A Douglas-Rachford Splitting Method for Solving Equilibrium Problems

TL;DR

This paper introduces a Douglas-Rachford splitting algorithm for equilibrium problems involving bifunctions, establishing equivalence with monotone operator zeros and analyzing their connections.

Contribution

It develops a novel splitting method for equilibrium problems, linking them to monotone operator theory and extending solution techniques.

Findings

Proposed a new splitting algorithm for equilibrium problems.

Proved the equivalence between equilibrium problems and monotone operator zeros.

Analyzed the connections between monotone inclusions and equilibrium problems.

Abstract

We propose a splitting method for solving an equilibrium problem involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of two appropriate maximally monotone operators. Our algorithm is a consequence of the Douglas--Rachford splitting applied to this auxiliary monotone inclusion. Connections between monotone inclusions and equilibrium problems are studied.