A bound for the number of different basic solutions generated by the simplex method

• Published in: Mathematical programming Vol. 137; no. 1-2; pp. 579 - 586
• Main Authors: Kitahara, Tomonari, Mizuno, Shinji
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2013; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Decision analysis; Linear programming; Markov processes; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Polynomials; Short Communication; Simplex method; Studies; Theoretical; Upper bounds; Variables; Vectors (mathematics); Simplex method; Linear programming; Basic feasible solutions; 90C05; 90C49; Number of iterations; Strong polynomiality; 90C08
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-011-0482-y

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 (LP) 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 problem is primal nondegenerate, it becomes a bound for the number of iterations. The result includes strong polynomiality for Markov Decision Problem by Ye ( http://www.stanford.edu/~yyye/simplexmdp1.pdf , 2010 ) and utilize its analysis. We also apply our result to an LP whose constraint matrix is totally unimodular and a constant vector b of constraints is integral.