On the facet pivot simplex method for linear programming II: a linear iteration bound

TL;DR

This paper introduces a facet pivot method for linear programming that guarantees solving the problem within at most n-d facet pivot iterations, addressing worst-case complexity inspired by the disproof of the Hirsch Conjecture.

Contribution

It proposes a new facet pivot algorithm that achieves a linear iteration bound for solving linear programming problems in the worst case.

Findings

Facet pivot method solves LP in at most n-d iterations

The method addresses worst-case complexity bounds

Inspired by the disproof of the Hirsch Conjecture

Abstract

The Hirsch Conjecture stated that any $d$-dimensional polytope with n facets has a diameter at most equal to $n - d$. This conjecture was disproved by Santos (A counterexample to the Hirsch Conjecture, Annals of Mathematics, 172(1) 383-412, 2012). The implication of Santos' work is that all {\it vertex} pivot algorithms cannot solve the linear programming problem in the worst case in $n - d$ vertex pivot iterations. In the first part of this series of papers, we proposed a {\it facet} pivot method. In this paper, we show that the proposed facet pivot method can solve the canonical linear programming problem in the worst case in at most $n-d$ facet pivot iterations. This work was inspired by Smale's Problem 9 (Mathematical problems for the next century, In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. Mathematics: frontiers and perspectives, American Mathematical Society, 271-294, 1999).