A generalization of the steepest-edge rule and its number of simplex iterations for a nondegenerate LP

TL;DR

This paper introduces a generalized $p$-norm rule as a pivoting strategy for the simplex method, providing upper bounds on iteration counts for nondegenerate LPs based on problem parameters.

Contribution

It proposes the $p$-norm rule as a new pivoting rule and derives iteration upper bounds for the simplex method using this rule and the steepest-edge rule.

Findings

Upper bounds depend on variables, constraints, and basic feasible solutions.

The $p$-norm rule generalizes the steepest-edge rule.

Bounds are applicable to nondegenerate LPs.

Abstract

In this paper, we propose a $p$-norm rule, which is a generalization of the steepest-edge rule, as a pivoting rule for the simplex method. For a nondegenerate linear programming problem, we show upper bounds for the number of iterations of the simplex method with the steepest-edge and $p$-norm rules. One of the upper bounds is given by a function of the number of variables, that of constraints, and the minimum and maximum positive elements in all basic feasible solutions.