On the facet pivot simplex method for linear programming

TL;DR

This paper introduces a new facet pivot simplex method for linear programming, showing promising numerical results and potential for polynomial-time solutions, with an available Matlab implementation.

Contribution

The paper proposes a novel facet pivot simplex method that outperforms traditional vertex pivot methods and may lead to polynomial-time algorithms for LP.

Findings

Numerical testing shows the new method is very promising.

Matlab implementation and benchmark tests are provided.

The method offers hope for polynomial pivot algorithms in LP.

Abstract

Dantzig's vertex pivot simplex method has been published for more than seven decades. Amazingly, it remains one of the most efficient methods to solve linear programming (LP) problem after numerous efforts trying to find some better methods. In this paper, we propose a facet pivot simplex method and demonstrate by numerical testing that the new method is very promising compared to the vertex pivot method. Since there is no polynomial pivot simplex algorithm for linear programming problems after many decades of effort, we hope that this new type of pivot algorithm will give us some hope to find a polynomial pivot simplex method for linear programming problems. A Matlab implementation of the facet pivot algorithm and Netlib benchmark test problems are available in Matlab file exchange website.