Backing PDHG into a Corner

TL;DR

This paper explores enhancing the PDHG algorithm to better approximate optimal solutions in linear programming, aiming to improve the efficiency of the crossover step and overall solution quality.

Contribution

It proposes a method to modify PDHG iterations to push solutions into optimal LP corners, reducing crossover step runtime.

Findings

PDHG can be guided into LP corners for better initial solutions.

Enhanced PDHG reduces crossover step time significantly.

Method improves overall efficiency of large LP problem solving.

Abstract

Recent enhancements to the Primal-Dual Hybrid Gradient (PDHG) algorithm have enabled GPUs to efficiently solve large linear programming problems, often faster than the long-dominant simplex and interior-point methods. The solutions found by PDHG are typically of much lower quality than those found by the alternatives, which can be remedied by following the PDHG iterations with a crossover step to obtain an accurate optimal basic solution. However, the cost of this highly sequential crossover step can be quite significant. This paper examines whether PDHG iterations can be enhanced to push the solution into a corner of the optimal LP face, thereby providing crossover a better starting point and hopefully reducing its runtime.