Peaceman-Rachford Splitting Method Converges Ergodically for Solving Convex Optimization Problems

TL;DR

This paper proves the convergence of the ergodic sequence generated by the Peaceman-Rachford splitting method with semi-proximal terms for convex optimization problems and demonstrates its superior performance over Douglas-Rachford methods in large-scale cases.

Contribution

It establishes the ergodic convergence of the PR splitting method with semi-proximal terms and shows its practical advantages through numerical experiments.

Findings

Ergodic sequence of PR method converges for convex problems.

Restarted ergodic PR outperforms DR in large-scale benchmarks.

PR method with semi-proximal terms is more effective for large-scale COPs.

Abstract

In this paper, we prove that the ergodic sequence generated by the Peaceman-Rachford (PR) splitting method with semi-proximal terms converges for convex optimization problems (COPs). Numerical experiments on the linear programming benchmark dataset further demonstrate that, with a restart strategy, the ergodic sequence of the PR splitting method with semi-proximal terms consistently outperforms both the point-wise and ergodic sequences of the Douglas-Rachford (DR) splitting method. These findings indicate that the restarted ergodic PR splitting method is a more effective choice for tackling large-scale COPs compared to its DR counterparts.