Restarted Halpern PDHG for Linear Programming

TL;DR

This paper introduces a restarted Halpern primal-dual hybrid gradient algorithm for linear programming that achieves accelerated convergence rates, improves complexity bounds, and is effectively implemented in a GPU-based solver with superior experimental performance.

Contribution

The paper presents a novel matrix-free primal-dual algorithm with accelerated convergence for LP, including infeasible cases, and extends it with a reflection operation for additional speedup.

Findings

Achieves optimal accelerated linear convergence on feasible LP.

Demonstrates accelerated identification and convergence without relying on the Hoffman constant.

Shows improved numerical performance in GPU-based LP solver on MIPLIB instances.

Abstract

In this paper, we propose and analyze a new matrix-free primal-dual algorithm, called restarted Halpern primal-dual hybrid gradient (rHPDHG), for solving linear programming (LP). We show that rHPDHG can achieve optimal accelerated linear convergence on feasible and bounded LP. Furthermore, we present a refined analysis that demonstrates an accelerated two-stage convergence of rHPDHG over the vanilla PDHG with an improved complexity for identification and an accelerated eventual linear convergence that does not depend on the conservative global Hoffman constant. Regarding infeasible LP, we show that rHPDHG can recover infeasibility certificates with an accelerated linear rate, improving the previous convergence rates. Furthermore, we discuss an extension of rHPDHG by adding reflection operation (which is dubbed as $\mathrm{r^2HPDHG}$), and demonstrate that it shares all theoretical guarantees of rHPDHG with an additional factor of 2 speedup in the complexity bound. Lastly, we build up a GPU-based LP solver using rHPDHG/$\mathrm{r^2HPDHG}$, and the experiments on 383 MIPLIB instances showcase an improved numerical performance compared cuPDLP.jl.