On the number of solutions generated by the simplex method for LP

Tomonari Kitahara, Shinji Mizuno

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We obtain upper bounds for the number of distinct solutions generated by the simplex method for linear programming (LP). One of the upper bounds is polynomial in the number of variables, the number of constraints, and the ratio of the maximum to the minimum positive components in all the basic feasible solutions. We show that they are good upper bounds for some special LP problems including those on 0-1 polytopes, those with totally unimodular matrices, and the Markov decision problems. We also show that the upper bounds are almost tight by using an LP instance on a 0-1 polytope and a simple variant of the Klee-Minty example.

Original languageEnglish
Title of host publicationOptimization and Control Techniques and Applications
EditorsYi Zhang, Honglei Xu, Kok Lay Teo, Honglei Xu
PublisherSpringer New York LLC
Pages75-90
Number of pages16
Volume86
ISBN (Electronic)9783662434031
DOIs
Publication statusPublished - Jan 1 2014
Externally publishedYes

Fingerprint

Simplex Method
Number of Solutions
Linear programming
Upper bound
Totally Unimodular Matrix
Polytopes
Polytope
Decision problem
Distinct
Polynomial

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Kitahara, T., & Mizuno, S. (2014). On the number of solutions generated by the simplex method for LP. In Y. Zhang, H. Xu, K. L. Teo, & H. Xu (Eds.), Optimization and Control Techniques and Applications (Vol. 86, pp. 75-90). Springer New York LLC. https://doi.org/10.1007/978-3-662-43404-8_4

On the number of solutions generated by the simplex method for LP. / Kitahara, Tomonari; Mizuno, Shinji.

Optimization and Control Techniques and Applications. ed. / Yi Zhang; Honglei Xu; Kok Lay Teo; Honglei Xu. Vol. 86 Springer New York LLC, 2014. p. 75-90.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Kitahara, T & Mizuno, S 2014, On the number of solutions generated by the simplex method for LP. in Y Zhang, H Xu, KL Teo & H Xu (eds), Optimization and Control Techniques and Applications. vol. 86, Springer New York LLC, pp. 75-90. https://doi.org/10.1007/978-3-662-43404-8_4
Kitahara T, Mizuno S. On the number of solutions generated by the simplex method for LP. In Zhang Y, Xu H, Teo KL, Xu H, editors, Optimization and Control Techniques and Applications. Vol. 86. Springer New York LLC. 2014. p. 75-90 https://doi.org/10.1007/978-3-662-43404-8_4
Kitahara, Tomonari ; Mizuno, Shinji. / On the number of solutions generated by the simplex method for LP. Optimization and Control Techniques and Applications. editor / Yi Zhang ; Honglei Xu ; Kok Lay Teo ; Honglei Xu. Vol. 86 Springer New York LLC, 2014. pp. 75-90
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