Efficient Methods for Solving Equilibrium and Fixed-Point Problems Using Accelerated Viscosity-Based Approximations

Abstract

In this article, we proposed enhanced extragradient methods for solving equilibrium problems in Hilbert spaces. The constraint set is defined as the solution set for a fixed-point problem with demicontractive mappings. The proposed methodologies center on a subgradient extragradient approach that combines an inertial technique, a self-adaptive procedure, and a viscosity approximation scheme. These algorithms use variable step sizes that are dynamically updated in accordance with previous iteration outcomes. A key advantage of the proposed methods is their ability to function eliminating the need for prior knowledge of Lipschitz constants or line search techniques. Instead, the step sizes are determined through simple calculations at each iteration. The convergence analysis is established under relaxed assumptions. Furthermore, we present a numerical study that compares the effectiveness of the proposed techniques to existing approaches.