A New Crossover Algorithm for LP Inspired by the Spiral Dynamic of PDHG

TL;DR

This paper reveals a spiral geometric pattern in PDHG for LP, and introduces a novel crossover algorithm inspired by this pattern to efficiently find vertex solutions, demonstrated through numerical experiments.

Contribution

The paper provides a geometric analysis of PDHG for LP and proposes a new crossover algorithm based on the spiral behavior, differing from traditional methods.

Findings

PDHG iterates exhibit a spiral pattern with a closed-form solution.

The spiral pattern includes rotation and forward movement components.

Numerical experiments show the effectiveness of the new crossover algorithm.

Abstract

Motivated by large-scale applications, there is a recent trend of research on using first-order methods for solving LP. Among them, PDLP, which is based on a primal-dual hybrid gradient (PDHG) algorithm, may be the most promising one. In this paper, we present a geometric viewpoint on the behavior of PDHG for LP. We demonstrate that PDHG iterates exhibit a spiral pattern with a closed-form solution when the variable basis remains unchanged. This spiral pattern consists of two orthogonal components: rotation and forward movement, where rotation improves primal and dual feasibility, while forward movement advances the duality gap. We also characterize the different situations in which basis change events occur. Inspired by the spiral behavior of PDHG, we design a new crossover algorithm to obtain a vertex solution from any optimal LP solution. This approach differs from traditional simplex-based crossover methods. Our numerical experiments demonstrate the effectiveness of the proposed algorithm, showcasing its potential as an alternative option for crossover.