Dual-Inertial Viscosity-Based Subgradient Extragradient Methods for Equilibrium Problems Over Fixed Point Sets

Abstract

This paper introduces a new algorithmic framework for solving pseudomonotone equilibrium problems and demicontractive fixed-point problems. Unlike conventional methods that incorporate a single inertial step, our approach employs dual inertia to accelerate convergence while preserving stability. The proposed method combines the viscosity approximation technique with the extragradient method to guarantee strong convergence. Initially, the extragradient method is applied under a Lipschitz continuity assumption on the equilibrium bifunction. Subsequently, this condition is relaxed by adopting a self-adaptive step size strategy, allowing variable step sizes to be updated iteratively based on the current iterates information. Notably, the algorithm operates without requiring prior knowledge of the Lipschitz constants or any line search procedures. Strong convergence is established under mild condition, and its applicability to variational inequality problems is demonstrated. Numerical experiments validate the effectiveness of the proposed approach, showcasing its capability to handle large-scale and complex problems efficiently while outperforming traditional single-inertia methods.