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Book Review

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MathSciNet review: 2731656
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Book Information:

Author: Richard W. Cottle
Title: The basic George B. Dantzig
Additional book information: Stanford University Press, Stanford, California, 2003, xvi + 378 pp., ISBN 978-0-8047-4834-6, $57.00, hardcover

Ilan Adler, Nimrod Megiddo, and Michael J. Todd, New results on the average behavior of simplex algorithms, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 378–382. MR 752803, DOI 10.1090/S0273-0979-1984-15317-5

M. L. Balinski. Mathematical programming: Journal, society, recollections. In History of Mathematical Programming, (J. K. Lenstra, A. H. G. Rinnooy Kan, and A. Schrijver, editors), Elsevier Science Publishers, 1991, pp. 5–18.

Aharon Ben-Tal and Arkadi Nemirovski, Lectures on modern convex optimization, MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2001. Analysis, algorithms, and engineering applications. MR 1857264, DOI 10.1137/1.9780898718829

Robert E. Bixby, Solving real-world linear programs: a decade and more of progress, Oper. Res. 50 (2002), no. 1, 3–15. 50th anniversary issue of Operations Research. MR 1885204, DOI 10.1287/opre.50.1.3.17780

Robert G. Bland, Donald Goldfarb, and Michael J. Todd, The ellipsoid method: a survey, Oper. Res. 29 (1981), no. 6, 1039–1091. MR 641676, DOI 10.1287/opre.29.6.1039

Karl-Heinz Borgwardt, The simplex method, Algorithms and Combinatorics: Study and Research Texts, vol. 1, Springer-Verlag, Berlin, 1987. A probabilistic analysis. MR 868467, DOI 10.1007/978-3-642-61578-8

George B. Dantzig, Linear programming and extensions, Princeton University Press, Princeton, N.J., 1963. MR 0201189

Jan Karel Lenstra, Alexander H. G. Rinnooy Kan, and Alexander Schrijver (eds.), History of mathematical programming, North-Holland Publishing Co., Amsterdam; Centrum voor Wiskunde en Informatica, Amsterdam, 1991. A collection of personal reminiscences. MR 1183952

Jacques Faraut and Adam Korányi, Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1446489

Martin Grötschel, László Lovász, and Alexander Schrijver, Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics: Study and Research Texts, vol. 2, Springer-Verlag, Berlin, 1988. MR 936633, DOI 10.1007/978-3-642-97881-4

Branko Grünbaum, Convex polytopes, Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. MR 0226496

Gil Kalai, Linear programming, the simplex algorithm and simple polytopes, Math. Programming 79 (1997), no. 1-3, Ser. B, 217–233. Lectures on mathematical programming (ismp97) (Lausanne, 1997). MR 1464768, DOI 10.1016/S0025-5610(97)00061-0

Gil Kalai and Daniel J. Kleitman, A quasi-polynomial bound for the diameter of graphs of polyhedra, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 315–316. MR 1130448, DOI 10.1090/S0273-0979-1992-00285-9

N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), no. 4, 373–395. MR 779900, DOI 10.1007/BF02579150

L. G. Hačijan, A polynomial algorithm in linear programming, Dokl. Akad. Nauk SSSR 244 (1979), no. 5, 1093–1096 (Russian). MR 522052

L. G. Hačijan, Polynomial algorithms in linear programming, Zh. Vychisl. Mat. i Mat. Fiz. 20 (1980), no. 1, 51–68, 260 (Russian). MR 564776

J. Lasserre. Moments, Positive Polynomials and Their Applications. Imperial College Press, London, 2010.

Arkadi S. Nemirovski and Michael J. Todd, Interior-point methods for optimization, Acta Numer. 17 (2008), 191–234. MR 2436012, DOI 10.1017/S0962492906370018

Y. E. Nesterov and A. S. Nemirovski. Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms. SIAM Publications. SIAM, Philadelphia, USA, 1993.

James Renegar, A mathematical view of interior-point methods in convex optimization, MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2001. MR 1857706, DOI 10.1137/1.9780898718812

A. Schrijver. On the history of combinatorial optimization (til 1960). In Handbook of Discrete Optimization (K. Aardal, G. Nemhauser, and R. Weismantel, editors), Elsevier Science Publishers, 2005, pp. 1–68.

Daniel A. Spielman and Shang-Hua Teng, Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time, J. ACM 51 (2004), no. 3, 385–463. MR 2145860, DOI 10.1145/990308.990310

D. Spielman and S.-H. Teng. Smoothed analysis: An attempt to explain the behavior of algorithms in practice. Communications of the ACM, 52:76–84, 2009.

M. J. Todd, Semidefinite optimization, Acta Numer. 10 (2001), 515–560. MR 2009698, DOI 10.1017/S0962492901000071

Michael J. Todd, The many facets of linear programming, Math. Program. 91 (2002), no. 3, Ser. B, 417–436. ISMP 2000, Part 1 (Atlanta, GA). MR 1888985, DOI 10.1007/s101070100261

Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1

Review Information:

Reviewer: Michael J. Todd
Affiliation: Cornell University
Email: mjt7@cornell.edu

Journal: Bull. Amer. Math. Soc. 48 (2011), 123-129
DOI: https://doi.org/10.1090/S0273-0979-2010-01303-3
Published electronically: May 19, 2010
Review copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.