Solving Linear Programs with Fast Online Learning Algorithms

TL;DR

This paper introduces fast, matrix-multiplication-free first-order algorithms for approximate linear programming, leveraging online learning techniques and variable duplication to improve efficiency and integration with large-scale LP solvers.

Contribution

It adapts online linear programming algorithms for offline problems, introduces a variable-duplication technique, and integrates online methods into column generation schemes for large-scale LPs.

Findings

Algorithms avoid matrix multiplication, increasing speed.

Variable duplication reduces optimality gap and constraint violation.

Methods serve as approximate solvers or initialization routines.

Abstract

This paper presents fast first-order methods for solving linear programs (LPs) approximately. We adapt online linear programming algorithms to offline LPs and obtain algorithms that avoid any matrix multiplication. We also introduce a variable-duplication technique that copies each variable $K$ times and reduces the optimality gap and constraint violation by a factor of $\sqrt{K}$. Furthermore, we show how online algorithms can be effectively integrated into sifting, a column generation scheme for large-scale LPs. Numerical experiments demonstrate that our methods can serve as either an approximate direct solver, or an initialization subroutine for exact LP solving.