A double-pivot simplex algorithm and its upper bounds of the iteration numbers

TL;DR

This paper introduces a double-pivot simplex algorithm with new upper bounds on iteration numbers, demonstrating strong polynomial time solutions for certain LP problems and superior performance over Dantzig's method on large Klee-Minty instances.

Contribution

The paper proposes a novel double-pivot simplex method with tight iteration bounds, enabling efficient solutions for specific LP problems and outperforming traditional methods on large-scale tests.

Findings

The double-pivot method solves certain LPs in strongly polynomial time.

The iteration bounds are shown to be very tight using a variant of Klee-Minty cube.

Numerical tests demonstrate superior performance over Dantzig's simplex on large problems.

Abstract

In this paper, a double-pivot simplex method is proposed. Two upper bounds of iteration numbers are derived. Applying one of the bounds to some special linear programming (LP) problems, such as LP with a totally unimodular matrix and Markov Decision Problem (MDP) with a fixed discount rate, indicates that the double-pivot simplex method solves these problems in a strongly polynomial time. A variant of Klee-Minty cube is used to show that the estimated bounds of the iteration numbers are very tight. Numerical test on three variants of Klee-Minty cubes is performed for the problems with sizes as big as $200$ constraints and $400$ variables. Dantzig's simplex method cannot handle Klee-Minty cube problem with $200$ constraints because it needs about $2^{200} \approx 10^{60}$ iterations. But the proposed algorithm performs extremely good for all three variants.