Bounds for the number of basic feasible solutions generated by the simplex method with the largest distance rule

TL;DR

This paper derives upper bounds on the number of basic feasible solutions generated by the simplex method using the largest distance rule, extending existing analytical frameworks without requiring nondegeneracy assumptions.

Contribution

It extends the analytical framework for bounding simplex method solutions to the largest distance rule, involving a geometric parameter and removing nondegeneracy constraints.

Findings

Derived upper bounds involving a geometric parameter β

Extended existing bounds to the largest distance rule

Analysis does not require nondegeneracy assumption

Abstract

In this paper, we analyze the simplex method with the largest distance rule and derive upper bounds on the number of different basic feasible solutions generated. The pivoting rule was proposed by Pan [10], and in some cases, it was reported to be more efficient than the renowned steepest edge rule. We show that the analytical framework developed by Kitahara and Mizuno can be extended to this rule, despite its structural differences from previously studied pivoting rules. The resulting bounds involve a geometric parameter $\beta$ determined by the column norms of the constraint matrix. In addition, our analysis does not require a nondegeneracy assumption.