Modified inertial subgradient extragradient with auxiliary parameters and parallel viscosity algorithm for minimization problem induced by bounded linear operator over common solution of fixed points of nonexpansive mappings and pseudomonotone equilibrium problems

Abstract

This paper introduces a modification to the inertial subgradient extragradient algorithm by incorporating auxiliary parameters for updating, along with dynamic regularization coefficient, including the parallel viscosity algorithm. The aim is to find an element in the common solution set of fixed points in a finite family of nonexpansive mappings and Lipschitz-type continuous pseudomonotone equilibrium problems. This element also serves as the unique solution to a minimization problem induced by a bounded linear operator and contraction mapping in the context of a real Hilbert space. The efficiency of the proposed algorithm is influenced by the introduced auxiliary parameters, which are intended to leverage the value of the considered objective bifunction at each iteration, along with the advantages of the designed regularization coefficient, which is self-adaptive and utilizes a straightforward rule for automatic updates. The update rule avoids enforcing monotonic behavior on the dynamic regularization coefficient and does not require prior knowledge of the Lipschitz constants of the bifunction. This flexibility increases the algorithm's applicability for solving a wider range of practical problems. The discussions on the numerical experiments for Nash-Cournot models and image restoration problems are also provided to illustrate the computational effectiveness of the introduced algorithm.