Modified golden ratio algorithms for solving equilibrium problems

TL;DR

This paper introduces a novel explicit proximal-like algorithm for equilibrium problems with pseudomonotone bifunctions, achieving convergence without requiring Lipschitz constants and demonstrating an R-linear rate under strong pseudomonotonicity.

Contribution

The paper presents a new algorithm that does not need Lipschitz constants and proves its convergence and rate, advancing equilibrium problem-solving methods.

Findings

Algorithm converges without explicit Lipschitz constants.

Achieves R-linear convergence under strong pseudomonotonicity.

Numerical results show competitive performance.

Abstract

In this paper an explicit algorithm is proposed for solving an equilibrium problem whose associated bifunction is pseudomonotone and satisfies a Lipschitz-type condition. Contrary to many algorithms, our algorithm is done without using explicitly the Lipschitz constants of bifunction although its convergence is obtained under such that condition. The introduced method is a form of proximal-like method whose steplengths are explicitly generated at each iteration without using any linesearch procedure. First we prove the convergence of the algorithm, and after we establish its $R$-linear rate of convergence under the assumption of strong pseudomonotonicity of the bifunction. Afterwards several numerical results are displayed to illustrate and to compare the behavior of the new algorithm with other ones.