The many facets of linear programming

Mathematical Programming, Series B

Volumn 91, Issue 3, 2002, Pages 417-436

  Todd, Michael J ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

Ellipsoid method; History; Interior point methods; Linear programming; Simplex method

Indexed keywords

COMPUTATIONAL COMPLEXITY; COMPUTATIONAL METHODS; HISTORY; OPTIMIZATION;

ELLIPSOID METHOD; INTERIOR POINT METHODS; SIMPLEX METHOD;

LINEAR PROGRAMMING;

EID: 0010691873     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/s101070100261     Document Type: Review

References (108)

1. Adler, I. (1983): The expected number of pivots needed to solve parametric linear programs and the efficiency of the self-dual simplex method. Technical report. Department of Industrial Engineering and Operations research, University of California, Berkeley, CA
2. 2)) expected number of steps. Journal of Complexity 3, 372-387
3. Adler, I., Megiddo, N. (1985): A simplex algorithm where the average number of steps is bounded between two quadratic functions of the smaller dimension. Journal of the ACM 32, 871-895
4. Agmon, S. (1954): The relaxation method for linear inequalities. Canadian Journal of Mathematics 6, 382-392
5. Anstreicher, K.M. (1996): Potential reduction algorithms. In: Terlaky, T., ed., Interior Point Methods in Mathematical Programming. Kluwer Academic Publishers, Dordrecht, The Netherlands
6. Balas, E., Padberg, M. (1972): On the set covering problem. Operations Research 20, 1152-1161
7. Balinski, M.L. (1984): The Hirsch conjecture for dual transportation polyhedra. Mathematics of Operations Research 4, 629-633
8. Balinski, M.L. (1991): Mathematical programming: Journal, society, recollections. In: Lenstra, J.K., Rinnooy Kan, A.H.G., Schrijver, A., eds., History of Mathematical Programming, pp. 5-18. Elsevier Science Publishers
9. Benson, S.J., Ye, Y., Zhang, X. (1999): Mixed linear and semidefinite programming for combinatorial and quadratic optimization. Optimization Methods and Software 11, 515-544
10. Benson, S.J., Ye, Y., Zhang, X. (2000): Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization 10(2), 443-461
11. Bixby, R.E. (1992): Implementing the simplex method: The initial basis. ORSA Journal on Computing 4, 267-284
12. Bixby, R.E. (1994): Progress in linear programming. ORSA Journal on Computing 6(1), 15-22
13. Bixby, R.E., Fenelon, M., Gu, Z., Rothberg, E., Wunderling, R. (2000): MIP: Theory and practice - closing the gap. In: Powell, M.J.D., Scholtes, S., eds., System Modelling and Optimization: Methods, Theory, and Applications, pp. 19-49. Kluwer Academic Publishers
14. Bland, R., Goldfarb, D., Todd, M. (1981): The ellipsoid method: A survey. Operations Research 29, 1039-1091
15. Borgwardt, K. (1982): The average number of pivot steps required by the simplex method is polynomial. Zeitschrift für Operations Research 26, 155-177
16. Borgwardt, K.H. (1986): The Simplex Method: A Probabilistic Analysis. Springer-Verlag, Berlin
17. Borgwardt, K.H. (1990): Probabilistic analysis of the simplex method. In: Lagarias, J.C., Todd, M.J., eds., Mathematical Developments Arising from Linear Programming. American Mathematical Society
18. Borgwardt, K.H. (1999): A sharp upper bound for the expected number of shadow vertices in LP-polyhedra under orthogonal projection on two-dimensional planes. Mathematics of Operations Research 24(3), 544-603. A corrected version of Figure 1 appears in Volume 24(4), 925-984
19. Brown, G.W. (1951): Iterative solution of games by fictitious play. In: Koopmans, T.C., ed., Activity Analysis of Production and Allocation, pp. 374-376. Wiley, New York
20. Brown, G.W., Koopmans, T.C. (1951): Computational suggestions for maximizing a linear function subject to linear inequalities. In: Koopmans, T.C., ed., Activity Analysis of Production and Allocation, pp. 377-380. Wiley, New York
21. Brown, G.W., von Neumann, J. (1950): Solutions of games by differential equations. In: Kuhn, H.W., Tucker, A.W., eds., Contributions to the Theory of Games, pp. 73-79. Princeton University Press, Princeton
22. Burrell, B.P., Todd, M.J. (1985): The ellipsoid method generates dual variables. Mathematics of Operations Research 10, 688-700
23. Chang, S.Y., Murty, K.G. (1989): The steepest descent gravitational method for linear programming. Discrete Applied Mathematics 25, 211-239
24. Charnes, A., Cooper, W.W. (1962): On some works of Kantorovich, Koopmans and others. Management Science 8, 246-263
25. Christof, T., Jünger, M., Reinelt, G. (1991): A complete description of the traveling salesman polytope on 8 nodes. Operations Research Letters 10, 497-500
26. Clarkson, K.L. (1995): Las Vegas algorithms for linear and integer programming when the dimension is small. Journal of the ACM 42, 488-499
27. Dantzig, G.B. (1951): Maximization of a linear function of variables subject to linear inequalities. In: Koopmans, T.C., ed., Activity Analysis of Production and Allocation, pp. 339-347. Wiley, New York
28. Dantzig, G.B. (1955): Upper bounds, secondary constraints, and block triangularity in linear programming. Econometrica 23, 174-183
29. Dantzig, G.B. (1963): Linear Programming and Extensions. Princeton University Press, Princeton, NJ
30. Dantzig, G.B. (1991): Linear programming. In: Lenstra, J.K., Rinnooy Kan, A.H.G., Schrijver, A., eds., History of Mathematical Programming, pp. 19-31. Elsevier Science Publishers
31. Dantzig, G.B., Wolfe, P. (1961): The decomposition algorithm for linear programming. Econometrica 29, 767-778
32. De Leone, R., Mangasarian, O.L. (1988): Serial and parallel solution of large scale linear programs by augmented Lagrangian successive overrelaxation. In: Kurzhanski, A., Neumann, K., Pallaschke, D., eds., Optimization, Parallel Processing and Applications, pp. 103-124. Berlin, Springer-Verlag. Lecture Notes in Economics and Mathematical Systems 304
33. Dikin, I.I. (1967): Iterative solution of problems of linear and quadratic programming (in Russian). Doklady Akademiia Nauk SSSR 174, 747-748.
34. English Translation: Soviet Mathematics Doklady, 1967, Volume 8, pp. 674-675
35. Dikin, I.I. (1974): On the speed of an iterative process (in Russian). Upravlaemye Sistemy 12, 54-60
36. Fiacco, A.V., McCormick, G.P. (1968): Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley Reprinted as Volume 4 of the SIAM Classics in Applied Mathematics Series, 1990
37. Forrest, J.J., Goldfarb, D. (1992): Steepest-edge simplex algorithms for linear programming. Mathematical Programming 57, 341-374
38. Frisch, R.A.K. (1955): The logarithmic potential method of convex programming. Technical report, University Institute of Economics, Oslo, Norway
39. Gärtner, B., Welzl, E. (1996): Linear programming - randomization and abstract frameworks. In: Proceedings of the 13th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 1064, pp. 669-687. Springer-Verlag, Berlin, Germany
40. Gay, D.M. (1985): Electronic mail distribution of linear programming test problems. Mathematical Programming Society Committee on Algorithms Newsletter 13, 10-12. Problems available from ftp://netlib2.cs.utk.edu/lp/data/
41. Gill, P.E., Murray, W., Saunders, M.A., Tomlin, J.A., Wright, M.H. (1986): On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method. Mathematical Programming 36, 183-209
42. Goemans, M.X. (1995): Worst-case comparison of valid inequalities for the TSP. Mathematical Programming 69, 335-349
43. Goffin, J.-L. (1980): The relaxation method for solving systems of linear inequalities. Mathematics of Operations Research 5, 388-414
44. Goffin, J.-L. (1984): Variable metric relaxation methods, Part II: The ellipsoid method. Mathematical Programming 30, 147-162
45. Gonzaga, C.C. (1992): Path following methods for linear programming. SIAM Review 34(2), 167-224
46. Grötschel, M., Lovasz, L., Schrijver, A. (1988): Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, Germany
47. Grunbaum, B. (1967): Convex Polytopes. Wiley, New York
48. Haimovich, M. (1983): The simplex method is very good! On the expected number of pivot steps and related properties of random linear programs. Technical report, Columbia University
49. Hoffman, A.J. (1991): Linear programming at the National Bureau of Standards. In: Lenstra, J.K., Rinnooy Kan, A.H.G., Schrijver, A., eds., History of Mathematical Programming, pp. 62-64. Elsevier Science Publishers
50. Hoffman, A.J., Mannos, M., Sokolowsky, D., Wiegmann, N. (1953): Computational experience in solving linear programs. Journal of SIAM 1, 17-33
51. How to make a CD disco ball. See http://www.ehow.com/eHow/eHow/0,1053, 4570,00. html
52. Image of a quartz crystal. See http://www.halcyon.com/nemain/gallery/ Quartz_43.html
53. Kalai, G. (1992): A subexponential randomized simplex algorithm. In: Proceedings of the 24th ACM Symposium on Theory of Computing, pp. 475-482
54. Kalai, G. (1997): Linear programming, the simplex algorithm and simple polytopes. Mathematical Programming 79, 217-233
55. Kalai, G., Kleitman, D.J. (1992): A quasi-polynomial bound for diameter of graphs of polyhedra. Bulletin of the American Mathematical Society 24, 315-316
56. Kantorovich, L.V. (1939): Mathematical methods in the organization and planning of production (in Russian). Publication House of the Leningrad State University 68 pp.
57. English Translation: Management Science, Volume 6 (1960), pp. 363-422
58. Karmarkar, N.K. (1984): A new polynomial-time algorithm for linear programming. Combinatorica 4, 373-395
59. Khachiyan, L.G. (1979): A polynomial algorithm in linear programming (in Russian). Doklady Akademiia Nauk SSSR 224, 1093-1096.
60. English Translation: Soviet Mathematics Doklady, Volume 20, 191-194
61. Khachiyan, L.G. (1980): Polynomial algorithms in linear programming (in Russian). Zhurnal Vychisitel'noi Matematiki i Matematicheskoi Fiziki, 20, 51-68.
62. E nglish Translation: U.S.S.R. Computational Mathematics and Mathematical Physics, Volume 20, 53-72
63. Klee, V., Kleinschmidt, P. (1987): The d-step conjecture and its relatives. Mathematics of Operations Research 12, 718-755
64. Klee, V., Minty, G.J. (1972): How good is the simplex algorithm? In: Shisha, O., ed., Inequalities, III, pp. 159-175. Academic Press
65. Klee, V., Walkup, D.W. (1967): The d-step conjecture for polyhedra of dimension d < 6. Acta Mathematica 133, 53-78
66. Kleinschmidt, P. (1994): The diameter of polytopes and related applications. In: Bisztriczky, T., McMullen, P., Schneider, R., Weiss, I.A., eds., Polytopes: Abstract, Convex, and Computational, pp. 467-492. Kluwer Academic Publishers, Dordrecht, The Netherlands
67. Kojima, M., Mizuno, S., Yoshise, A. (1991): An O(√n L)-iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming 50, 331-342
68. Koopmans, T.C., ed.. (1951): Activity Analysis of Production and Allocation. Wiley, New York
69. Kuhn, H.W. (1991): On the origin of the Hungarian method. In: Lenstra, J.K., Rinnooy Kan, A.H.G., Schrijver, A., eds., History of Mathematical Programming, pp. 77-81. Elsevier Science Publishers
70. Kuhn, H.W., Quandt, R.E. (1963): An experimental study of the simplex method. In: Metropolis, N.C. et al., ed., Experimental Arithmetic, High-Speed Computing and Mathematics, pp. 107-124. American Mathematical Society
71. Larman, D.G. (1970): Paths on polytopes. Proceedings of the London Mathematical Society 20, 161-178
72. Lawler, E.L. (1980): The great mathematical Sputnik of 1979. The Mathematical Intelligencer 2, 190-198
73. Lemke, C.E. (1954): The dual method of solving the linear programming problem. Naval Research Logistics Quarterly 1, 36-47
74. Lemke, C.E. (1961): The constrained gradient method of linear programming. Journal of SIAM 9, 1-17
75. Liao, A., Todd, M.J. (1996): Solving LP problems via weighted centers. SIAM Journal on Optimization 6, 933-960
76. Lustig, I.J., Marsten, R.E., Shanno, D.F. (1990): The primal-dual interior point method on the Cray supercomputer. In: Coleman, T.F., Li, Y., eds., Large-Scale Numerical Optimization, pp. 70-80. SIAM
77. Matousek, J., Sharir, M., Welzl, E. (1996): A subexponential bound for linear programming. Algorithmica 16, 498-516
78. Megiddo, N. (1984): Linear programming in linear time when the dimension is fixed. Journal of the Association for Computing Machinery 31(1), 114-127
79. Megiddo, N. (1987): On the complexity of linear programming. In: Bewley, T., ed., Advances in Economic Theory, pp. 225-268. Cambridge University Press, Cambridge
80. Motzkin, T.S., Schoenberg, I.J. (1954): The relaxation method for linear inequalities. Canadian Journal of Mathematics 6, 393-404
81. Naddef, D. (1989): The Hirsch conjecture is true for (0,1)-polytopes. Mathematical Programming 45, 109-110
82. Nesterov, Y.E., Nemirovski, A.S. (1993): Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms. SIAM Publications. SIAM, Philadelphia, USA
83. Padberg, M.W., Rao, M.R. (1974): The traveling salesman problem and a class of polyhedra of diameter two. Mathematical Programming 7, 32-45
84. Renegar, J. (1988): A polynomial-time algorithm based on Newton's method for linear programming. Mathematical Programming 40, 59-93
85. Robinson, J. (1951): An iterative method of solving a game. Annals of Mathematics 54, 296-301
86. Rosen, J.B. (1961): The gradient projection method for nonlinear programming. Part I: Linear constraints. Journal of SIAM 8, 181-217
87. Schrijver, A. (1986): Theory of Linear and Integer Programming. John Wiley, Chichester
88. Shor, N.Z. (1977): Cut-off method with space extension in convex programming problems (in Russian). Kibernetika 13, 94-95.
89. English translation: Cybernetics 13(1), 94-96
90. Smale, S. (1983): On the average speed of the simplex method in linear programming. Mathematical Programming 27, 241-262
91. Tanabe, K. (1988): Centered Newton method for mathematical programming. In: Iri, M., Yajima, K., eds., System Modelling and Optimization: Proceedings of the 13th IFIP Conference, Tokyo, Japan, Aug./Sept. 1987, pp. 197-206. Springer-Verlag, Berlin, Germany, Volume 113 of Lecture Notes in Control and Information Sciences
92. Todd, M.J. (1980): The monotonic bounded Hirsch conjecture is false for dimension at least four. Mathematics of Operations Research 5, 599-601
93. Todd, M.J. (1986): Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems. Mathematical Programming 35, 173-192
94. Todd, M.J. (1988): Polynomial algorithms for linear programming. In: Eiselt, H.A., Pederzoli, G., eds., Advances in Optimization and Control, pp. 49-66. Springer-Verlag, Berlin
95. Todd, M.J. (1991): Probabilistic models for linear programming. Mathematics of Operations Research 16, 671-693. Erratum: see Mathematics of Operations Research 23 (1998), 767-768
96. Todd, M.J. (1997): Potential-reduction methods in mathematical programming. Mathematical Programming 76, 3-45
97. Todd, M.J., Ye, Y. (1990): A centered projective algorithm for linear programming. Mathematics of Operations Research 15(3), 508-529
98. Todd, M.J., Ye, Y. (1996): A lower bound on the number of iterations of long-step and polynomial interior-point methods for linear programming. Annals of Operations Research 62, 233-252
99. von Neumann, J. (1954): A numerical method to determine optimal strategy. Naval Research Logistics Quarterly 1, 109-115
100. von Neumann. J. (1963): Discussion of a maximum problem. In: Taub, A.H., ed., John von Neumann: Collected Works, Volume VI, pp. 89-95. Pergamon Press, Oxford
101. Wolfe, P., Cutler, L. (1963): Experiments in linear programming. In: Graves, R.L., Wolfe, P., eds., Recent Advances in Mathematical Programming, pp. 177-200. McGraw-Hill, New York
102. Wright, S. (1996): Primal-dual Interior Point Methods. SIAM, Philadelphia
103. 3 L) potential reduction algorithm for linear programming. Mathematical Programming 50(2), 239-258
104. Yudin, D.B., Nemirovski, A.S. (1976): Informational complexity and efficient methods for the solution of convex extremal problems (in Russian). Ékonomika i Matematicheskie metody 12, 357-369.
105. English translation: Matekon 13(2), 3-25
106. Ziegler, G. (1995): Lectures on Polytopes. Springer-Verlag, Berlin
107. Zoutendijk, G. (1960): Methods of Feasible Directions. Elsevier, Amsterdam
108. Amenta, N., Ziegler, G. (1999): Deformed products and maximal shadows of polytopes. In: Chazelle, B., Goodman, J.E., Pollack, R., eds., Advances in Discrete and Computational Geometry, pp. 57-90. American Mathematical Society