n, log n log log n) and that, surprisingly, this highly-technical result implied that clique could not be approximated to any constant factor unless NP ⊆ DTIME[nO(log log n) ]. This is a weaker conclusion than P = NP, but not much more believable, and in any case, the implication was strengthened to P = NP in early 1992, when Arora and Safra [7] showed that NP = PCP(log n, log n). Shortly thereafter, Arora, Lund, Motwani, Sudan, and Szegedy [5] improved this to NP = PCP(log n, 1), which had even stronger consequences for approximation. In particular, it implied that many famous problems could not have PTASs, including max 2-sat, vertex cover, and the triangle-inequality TSP. There is not room here to give the details of the proofs of these results or all the references, but the key idea was to produce a gap construction for the problem in question, based on the relation between the random bits used by the verifier in a PCP for 3sat, and the proof bits at the addresses determined by those random bits. For a contemporaneous survey, providing details and references, see [34]. In the twenty years since these breakthrough results, there has been an explosion of inapproximability results exploiting variants and strengthenings of the original PCP results, and based on a variety of strengthenings of the hypothesis that P 6= NP. For surveys, see for instance [36, 54]. Today we know that clique cannot be approximated to a factor n1−ǫ for any constant ǫ > 0 unless P = NP [56]. We also know that the Greedy algorithm for set cover, mentioned above, cannot be bettered (except in lower-order terms) unless NP ⊆ DTIME[nO(log log n) ] [19]. Other hypotheses under which hardness of approximation results have been k proved include NP * DTIME[nO(log log log n) ], NP * ∪k>0 DTIME[nlog n ], NP ǫ * ∩ǫ>0 DTIME[2n ], and NP * BPP, the latter a class of problems solvable by randomized algorithms in polynomial time. Currently, the most popular hypothesis, however, is the “Unique Games Conjecture” (UGC) of Subhash Khot [39]. Suppose we are given a prime q, a small ǫ > 0, and a list of equations of the form xj − xk = ch (mod q) in variables xi and constants ch . The conjecture says that it is NP-hard to distinguish between the case where at least a fraction 1 − ǫ of the equations can be simultaneously satisfied and the case when no more than a fraction ǫ of the equations can – a very large gap. As with the PCP results, this conjecture initially came from a problem involving multiple prover systems, and it was in this context that it obtained its name. The reason this rather specialized hypothesis has garnered attention is that it implies that for many important problems, our currently best approximation algorithms cannot be improved upon unless P = N P . For instance, no Documenta Mathematica

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polynomial-time approximation algorithm for vertex cover can guarantee better than the factor of 2 already guaranteed by several simple approximation algorithms [9]. Similarly, the Goemans-Williamson algorithm [24] for max cut, which exploits semidefinite programming and randomized rounding and has a worst-case guarantee of (2/π)/(min0 0 for all x ∈ Rn ) a constructive method was developed by Habicht [3]. It is based on Polya’s theorem [13] which states that for each hom*ogeneous polynomial p such that p(x1 , . . . , xn ) > 0 for all x1 ≥ 0, · · · , xn ≥ 0 and (x1 , . . . , xn ) 6= 0, there exists a natural number N such that all coefficients of the polynomial (x1 + · · · + xn )N p are positive. A quantitative version of Polya’s theorem providing a lower estimate for the number N in terms of p was recently given by Powers and Reznick [14]. There is also a quantitative version of Hilbert’s 17th problem which asks how many squares are needed. In mathematical terms it can be formulated in terms of the pythagoras number. For a ring K, the pythagoras number p(K) is the smallest natural number m such that each finite sum of squares of elements of K is a sum of m squares. If there is no such number m we set p(K) = ∞. Clearly, p(R[x]) = p(R(x)) = 2. Recall that Hilbert [5] had shown that p(R(x, y)) ≤ 4. The landmark result on the quantitative version of Hilbert’s 17th problem was published in 1967 by A. Pfister [11] who proved Theorem 2. p(R(x1 , · · · , xn )) ≤ 2n . Documenta Mathematica

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That is, by Theorems 1 and 2, each nonnegative polynomial f ∈ R[x1 , . . . , xn ] is a sum of at most 2n squares of rational functions. Pfister’s proof was based on the theory of multiplicative forms (see, e.g., [12]), now also called Pfister forms. The next natural question is: What is value of the number p(R(x1 , . . . , xn ))? For n ≥ 3 this is still unknown! It is not difficult to prove that the sum 1 + x21 + · · · + x2n of n + 1 squares is not a sum of m squares with m < n + 1. Therefore n + 1 ≤ p(R(x1 , . . . , xn )) ≤ 2n . Using the theory of elliptic curves over algebraic function fields it was shown in [2] that Motzkin’s polynomial is not a sum of 3 squares. Hence p(R(x1 , x2 )) = 4. Artin’s theorem triggered many further developments. The most important one in the context of optimization is to look for polynomials which are nonnegative on sets defined by polynomial inequalities rather than the whole Rn . To formulate the corresponding result some preliminaries are needed. Let us write P2 for the cone of finite sums of squares of polynomials from R[x1 , . . . , xn ]. n In what follows we suppose that F = {f1 , . . . , fk } is a finite subset of R[x1 , . . . , xn ]. In real algebraic geometry two fundamental objects are associated with F . These are the basic closed semialgebraic set KF = {x ∈ Rn : f1 (x) ≥ 0, · · · , fk (x) ≥ 0} and the preorder TF :=

X

f1ε1 · · · fkεk σε ; σε ∈

εi ∈{0,1}

X2 . n

Note that the preorder TF depends on the set F of generators for the semialgebraic set KF rather than the set KF itself. Obviously, all polynomials from TF are nonnegative on the set KF , but in general TF does not exhaust the nonnegative polynomials on KF . The Positivstellensatz of Krivine-Stengle describes all nonnegative resp. positive polynomials on the semialgebraic set KF in terms of quotients of elements of the preorder TF . Theorem 3. Let f ∈ R[x1 , . . . , xn ]. (i) f (x) ≥ 0 for all x ∈ KF if and only if there exist p, q ∈ TF and m ∈ N such that pf = f 2m + q. (ii) f (x) > 0 for all x ∈ KF if and only if there are p, q ∈ TF such that pf = 1 + q. This theorem was proved by G. Stengle [19], but essential ideas were already contained in J.-L. Krivine’s paper [8]. In both assertions (i) and (ii) the ‘if’ parts are almost trivial. Theorem 3 is a central result of modern real algebraic geometry. Proofs based on the Tarski-Seidenberg transfer principle can be found in [15] and [9]. Documenta Mathematica

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Let us set f1 = 1 and k = 1 in Theorem 3(i). Then KF = Rn and TF = P2 n . Hence in this special case Theorem 3(i) gives Artin’s Theorem 1. The Krivine–Stengle Theorem 3(i) expresses the nonnegative polynomial f on KF as a quotient of the two polynomials f 2m + q and p from the preorder TF . Simple examples show that the denominator polynomial p cannot be avoided in general. For instance, if f1 = 1, k = 1, the Motzkin polynomial M is P2 nonnegative on KF = Rn , but it is not in the preorder TF = n . Replacing ˜ (x, y) := x4 y 2 + x2 y 4 + 1 − x2 y 2 we even get a strictly M by the polynomial M ˜ (x, y) ≥ 26 for all (x, y) ∈ R2 .) positive polynomial of this kind. (One has M 27 2 3 Letting f1 = (1 − x ) , k = n = 1, the semialgebraic set KF is the interval [−1, 1] and the polynomial f = 1−x2 is obviously nonnegative on KF . Looking at the orders of zeros of f at ±1 one concludes easily that f is not in TF . In view of these examples it seems to be surprising that strictly positive polynomials on a compact basic closed semialgebraic set always belong to the preorder. This result is the Archimedean Positivstellensatz which was proved by the author [17] in 1991. Theorem 4. Suppose that f ∈ R[x1 , . . . , xn ]. If the set KF is compact and f (x) > 0 for all x ∈ KF , then f ∈ TF . The original proof given in [17] (see also [18], pp. 344–345) was based on the solution of the moment problem for compact semialgebraic sets. The first algebraic proof of Theorem 4 was found by T. W¨ormann [20], see, e.g., [15] or [9]. P2 By definition the preorder TF is the sum of sets f1ε1 · · · fkεk n . It is natural to ask how many terms of this kind are really needed. This question is answered by a result of T. Jacobi and A. Prestel in 2001. Let g1 , . . . , glk denote the first lk := 2k−1 + 1 polynomials of the following row of mixed products with no repeated factors of the generators f1 , . . . fk : 1, f1 , . . . , fk , f1 f2 , f1 f3 , . . . , fk−1 fk , f1 f2 f3 , . . . , fk−2 fk−1 fk , f1 f2 · · · fk . P2 Let SF be the sum of sets gj n , where j = 1, . . . , lk . Then Jacobi and Prestel [7] proved the following Theorem 5. If KF is compact and f ∈ R[x1 , . . . , xn ] satisfies f (x) > 0 for all x ∈ KF , then f ∈ SF . briefly k = 3, then lk = P 5 and SF = PWe P2 discuss P2 this P2result. PIf 2 2 2 +f +f +f +f f , that is, the sets g 1 2 3 1 2 n n n n n n for g = f1 f3 , f2 f3 , f1 f2 f3 do not enter in the definition of SF . If k = 4, then no products of three or four generators occur in the definition of SF . Thus, if k ≥ 3, Theorem 5 is an essential strengthening of Theorem 4. References ¨ [1] E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. math. Sem. Hamburg 5(1927), 110–115. Documenta Mathematica

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[2] J.S.W. Cassels, W.J. Ellison and A. Pfister, On sums of squares and on elliptic curves over function fields, J. Number Theory 3(1971), 125–49. ¨ [3] W. Habicht, Uber die Zerlegung strikte definiter Formen in Quadrate, Comment. Math. Math. 12(1940), 317–322. ¨ [4] D. Hilbert, Uber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32(1888), 342–350. ¨ [5] D. Hilbert, Uber tern¨ are definite Formen, Acta Math. 17 (1893), 169–197. [6] D. Hilbert, Hermann Minkowski. Ged¨achtnisrede, Math. Ann. 68(1910), 445–471. [7] T. Jacobi and A. Prestel, Distinguished representations of strictly positive polynomials, J. reine angew. Math. 532(2001), 223–235. [8] J.-L. Krivine, Anneaux preordennes, J. Analyse Math. 12(1964), 307–326. [9] M. Marshall, Positive Polynomials and Sums of Squares, Math. Surveys and Monographs 146, Amer. Math. Soc., 2008. [10] T.S. Motzkin, The arithmetic-geometric inequality. In: Proc. Symposium on Inequalities, edited by O. Shisha, Academic Press, New York, 1967, pp. 205–224. [11] A. Pfister, Zur Darstellung definiter Formen als Summe von Quadraten, Invent. Math. 4(1967), 229–237. [12] A. Pfister, Quadratic Forms and Applications in Algebraic Geometry and Topology, London Math. Soc. Lect. Notes 217, Cambridge, 1995. ¨ [13] G. Polya, Uber positive Darstellung von Polynomen, Vierteljschr. Naturforsch. Ges. Z¨ urich 73(1928), 141–145. [14] V. Powers and B. Reznick, A new bound for Polya’s theorem with applications to polynomials positive on polyhedra, J. Pure Applied Algebra 164(2001), 221–229. [15] A. Prestel and C. N. Delzell, Positive Polynomials, Springer-Verlag, Berlin, 2001. [16] B. Reznick, On Hilbert’s construction of positive polynomials, Preprint, 2007. [17] K. Schm¨ udgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289(1991), 203–206. [18] K. Schm¨ udgen, Noncommutative real algebraic geometry – some basic concepts and first ideas. In: Emerging Appl. Algebraic Geometry, edited by M. Putinar and S. Sullivant, Springer-Verlag, Berlin, 2009, pp. 325–350. Documenta Mathematica

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[19] G. Stengle, A Nullstellensatz and a Positivstellensatz in semialgebraic geometry, Math. Ann. 207( 1974), 87–97. [20] T. W¨ ormann, Strikt positive Polynome in der semialgebraischen Geometrie, Dissertation, Universit¨at Dortmund, 1998.

Konrad Schm¨ udgen Mathematisches Institut Universit¨ at Leipzig Johannisgasse 26 04103 Leipzig [emailprotected]

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From Kepler to Hales, and Back to Hilbert Michael Joswig

2010 Mathematics Subject Classification: 01A65 (52B17, 05B40, 03B35) Keywords and Phrases: Sphere packing, Kepler conjecture, formal proofs In layman’s terms the Kepler Conjecture from 1611 is often phrased like “There is no way to stack oranges better than greengrocers do at their fruit stands” and one might add: all over the world and for centuries already. While it is not far from the truth this is also an open invitation to a severe misunderstanding. The true Kepler Conjecture speaks about infinitely many oranges while most grocers deal with only finitely many. Packing finitely many objects, for instance, within some kind of bin, is a well-studied subject in optimization. On the other hand, turning the Kepler Conjecture into a finite optimization problem was a first major step, usually attributed to L´aszl´o Fejes T´oth [5]. Finally, only a little bit less than 400 years after Johannes Kepler, Thomas C. Hales in 1998 announced a complete proof which he had obtained, partially with the help of his graduate student Samuel P. Ferguson [7]. There are many very readable introductions to the proof, its details, and the history, for instance, by Hales himself [8] [10]. Here I will make no attempt to compete with these presentations, but rather I would like to share an opinion on the impact of the Kepler Conjecture and its history for mathematics in general. 1

Packing Spheres

Yet we should start with the formal statement. In the following we will encode a packing of congruent spheres in 3-space by collecting their centers in a set Λ ⊂ R3 . If B(x, r) is the ball with center x ∈ R3 and radius r > 0 and if c > 0 is the common radius of the spheres in the packing then 3 X vol(B(0, r) ∩ B(x, c)) , δ(r, Λ) = 4πr3 x∈Λ

the fraction of the ball B(0, r) covered by the balls in the packing Λ, is the finite packing density of Λ with radius r centered at the origin. Now the upper limit δ(Λ) = limr→∞ δ(r, Λ) does not depend on the constant c, and it is called the packing density of Λ. Documenta Mathematica

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Theorem (Kepler Conjecture). The packing density δ(Λ) of any sphere packing Λ in R3 does not exceed π √ ≈ 0.74048 . 18 It remains to explain where the oranges are. The standard pattern originates from starting with three spheres whose centers form a regular triangle and putting another on top such that it touches the first three. This can be extended indefinitely in all directions. One way of describing this sphere packing in an encoding like above is the following: Λfcc = { a(1, 0, 0) + b(0, 1, 0) + c(1, 1, 1) | a, b, c ∈ Z} , This amounts to tiling √ 3-space with regular cubes of side length 2 and placing spheres of radius 1/ 2 on the vertices as well as on the mid-points of the facets of each cube. This is why Λfcc is called the face-centered cubical packing. Figure 1 (left) shows 14 spheres (significantly shrunk for better visibility) in the cube, the black edges indicate spheres touching. To determine the packing density it suffices to measure a single fundamental domain, that is, one of the cubes. Each sphere at a vertex contributes 1/8 to each of the eight cubes which contain it while each sphere on a facet contributes 1/2. We obtain δ(Λfcc ) = (8 ·

1 1 4π 2π 1 π 1 π +6· )· √ 3 · 3 = 4· √ · = √ = √ . 8 2 3( 2) 2 8 3 2 3 2 18

One thing which is remarkable about the Kepler Conjecture is that the optimum is attained at a lattice packing, that is a sphere packing whose centers form a Z3 -isomorphic subgroup of the additive group of R3 . This means that the optimum is attained for a packing with a great deal of symmetry while the statement itself does not mention any symmetry. It was already known to Carl Friedrich Gauß that Λfcc is optimal among all lattice packings, but the challenge for Hales to overcome was to show that there is no non-lattice packing which is more dense. As already mentioned I will not try to explain the proof, not even its overall structure, but I would like to point out a few aspects. What also contributes to the technical difficulty is √ that Λfcc is by no means the only sphere packing with the optimal density π/ 18. There are infinitely many others, including another well-known example which is called the hexagonal-close packing. This means that the naively phrased optimization problem sup δ(Λ) Λ is a sphere packing in R3 (1)

has infinitely many optimal solutions. A key concept in discrete geometry is the Voronoi diagram of a set Λ of points, say in R3 . The Voronoi region of x ∈ Λ is the set of points in R3 which is at least as close to x as to any other point in Λ. This notion makes sense for Documenta Mathematica

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Figure 1: 14 balls of Λfcc in a cube and corresponding Voronoi regions finite as well as infinite sets Λ. If Λ is finite or if the points are “sufficiently spread out” then the Voronoi regions are convex polyhedra. The Voronoi diagram is the polyhedral complex formed from these polyhedra. For example, the Voronoi region of any point in the face-centered cubical lattice Λfcc is a rhombic dodecahedron, a 3-dimensional polytope whose twelve facets are congruent rhombi. Figure 2 shows the rhombic dodecahedron, and Figure 1 (right) shows how it tiles the space as Voronoi regions of Λfcc . Some 2-dimensional cells (facets of Voronoi regions) are also shown in Figure 1 (left) to indicate their relative position in the cube. Here comes a side-line of √ the story: The volume of the rhombic dodecahedron with inradius one equals 32 ≈ 5.65685, and this happens to be slightly larger than the volume of the regular dodecahedron of inradius one, which amounts to q √ 10 130 − 58 5 ≈ 5.55029 . A potential counter-example to √ the Kepler Conjecture would have Voronoi regions of volume smaller than 32. The statement that, conversely, each unit sphere packing should have Voronoi regions of volume at least the volume of the regular dodecahedron of inradius one, is the Dodecahedral Conjecture of L. Fejes T´ oth from 1943. This was proved, also in 1998, also by Hales together with Sean McLaughlin [12, 13]. Despite the fact that quantitative results for one of the conjectures imply bounds for the other, the Kepler Conjecture does not directly imply the Dodecahedral Conjectures or conversely. Not surprisingly, however, the proofs share many techniques. We now come back to the Kepler Conjecture. The reduction of the infinitedimensional optimization problem (1) to finite dimensions is based on these Voronoi regions. The observation of L. Fejes T´oth in 1953 was that in an optimal sphere packing only finitely many different combinatorial types of Voronoi regions can occur. This resulted in a non-linear optimization problem over a compact set. Hales simplified this non-linear problem using linear approximations. In this manner each candidate for a sphere packing more dense than the face-centered cubical packing gives rise to a linear program. Its infeasibilDocumenta Mathematica

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Figure 2: Rhombic dodecahedron ity refutes the potential counter-example. This idea was improved and further extended by Hales and his co-authors such that this approach resulted in a managable computation, albeit an enormous one. What differs mathematics fundamentally from other fields of science is the concept of a proof. A sequence of statements which establish the claim in a step-by-step manner by applying the rules of logic to trace the result back to a set of axioms. Once the proof is there the result holds indefinitely. The traditional way to accept a proof is to have it scrutinized by peers who review the work prior to publication in a mathematical journal. While neither the author of a proof nor its reviewers are perfect it is rather rare that results are published with a severe error. The mathematical community was content with this proof paradigm for more than 100 years, since the logical foundations of mathematics were laid at the turn from the 19th to the 20th century. The main impact of Hales’ proof to mathematics in its generality is that it is about to change this paradigm, most likely forever. After obtaining his computer-based proof Hales submitted his result to the highly esteemed journal Annals of Mathematics. The journal editors initiated the reviewing process which involved a team of more than a dozen experts on the subject, lead by G´ abor Fejes T´oth, the son of L´aszl´o Fejes T´oth. It took more than seven years until an outline version of the proof was finally accepted and published [9]. To quote the guest editors of a special volume of Discrete & Computational Geometry on more details of the proof, G´abor Fejes T´oth and Jeffrey C. Lagarias [4]: The main portion of the reviewing took place in a seminar run at E¨ otvos University over a 3 year period. Some computer experiments were done in a detailed check. The nature of this proof, consisting in part of a large number of inequalities having little internal structure, and a complicated proof tree, makes it hard for humans to check every step reliably. Detailed checking of specific Documenta Mathematica

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assertions found them to be essentially correct in every case tested. The reviewing process produced in the reviewers a strong degree of conviction of the essential correctness of this proof approach, and that the reduction method led to nonlinear programming problems of tractable size. [. . . ] The reviewing of these papers was a particularly enormous and daunting task. The standard paradigm for establishing proofs in mathematics was stretched beyond its limits. There is also a personal aspect to this. Hales and his coauthors had devoted a lot to the proof, and after waiting for a very long time they had their papers published but only with a warning. The referees had given up on the minute details and said so in public. The referees cannot be blamed in any way, to the contrary, their effort was also immense. This was widely acknowledged, also by Hales. But for him to see his results published with the written hint that, well, a flaw cannot be entirely excluded, must have been quite harsh nonetheless. 2

The Subsequent Challenge

It was David Hilbert who initiated a quest for provably reliable proofs in the 1920s. Ideally, he thought, proofs should be mechanized. The first trace to what later became famous as the “Hilbert Program” is maybe the following quote [16, p. 414]: Diese speziellen Ausf¨ uhrungen zeigen [. . .], wie notwendig es ist, das Wesen des mathematischen Beweises an sich zu studieren, wenn man solche Fragen, wie die nach der Entscheidbarkeit durch endlich viele Operationen mit Erfolg aufkl¨aren will.1 Hilbert’s work on this subject resulted in two books with his student Paul Bernays [17, 18]. It is widely believed that the incompleteness theorems of Kurt G¨ odel [6] put an end to Hilbert’s endeavor. However, this is not completely true. After his proof was published with disclaimers Hales set out to start the Flyspeck project [2]. Its goal is to establish a formal proof of the Kepler Conjecture, quite to Hilbert’s liking. The idea is to formalize the proof in a way that it can be verified by a theorem prover. Hales settled for John Harrison’s HOL Light [14] and now also uses Coq [1] as well as Isabelle [20]. A theorem prover like HOL Light is a program which takes a human-written proof and validates that the rules of propositional logic are correctly applied to obtain a chain of arguments from the axioms to the claim, without any gap. In this way a theorem prover assists the mathematician in proving rather than finding a proof on its own. Of course, such a theorem prover itself is a 1 These

special arguments show [. . .], how necessary it is to study the genuine nature of the mathematical proof, if one wants to clarify questions like the decidability by finitely many operations. Documenta Mathematica

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piece of software which is written by humans. So, where is the catch? The actual core of a theorem prover is very small, small enough to be verified by a human, and this core verifies the rest of the system in a bootstrapping like fashion. This is already much better in terms of reliability. Moreover, if this is not enough, it is even possible to use several independent theorem provers for mutual cross-certification. This way theorem provers help to establish proofs in mathematics with a reliability unprecedented in the history of the subject. For an introduction to automated theorem proving see [21]. To get an idea how such a formal proof may look alike, for example, here is √ the HOL Light proof [15, p. 75] that 2 is irrational: let NSQRT_2 = prove (‘!p q. p * p = 2 * q * q ==> q = 0‘, MATCH_MP_TAC num_WF THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o AP_TERM ‘EVEN‘) THEN REWRITE_TAC[EVEN_MULT; ARITH] THEN REWRITE_TAC[EVEN_EXISTS] THEN DISCH_THEN(X_CHOOSE_THEN ‘m:num‘ SUBST_ALL_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPECL [‘q:num‘; ‘m:num‘]) THEN ASM_REWRITE_TAC[ARITH_RULE ‘q < 2 * m ==> q * q = 2 * m * m ==> m = 0 (2 * m) * 2 * m = 2 * q * q ==> 2 * m ~(A |-- Not G))

3

Conclusion

A minimalistic way to tell the story about the Kepler Conjecture is: “Kepler meets Hilbert twice”. The first encounter is Hilbert’s 1900 address in Paris, where he specifically mentioned the Kepler Conjecture in his 18th problem. This way the Kepler Conjecture was ranked among the most eminent mathematical problems of the time. Later, at various stages in the history of the proof several different flavors of mathematical software systems played and still play a key role. The downside of the current state of affairs is that a computer based proof seems to be unavoidable. The upside, however, is that a reliable version of such a machine-assisted proof is, in fact, possible. Quite close to what Hilbert had imagined. Acknowledgment: I would like to thank Martin Henk and G¨ unter M. Ziegler for helpful comments. References [1] The Coq proof assistant, http://coq.inria.fr/. [2] The Flyspeck project, http://code.google.com/p/flyspeck/. [3] Robert S. Boyer, Matt Kaufmann, and J. Strother Moore, The Boyer– Moore theorem prover and its interactive enhancement, Comput. Math. Appl. 29 (1995), no. 2, 27–62. MR 1314243 (95i:68115) [4] G´ abor Fejes Toth and Jeffrey C. Lagarias, Guest editors’ foreword [The Kepler conjecture by Thomas C. Hales, with Samuel P. Ferguson], Discrete Comput. Geom. 36 (2006), no. 1, 1–3. MR 2229656 [5] L´ aszl´ o Fejes T´ oth, Lagerungen in der Ebene, auf der Kugel und im Raum, Die Grundlehren der mathematischen Wissenschaften, Band 65, SpringerVerlag, Berlin, 1953, 2nd ed. 1972. MR 0057566 (15,248b), 0353117 (50 #5603) ¨ [6] Kurt G¨ odel, Uber formal unentscheidbare S¨ atze der Principia Mathematica und verwandter Systeme I, Monatsh. Math. Phys. 38 (1931), no. 1, 173– 198. MR 1549910 [7] Thomas C. Hales, The Kepler conjecture, arXiv: http://front.math. ucdavis.edu/math.MG/9811078. Documenta Mathematica

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[8]

, Canonballs and honeycombs, Notices Amer. Math. Soc. 47 (2000), no. 4, 440–449.

[9]

, A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), no. 3, 1065–1185. MR 2179728 (2006g:52029)

[10]

, Historical overview of the Kepler conjecture, Discrete Comput. Geom. 36 (2006), no. 1, 5–20. MR 2229657 (2007d:52021)

[11] Thomas C. Hales, John Harrison, Sean McLaughlin, Tobias Nipkow, Steven Obua, and Roland Zumkeller, A revision of the proof of the Kepler conjecture, Discrete Comput. Geom. 44 (2010), no. 1, 1–34. MR 2639816 [12] Thomas C. Hales and Sean McLaughlin, A proof of the dodecahedral conjecture, arXiv: http://front.math.ucdavis.edu/math.MG/9811079. [13]

, The dodecahedral conjecture, J. Amer. Math. Soc. 23 (2010), no. 2, 299–344. MR 2601036 (2011d:52037)

[14] John Harrison, http://www.cl.cam.ac.uk/~jrh13/hol-light/. [15]

, HOL Light tutorial (for version 2.20), http://www.cl.cam.ac. uk/~jrh13/hol-light/tutorial_220.pdf, 2011.

[16] David Hilbert, Axiomatisches Denken, Math. Ann. 78 (1917), no. 1, 405– 415. MR 1511909 [17] David Hilbert and Paul Bernays, Grundlagen der Mathematik. Vol. I, J. W. Edwards, Ann Arbor, Michigan, 1944, 2nd ed. Springer-Verlag, Berlin, 1968. MR 0010509 (6,29a), 0237246 (38 #5536) [18]

, Grundlagen der Mathematik. Vol. II, J. W. Edwards, Ann Arbor, Michigan, 1944, 2nd ed. Springer-Verlag, Berlin, 1970. MR 0010510 (6,29b), 0272596 (42 #7477)

[19] Christian Marchal, Study of the Kepler’s conjecture: the problem of the closest packing, Math. Z. 267 (2011), no. 3-4, 737–765. MR 2776056 (2012b:52032) [20] Larry Paulson, Tobias Nipkow, and Makarius Wenzel, http://www.cl. cam.ac.uk/research/hvg/Isabelle/. [21] Freek Wiedijk, Formal proof — getting started, Notices Amer. Math. Soc. 55 (2008), no. 11, 1408–1414. Michael Joswig Fachbereich Mathematik TU Darmstadt 64289 Darmstadt Germany [emailprotected] Documenta Mathematica

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Vilfredo Pareto and Multi-objective Optimization Matthias Ehrgott

2010 Mathematics Subject Classification: 90C29 Keywords and Phrases: Multi-objective optimization, Pareto optimality A multi-objective optimization problem consists in the simultaneous optimization of p objective functions f1 , . . . , fp subject to some constraints, which I will just write as x ∈ X , where X is a subset of Rn . It is usually assumed that there does not exist any x ∈ X such that all functions fk attain their minimima at x. Hence, due to the absence of a total order on Rp , it is necessary to define the minimization with respect to partial orders. So let Y := {f (x) : x ∈ X } be the set of outcome vectors. To compare elements of Y, I will follow the definition of Koopmans (1951). Let y 1 , y 2 ∈ Y. Then y 1 ≦ y 2 if and only if yk1 ≦ yk2 for all k = 1, . . . p; y 1 ≤ y 2 if and only if y 1 ≦ y 2 , but y 1 6= y 2 and y 1 < y 2 if and only if yk1 < yk2 for all k = 1, . . . p. It is here that Pareto makes his appearance. In countless books and articles on multi-objective optimization, one can find a definition like this: Definition 1. Let X ⊂ Rn be a non-empty set of feasible solutions and f = (f1 , . . . fp ) : Rn → Rp be a function. Feasible solution x ˆ ∈ X is called a Pareto optimal solution of the multi-objective optimization problem min{f (x) : x ∈ X }

(1)

if and only if there does not exist any x ∈ X such that f (x) ≤ f (ˆ x). Sometimes Pareto optimality is defined with respect to outcome vectors. Definition 2. Let Y ∈ Rp be a non-empty set of outcome vectors. Outcome vector yˆ ∈ Y is called Pareto optimal if and only if there does not exist any y ∈ Y such that y ≤ yˆ. Where does the name Pareto optimal come from? Vilfredo Pareto and Francis Ysidro Edgeworth are often called as the fathers of multi-objective optimization. Sentences like the “introduction of the Pareto optimal solution in 1896” (Chen et al., 2005, p. VII); “The concept of noninferior solution was introduced at the turn of the century [1896] by Pareto, a prominent economist” (Chankong and Haimes, 1983, p. 113); “Edgeworth and Pareto were probably Documenta Mathematica

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the first who introduced an optimality concept for such problems” (Jahn, 2004, p. 113); “wurden besonders von F.Y. Edgeworth (1845–1926) and V. Pareto (1848–1929 [sic!]) hinreichende Bedingungen f¨ ur Paretomaximalit¨at bzw. Gleichgewichtsbedingungen angegeben.” (G¨opfert and Nehse, 1990, p. 9) or “The foundations are connected with the names of Vilfredo Pareto (1848–1923) and Francis Ysidro Edgeworth (1845–1926)” (L¨ohne, 2011, p. 1) abound in textbooks. The International Society on Multiple Criteria Decision Making bestows the Edgeworth–Pareto award “upon a researcher who, over his/her career, has established a record of creativity to the extent that the field of MCDM would not exist in its current form without the far-reaching contributions from this distinguished scholar”, see http://www.mcdmsociety.org/intro.html#Awards. Edgeworth was an influential Professor of Economics at King’s College London and from 1891 Professor of Political Economy at Oxford University. In his best known book Mathematical Psychics (Edgeworth, 1881) he applied formal mathematics to decision making in economics. He developed utility theory, introducing the concept of indifference curve and is best known for the Edgeworth box. But because multi-objective optimization is concerned with Pareto optimality rather than Edgeworth optimality, this story focuses on his contemporary. Fritz Wilfried Pareto According to Yu (1985, p. 49) Pareto “was a famous Italian engineer” but he is certainly much better known as an economist. The following information is taken from Stadler (1979) and the wikipedia entry (http://en.wikipedia. org/wiki/Vilfredo_Pareto) on Pareto. Vilfredo Federico Damaso Pareto was born on 15 July 1848 in Paris as Fritz Wilfried Pareto, son of a French woman and an Italian civil engineer, who was a supporter of the German revolution of 1848. His name was changed to the Italian version when his family moved back to Italy in 1855 (or 1858). In 1870 he graduated from Polytechnic Institute of Turin with a dissertation entitled “The Fundamental Principles of Equilibrium in Solid Bodies”. He then worked as an engineer and manager for an Italian railway company. He was very politically active, an ardent supporter of free market economy. He obtained a lecturer position in economics and management at the University of Florence in 1886 (according to wikipedia). Eventually he resigned from his positions in 1889. During the 1880s he became acquainted with leading economists of the time and he published many articles by 1893 (not all academic, though). In 1893 he moved to Lausanne where he lectured at the University of Lausanne and became the successor of L´eon Walras as Professor of Political Economy. In his later years he mainly worked in Sociology. Vilfredo Pareto died at C´el´egny, Switzerland, on 19 August 1923. The University of Lausanne still has a Centre d’´etudes interdisciplinaires Walras Pareto (http://www.unil.ch/cwp). Apart from Pareto optimality, Pareto’s name is attached to the Pareto principle (or 80–20 rule), observing in 1906 that 80% of the property in Italy was owned by Documenta Mathematica

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Figure 1: Vilfredo Pareto 1848–1923 (Picture scanned from the second French edition of Pareto (1906) published in 1927.) 20 % of the population and the Pareto distribution, a power law probability distribution. Pareto Optimality The origin of the term Pareto optimality goes back to the following text from Pareto (1906, Chapter VI, Section 33). Principeremo col definire un termine di cui `e comodo fare uso per scansare lungaggini. Diremo che i componenti di una collettivit`a godono, in una certa posizione, del massimo di ofelimit`a, quando `e impossibile allontanarsi pochissimo da quella posizione giovando, o nuocendo, a tutti i componenti la collettivit`a; ogni piccotissimo spostamento da quella posizione avendo necessariamente per effetto di giovare a parte dei componenti ta collettivit`a e di nuocere ad altri. Or in the English translation (Pareto, 1971, p. 261): Documenta Mathematica

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Matthias Ehrgott We will begin by defining a term which is desirable to use in order to avoid prolixity. We will say that the members of a collectivity enjoy maximum ophelimity in a certain position when it is impossible to find a way of moving from that position very slightly in such a manner that the ophelimity enjoyed by each of the individuals of that collectivity increases or decreases. That is to say, any small displacement in departing from that position necessarily has the effect of increasing the ophelimity which certain individuals enjoy, and decreasing that which others enjoy, of being agreeable to some and disagreeable to others.

Of course, Pareto here refers to the distribution of utility (ophelimity) among individuals in an economy rather than solutions of an optimization problem. Multi-objective optimization or mathematical optimization in general as we know it today, did not exist during Pareto’s lifetime, it only developed in the 1940s. And it is some of the founding works of Operations Research and optimization that need to be cited here. Nobel Laureate in Economics T.C. Koopmans (1951) formally studied production as a resource allocation problem and the combination of activities to represent the output of commodities as a function of various factors. In this work he introduced the following definition of efficient vector (p. 60). “A point y in the commodity space is called efficient if it is possible [i.e., if y ∈ (A)], and if there exists no possible point y¯ ∈ (A) such that y¯ − y ≥ 0.” Note that (A) is what I called Y in Definition 2, i.e., possible means that there is some x such that y = Ax. Koopmans does hence only talk about efficient vectors in terms of the outcome set. He proves necessary and sufficient conditions for efficiency, but he does not refer to Pareto, nor does he talk about Pareto optimal solutions as in Definition 1 – instead he refers to “an activity vector x (that) shall lead to an efficient point y = Ax”. Another classic reference in optimization is the seminal paper by Kuhn and Tucker (1951). They refer to the “vector maximum of Koopmans’ efficient point type for several concave functions g1 (x), . . . , gp (x)”. This seems to be the earliest reference to the optimization of several functions in mathematics. Kuhn and Tucker cite Koopmans (1951) when they talk about vector maximum. They also apply the term efficient to the solutions of vector optimization problems (i.e., in decision space) and introduce the notion of proper efficiency. But, again, no mention of Pareto. Kuhn and Tucker (1951) cite another Nobel Laureate in Economics who contributed to the foundations of multi-objective optimization, Kenneth J. Arrow. Arrow discusses Pareto extensively in his economical work and statements of the impossibility theorem today usually refer to Pareto optimality as one of the axioms that cannot be jointly satisfied by a social choice function, but this term does not appear in Arrow’s original formulation (Arrow, 1951). Arrow’s important contribution to multi-objective optimization (Arrow et al., 1953) starts as follows “A point s of a closed convex subset S of k-space is admissible if there is no t ∈ S with ti ≤ si for all i = 1, . . . , k, t 6= s.” This is, of course, the same as Documenta Mathematica

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Koopmans’ definition of efficient point (whose paper Arrow et al. (1953) cite), and again is relevant in the outcome set of a multi-objective problem rather than the set of feasible solutions – no trace of Pareto here, either. There are a number of other definitions of Pareto optimal, efficient, or admissible points. Zadeh (1963) defines “A system S0 ∈ C is noninferior in C if the intersection of C and Σ> (S0 ) is empty.” Σ> (S0 ) is the set of all systems which are better than S0 with respect to a partial order ≥. Chankong and Haimes (1983) later use the same definition. While Zadeh cites Koopmans and Kuhn and Tucker, Pareto remains notably absent. The final term that is common today is that of a nondominated point. Multiobjective Optimization and Economics When did the term Pareto optimal first appear in the literature? As we have seen, it was not used in early mathematical works on multi-objective optimization. The answer is once again in economics. Little (1950, p. 87) in a discussion of the distribution of income (i.e., in the same context as Pareto himself) uses the term Pareto ‘optimum’ (with the quotation marks). The origin of the term is, therefore, clearly found in economics. It has then apparently mostly been used in economics, appearing in journals such as Public Choice and Journal of Economic Theory. As shown above, it was not used by the economists who are credited with having contributed to the origins of the mathematical theory of multi-objective optimization, but migrated to mathematics later on. The first journal articles that I could find are Basile and Vincent (1970) and Vincent and Leitmann (1970). These articles also used the term undominated as an alternative. This then turned to nondominated in Yu and Leitmann (1974). Economics had a strong influence on the early history of multi-objective optimization, especially Pareto’s original definition of the term maximum ophelimity and the origin of the term Pareto optimum in Little (1950). The move into mathematics and optimization coincides with the mathematization of economics by scholars such as Koopmans and Arrow and finally the introduction of the topic into mathematical optimization by Kuhn and Tucker. It seems to have taken quite a while for Pareto’s name to appear in the mathematical optimization literature. The consequence of the history of Pareto optimality outlined above, is that at present there are quite a few terms (efficient, noninferior, nondominated, admissible, Pareto optimal) that express the same idea. Since multi-objective optimization often distinguishes between decision vectors x ∈ X and outcome vectors y ∈ Y, one can find a large number of combinations of these terms in the literature used in parallel today, such as Pareto optimal decisions and efficient outcomes. It turns out that the history of multi-objective optimization (vector optimization) is quite an interesting read, and I would like to refer interested readers to Stadler (1979) as a starting point. The history of multiple criteria deciDocumenta Mathematica

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sion making in general is the topic of the book K¨oksalan et al. (2011). These works also consider roots of multi-objective optimization in game theory and the theory of ordered spaces and vector norms. References K. J. Arrow. Social Choice and Individual Values. Cowles Commission for Research in Economics Monograph No. 12. John Wiley & Sons, New York, 1951. K. J. Arrow, E. W. Barankin, and D. Blackwell. Admissible points of convex sets. In H.W. Kuhn and A.W. Tucker, editors, Contributions to the Theory of Games, volume 2, pages 87–91. Princeton University Press, Princeton, 1953. G. Basile and T. L. Vincent. Absolutely cooperative solution for a linear, multiplayer differential game. Journal of Optimization Theory and Applications, 6:41–46, 1970. V. Chankong and Y. Y. Haimes. Multiobjective Decision Making – Theory and Methodology. Elsevier Science, New York, 1983. G. Chen, X. Huang, and X. Yang. Vector Optimization – Set-Valued and Variational Analysis, volume 541 of Lecture Notes in Economics and Mathematical Systems. Springer Verlag, Berlin, 2005. F. Y. Edgeworth. Mathematical Psychics. C. Kegan Paul & Co., London, 1881. A. G¨ opfert and R. Nehse. Vektoroptimierung, volume 74 of MathematischNaturwissenschaftliche Bibliothek. BSB B.G. Teubner Verlagsgesellschaft, Leipzig, 1990. J. Jahn. Vector Optimization – Theory, Applications, and Extensions. Springer Verlag, Berlin, 2004. M. K¨ oksalan, J. Wallenius, and S. Zionts. Multiple Criteria Decision Making – From Early History to the 21st Century. World Scientific Publishing, Singapore, 2011. T. C. Koopmans. Analysis of production as an efficient combination of activities. In T.C. Koopmans, editor, Activity Analysis of Production and Allocation, Cowles Commission for Research in Economics Monograph No. 13, pages 33–97. John Wiley & Sons, New York, 1951. H. W. Kuhn and A. W. Tucker. Nonlinear programming. In J. Neyman, editor, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pages 481–492. University of California Press, Berkeley, 1951. Documenta Mathematica

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I. M. D. Little. A Critique of Welfare Economics. The Clarendon Press, Oxford, 1950. A. L¨ ohne. Vector Optimization with Infimum and Supremum. Springer Verlag, Berlin, 2011. V. Pareto. Manuale di Economia Politica. Societ`a Editrice Libraria, Milan, 1906. V. Pareto. Manual of Political Economy. Augustus M. Kelley Publishers, New York, 1971. W. Stadler. A survey of multicriteria optimization or the vector maximum problem, Part I: 1776-1960. Journal of Optimization Theory and Applications, 29:1–52, 1979. T. L. Vincent and G. Leitmann. Control-space properties of cooperative games. Journal of Optimization Theory and Applications, 6:91–113, 1970. P. L. Yu. Multiple Criteria Decision Making: Concepts, Techniques and Extensions. Plenum Press, New York, 1985. P. L. Yu and G. Leitmann. Compromise solutions, domination structures, and Salukvadze’s solution. Journal of Optimization Theory and Applications, 13: 362–378, 1974. L. A. Zadeh. Optimality and non-scalar-valued performance criteria. IEEE Transactions on Automatic Control, 8:59–60, 1963.

Matthias Ehrgott Department of Engineering Science The University of Auckland New Zealand [emailprotected]

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Optimisation and Utility Functions Walter Schachermayer

2010 Mathematics Subject Classification: 91B16, 91B24 Keywords and Phrases: Portfolio optimisation, utility functions The story begins in St. Petersburg in 1738. There Daniel Bernoulli proposed a solution to the “St. Petersburg Paradox” by introducing the notion of a utility function. The problem is formulated in somewhat flowery terms as a game. It was proposed by Nicholas Bernoulli, a cousin of Daniel, in a letter from 1713 to Pierre Raymond de Montmort. Suppose I offer you a random sum of money where the amount is determined from subsequent tosses of a fair coin in the following way. The payoff equals 2n ducats if the first heads appears on the n’th toss. Of course, this event has probability 2−n , so that the expected value of the payoff equals 1 2

×2+

1 4

× 4 + ... +

1 n 2n 2

+ . . . = ∞.

(1)

Here is the question: how much would you be willing to pay to me as a fixed price for obtaining this kind of lottery ticket? It is instructive to discuss this question with students in a class and to ask for bids. One rarely gets a bid higher than, say, 10 ducats. This is remarkably far away from the expected payoff of the game which is infinity. Clever students quickly ask a crucial question: are we allowed to play this game repeatedly? This would change the situation dramatically! The law of large numbers, which was already well understood in Daniel Bernoulli’s times, at least in its weak form, tells you that in the long run the average win per game would indeed increase to infinity. Hence in this case, clever students would be willing to pay quite an elevated fixed price for the game. But the flavor of the problem is that you are only offered to play the game once. How to determine a reasonable value of the game? Daniel Bernoulli proposed not to consider the nominal amount of money but rather to transform the money scale onto a different scale, namely the utility which a person draws from the money. For a good historic account we refer to [4]. Daniel Bernoulli proposed to take U (x) := log(x) as a measure of the utility of having an amount of x ducats. And he gives good reasons for this choice: think of a person, an “economic agent” in todays economic lingo, who Documenta Mathematica

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manages to increase her initial wealth w > 0 by 10%. Measuring utility by the logarithm then yields that the increase in utility is independent of w, namely 11 log( 11w 10 ) − log(w) = log( 10 ). Bernoulli therefore passes from the expected nominal amount (1) of the game to the expected utility of the wealth of an agent after receiving the random amount of the game, i.e., 1 2

log(w − c + 2) +

1 4

log(w − c + 4) + . . . +

1 2n

log(w − c + 2n ) + . . . ,

(2)

where w denotes the initial wealth of the agent and c the price she has to pay for the game. Of course, this sum now converges. For example, if w − c = 0, the sum equals log(4). This allows for the following interpretation: suppose the initial wealth of the agent equals w = 4. Then c = 4 would be a reasonable price for the game, as in this case the agent who uses expected log-utility as a valuation of the payoff, is indifferent between the following two possibilities: (1) not playing the game in which case the wealth remains at w = 4, yielding a certain utility of log(4). (2) Playing the game and paying c = 4 for this opportunity. This yields, by the above calculation, also an expected utility of log(4). The above method today is known as “utility indifference pricing”. We have illustrated it for initial wealth w = 4, as the calculations are particularly easy for this special value. But, of course, the same reasoning applies to general values of w. It is immediate to verify that this pricing rule yields a price c(w) in dependence of the initial wealth w which is increasing in w. In economic terms this means that, the richer an agent is, the more she is willing to pay for the above game. This does make sense economically. In any case, the introduction of utility functions opened a perspective of dealing with the “St. Petersburg Paradox” in a logically consistent way. Let us now make a big jump from 18’th century St. Petersburg to Vienna in the 1930’s. The young Karl Menger started with a number of even younger mathematical geniuses the “Mathematische Colloquium”. Participants were, among others, Kurt G¨ odel, Olga Taussky, Abraham Wald, Franz Alt. There also came international visitors, e.g., John von Neumann or Georg N¨obeling. In this colloquium a wide range of mathematical problems were tackled. Inspired by an open-minded banker, Karl Schlesinger, the Colloquium also dealt with a basic economic question: How are prices formed in a competitive economy? As a toy model think about a market place where “many” consumers can buy apples, bananas, and citruses from “many” merchants. We assume that the consumers are well informed, that they want to get the best deal for their money, and that there are no transaction costs. This assumption implies already that the prices πa , πb , πc of these goods have to be equal, for each merchant. Indeed, otherwise merchants offering higher prices than their competitors could not sell their fruits. For each of the consumers the market prices πa , πb , πc are given and, depending on their preferences and budgets, they make their buying decisions as functions of (πa , πb , πc ). On the other hand, the merchants decide on these Documenta Mathematica

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prices. For example, if the current prices are such that the apples are immediately sold out, while few people want to buy the bananas, it seems obvious that the price πa should go up, while πb should go down. This seems quite convincing if we only have apples and bananas, but if there are more than two goods, it is not so obvious any more how the prices for the apples and the bananas relate to the demand for citruses. This question was already treated some 50 years earlier by L´eon Walras, who was Professor of economics in Lausanne. He modeled the situation by assuming that each agent is endowed with an initial wealth w and a utility function U assigning to each combination (xa , xb , xc ) of apples, bananas, and citruses a real number U (xa , xb , xc ). For given prices (πa , πb , πc ), each of the agents optimises her “portfolio” (xa , xb , xc ) of apples, bananas, and citruses. In this setting, we call a system of prices (πa , πb , πc ) an equilibrium if “markets clear”, i.e., if for each of the three goods the total demand equals the total supply. The obvious question is: Is there an equilibrium? Is it unique? L´eon Walras transformed the above collection of optimisation problems, which each of the “many” agents has to solve for her specific endowment and utility function, into a set of equations by setting the relevant partial derivatives zero. And then he simply counted the resulting number of equations and unknowns and noted that they are equal. At this point he concluded – more or less tacitly – that there must be a solution which, of course, should be unique as one can read in his paper “Die Gleichungen des Tausches” from 1875. But, of course, in the 1930’s such a reasoning did not meet the standards of a “Mathematische Colloquium” any more. Abraham Wald noticed that the question of existence of an equilibrium has to be tackled as a fixed point problem and eventually reduced it to an application of Brouwer’s fixed point theorem. He gave a talk on this in the Colloquium and the paper was announced to appear in the spring of 1938. However, the paper was lost in the turmoil of the “Anschluss” of Austria, when the Colloquium abruptly ended, and most participants had other worries, namely organising their emigration. It was only after the war that this topic was brought up again with great success, notably by the eminent economists Kenneth Arrow and Gerard Debreu. Finally, we make one more big jump in time and space, this time to Boston in the late 1960’s. The famous economist Paul Samuelson at MIT had become interested in the problem of option pricing. Triggered by a question of Jim Savage, Paul Samuelson had re-discovered the dissertation of Louis Bachelier, entitled “Th´eorie de la sp´eculation”, which Bachelier had defended in 1900 in Paris. Henri Poincar´e was a member of the jury. In his dissertation Bachelier had introduced the concept of a “Brownian motion” (this is today’s terminology) as a model for the price process of financial assets. He thus anticipated the work of Albert Einstein (1905) and Marian Smoluchowski (1906) who independently applied this concept in the context of thermodynamics. Paul Samuelson proposed a slight variant of Bachelier’s model, namely

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putting the Brownian motion W on an exponential scale, i.e., dSt = St µdt + St σdWt ,

0 ≤ t ≤ T.

(3)

Here St denotes the price of a “stock” (e.g. a share of Google) at time t. The initial value S0 is known and the above stochastic differential equation models the evolution of the stock price in time. The parameter µ corresponds to the drift of the process, while σ > 0 is the “volatility” of the stock price, which models the impact of the stochastic influence of the Brownian motion W . This model is called the “Black-Scholes model” today, as Fisher Black and Myron Scholes managed in 1973 to obtain a pricing formula for options on the stock S which is solely based on the “principle of no arbitrage”. This result was obtained simultaneously by Robert Merton, a student of Paul Samuelson. The “Black-Scholes formula” earned Myron Scholes and Robert Merton a Nobel prize in Economics in 1997 (Fisher Black unfortunately had passed away already in 1995). Here we want to focus on a slightly different aspect of Robert Merton’s work, namely dynamic portfolio optimisation, which he started to investigate in the late sixties [3]. Imagine an investor who has the choice of investing either into a stock which is modeled by (3) above, or into a bond which earns a deterministic fixed interest rate, which we may assume (without loss of generality) to be simply zero. How much of her money should she invest into the stock and how much into the bond? The dynamic aspect of the problem is that the investor can – and, in fact, should – rebalance her portfolio in continuous time, i.e., at every moment. To tackle this problem, Merton fixed a utility function U : R+ → R modeling the risk aversion of the investor. A typical choice is the “power utility” U (x) =

xγ γ ,

x > 0,

(4)

where γ is a parameter in ] − ∞, 1[ \ {0}. Of course, the case γ = 0 corresponds to the logarithmic utility. One thus may well-define the problem of maximising the expected utility of terminal wealth at a fixed time T , where we optimise over all trading strategies. A similar problem can be formulated when you allow for consumption in continuous time. Here is the beautiful result by Robert Merton. Fixing the model (3) and the utility function (4), the optimal strategy consists of investing a fixed fraction m of one’s total wealth into the stock (and the remaining funds into the bond). The value m of this fraction can be explicitly calculated from the parameters appearing in (3) and (4). To visualize things suppose that m = 21 , so that the investor always puts half of her money into the stock and the other half into the bond. This implies that the investor sells stocks, when their prices go up, and buys them when they go down. A remarkable feature is that she should do so in continuous time which – in view of wellknown properties of Brownian trajectories – implies that the total volume of her trading is almost surely infinite, during each interval of time! Documenta Mathematica

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The method of Merton is dynamic programming. He defines the Hamilton– Jacobi–Bellman value-function corresponding to the above problem. In this setting he manages to explicitly solve the PDE which is satisfied by this valuefunction. Of course, this so-called “primal method” is not confined to the special setting analysed by Robert Merton. It can be – and was – extended to many variants and generalisations of the above situation. There is also a dual approach to this family of problems which was initiated in a different context by J.-M. Bismut [1]. In the Mathematical Finance community this approach is also called the “martingale method”. Speaking abstractly, Merton’s problem is just a convex optimisation problem over some infinite-dimensional set, namely the set of all “admissible” trading strategies. As is very wellknown, one may associate to each convex optimisation problem a “dual” problem, at least formally. The method consists in introducing (an infinite number of) Lagrange multipliers and to find a saddle point of the resulting Lagrangian function. This leads to an application of the minmax theorem. Eventually one has to optimize the Legendre transform of U over an appropriate “polar” set. To make this general route mathematically precise, one has to identify appropriate regularity conditions, which make sure that things really work as they should, e.g., existence and uniqueness of the primal and dual optimizer as well as their differential relations. In the present case, there are two aspects of regularity conditions: on the one hand side on the model of the stock price process, e.g., (3), and on the other hand on the choice of the utility function, e.g., (4). In order to develop a better understanding of the nature of the problem, from a mathematical as well as from an economic point of view, it is desirable to identify the natural regularity assumptions. Ideally, they should be necessary and sufficient for a good duality theory to hold true. In [2] this question was answered in the following way. As regards the choice of the model S for the stock price process, virtually nothing has to be assumed, except for its arbitrage freeness, which is very natural in the present context. As regards the utility function U one has to impose the condition of “reasonable asymptotic elasticity”, xU ′ (x) < 1, (5) lim sup U (x) x→∞ which is reminiscent of the ∆2 condition in the theory of Orlicz spaces. The name “asymptotic elasticity” comes from the fact that the derivative U ′ (x), normalised by U (x) and x as in (5), is called the “elasticity” of U in economics. To get a feeling for the significance of condition (5), note that for a concave, increasing function U the above limit is always less than or equal to 1. In the case of power utility (4) this limit equals γ < 1. Considering x U (x) = log(x) , for x > x0 , we find an example where the above limit equals 1, i.e., a utility function U which fails to have “reasonable asymptotic elasticity”. Documenta Mathematica

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It turns out that condition (5) is a necessary and sufficient condition for the duality theory to work in a satisfactory way. If it is violated, one can find a stock price process S – in fact a rather simple and regular one – such that the duality theory totally fails. On the other hand, if it holds true, the duality theory, as well as existence and uniqueness of the primal and dual optimiser etc, works out well, even for very general stock price processes S. There is a lot of further research on its way on related issues of portfolio optimisation. As an example, we mention the consideration of proportional transaction costs (e.g., Tobin tax) in the above problem of choosing an optimal dynamic portfolio. Of course, the most fruitful approach is the interplay between primal and dual methods. References [1] J.M. Bismut, Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404 (1973) [2] D. Kramkov and W. Schachermayer, The condition on the asymptotic elasticity of utility functions and optimal investment in incomplete markets. Annals of Applied Probability 9 3, 904–950 (1999) [3] R.C. Merton, Optimum consumption and portfolio rules in a continuoustime model. Journal of Economic Theory 3, 373–413 (1971) [4] Gordan Zitkovic, Utility Theory: Historical Perpectives. Encyclopedia of Quantitative Finance 4, 1860–1862 (2010)

Walter Schachermayer University of Vienna Faculty of Mathematics Nordbergstraße 15 1090 Vienna Austria [emailprotected]

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