Documenta Mathematica Optimization Stories - Mathematics

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42 Eberhard Knobloch References [1] EberhardKnobloch: LeonhardEuler1707–1783, Zum300.Geburtstageines langjährigen Wahlberliners. In: Mitteilungen der Deutschen Mathematiker- Vereinigung 15 (2007), 276–288. Eberhard Knobloch Berlin-Brandenburg Academy of Sciences and Humanities Technische Universität Berlin H 72 Straße des 17. Juni 135 10623 Berlin eberhard.knobloch@tu-berlin.de Documenta Mathematica · Extra Volume ISMP (2012) 37–42
Documenta Math. 43 Euler, Mei-Ko Kwan, Königsberg, and a Chinese Postman Martin Grötschel and Ya-xiang Yuan 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 anarticle[3]withthetitle“Solutio Problematis adGeometriamSitusPertinentis”, 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 Königsberg 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. Readersinterestedingraphtheorypapersofhistoricrelevanceshouldconsult [1] which contains a collection of 37 important articles, translated into English; [3] is the first one in this collection. Leonhard Euler: When did he solve the Königsberg bridges problem? 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 18 th 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 · Extra Volume ISMP (2012) 43–50
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Page 5 and 6: Richard W. Cottle William Karush an
Page 7 and 8: Documenta Math. 1 Preface When in d
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Page 15 and 16: Documenta Math. 9 Jiu Zhang Suan Sh
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Page 83 and 84: 2 A first counterexample Who Solved
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Pope Gregory, the Calendar, and Con
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Documenta Math. 95 Löwner-John Ell
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2 The ellipsoids Löwner-John Ellip
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Löwner-John Ellipsoids 99 For maxi
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H + Löwner-John Ellipsoids 101 t P
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Löwner-John Ellipsoids 103 any odd
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Löwner-John Ellipsoids 105 [23] P.
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Documenta Math. 107 A Brief History
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A Brief History 109 first-generatio
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A Brief History 111 developments ar
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A Brief History 113 in practice. Ka
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A Brief History 115 that LPs with s
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Version-to-Version Speedup 10 9 8 7
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A Brief History 119 [10] Crowder, H
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A Brief History 121 Robert E. Bixby
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Documenta Math. 123 Discrete Optimi
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Discrete Optimization Stories 125 I
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Documenta Math. 127 The Origins of
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Minimal Spanning Tree Algorithm 129
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Documenta Math. 143 The Coming of t
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The Sixties The Coming of the Matro
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The Coming of the Matroids 147 Figu
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The Coming of the Matroids 149 desc
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The Coming of the Matroids 151 In f
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The Coming of the Matroids 153 [10]
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Documenta Math. 155 On the History
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On the History of the Shortest Path
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Documenta Math. 169 On the History
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Transportation and Maximum Flow Pro
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Documenta Math. 181 Edmonds, Matchi
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The Birth of Polyhedral Combinatori
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Documenta Math. 199 Flinders Petrie
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The travelling salesman problem Fli
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Flinders Petrie 203 accuracy. The r
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Flinders Petrie 205 Figure 3: Paper
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Flinders Petrie 207 Figure 4: Petri
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Flinders Petrie 209 [8] F. R. Hodso
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Documenta Math. 211 D. Ray Fulkerso
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D. Ray Fulkerson and Project Schedu
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G 1 2 D. Ray Fulkerson and Project
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Documenta Math. 221 The Ongoing Sto
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The Ongoing Story of Gomory Cuts 22
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Acknowledgement The Ongoing Story o
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Documenta Math. 227 Markowitz and M
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Markowitz and Manne + Eastman + Lan
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Documenta Math. 239 Ronald Graham:
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Ronald Graham: Foundations of Onlin
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Machines Ronald Graham: Foundations
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Ronald Graham: Foundations of Onlin
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Documenta Math. 247 Continuous Opti
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Figure 1: Minkowski’s grave ( c
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Documenta Math. 251 Cauchy and the
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Cauchy and the Gradient Method 253
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Documenta Math. 255 William Karush
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William Karush and the KKT Theorem
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Documenta Math. 271 Nelder, Mead, a
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Nelder, Mead, and the Other Simplex
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Nelder, Mead, and the Other Simplex
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Documenta Math. 277 Subgradient Opt
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Subgradient Optimization in Nonsmoo
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Documenta Math. 291 A Science Ficti
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A Science Fiction Story in Nonsmoot
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Documenta Math. 301 Broyden Updatin
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Broyden Updating, the Good and the
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Documenta Math. 317 Carathéodory o
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Carathéodory on the Road to the Ma
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Documenta Math. 331 The Cold War an
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The Cold War and the Maximum Princi
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Documenta Math. 345 The Princess an
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yB y The Princess and Infinite-Dime
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x2 The Princess and Infinite-Dimens
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The Princess and Infinite-Dimension
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Documenta Math. 357 Computing Stori
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Documenta Math. 359 A Brief History
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A Brief History of NP-Completeness
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Documenta Math. 377 On the Evolutio
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Optimization Modeling Systems 379 T
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Optimization Modeling Systems 381 m
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Optimization Modeling Systems 383 C
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Optimization Modeling Systems 385 a
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Optimization Modeling Systems 387 e
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Documenta Math. 389 Who Invented th
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Who Invented the Reverse Mode of Di
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Documenta Math. 401 Gordon Moore an
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Gordon Moore and His Law 403 Figure
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Gordon Moore and His Law 405 jector
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!$ ? !$ > !$ & Gordon Moore and His
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Gordon Moore and His Law 415 [14] M
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Documenta Math. 417 More Optimizati
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Documenta Math. 419 Voronoi Diagram
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Voronoi Diagrams and Delaunay Trian
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a a b d Voronoi Diagrams and Delaun
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Documenta Math. 433 Around Hilbert
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Around Hilbert’s 17th Problem 435
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Around Hilbert’s 17th Problem 437
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Documenta Math. 439 From Kepler to
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From Kepler to Hales 441 Figure 1:
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From Kepler to Hales 443 assertions
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From Kepler to Hales 445 |- !A. con
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Documenta Math. 447 Vilfredo Pareto
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Vilfredo Pareto and Multi-objective
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Documenta Math. 455 Optimisation an
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Optimisation and Utility Functions
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Optimisation and Utility Functions
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