A reflected forward-backward splitting algorithmic framework

TL;DR

This paper introduces a unified reflected forward-backward splitting framework for solving monotone operator problems, with convergence guarantees and demonstrated effectiveness through numerical experiments.

Contribution

It presents a novel, unified algorithmic framework for monotone operators, simplifying convergence analysis and proposing heuristic strategies validated by experiments.

Findings

Unified convergence analysis under mild conditions

Effective heuristic strategies for matrix selection

Demonstrated success on regularized saddle-point problems

Abstract

In this paper, we propose a reflected forward-backward splitting algorithic framework for finding a zero of the sum of finitely many monotone op-erators, including maximally monotone operators, cocoercive operators, and monotone and Lipschitz continuous operators. We provide a unified convergence analysis under mild conditions, eliminating the need to analyze the convergence of each algorithm individually. The heuristic strategies for matrix selections are proposed through a numerical experiment, based on which a new algorithm is derived. A further numerical experiment on the regularized saddle-point problem is then presented to demonstrate the effectiveness of the proposed algorithm.