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A Bound for the Number of Different Basic Solutions Generated by the Simplex Method

Tomonari Kitahara (kitahara.t.ab***at***m.titech.ac.jp)
Shinji Mizuno (mizuno.s.ab***at***m.titech.ac.jp)

Abstract: In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions. When the primal problem is nondegenerate, it becomes a bound for the number of iterations. We show some basic results when it is applied to special linear programming problems. The results include strongly polynomiality of the simplex method for Markov Decision Problem by Ye and utilize its analysis.

Keywords: Simplex method, Linear programming, Iteration bound,

Category 1: Linear, Cone and Semidefinite Programming (Linear Programming )

Citation: To appear in Mathematical Programming

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Entry Submitted: 01/19/2011
Entry Accepted: 01/20/2011
Entry Last Modified: 07/25/2011

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