Accelerated Halpern Iteration with Projection and Contraction Methods for Solving Bi-Level Split Pseudomonotone Variational Inequalities in Hilbert Spaces

Abstract

The fixed point problems of nonlinear operators encompass a wide range of optimization and nonlinear analysis challenges, serving as powerful tools for addressing real-world issues in networks, transportation, ecology, image processing, recovery, and reconstruction. Motivated by these applications, we propose an accelerated Halpern-type iteration combined with projection and contraction methods to solve a specific bi-level split pseudomonotone variational inequality problem (BSVIP) involving fixed point problems in Hilbert spaces. Our approach redefines the BSVIP by incorporating non-empty fixed point sets of pseudocontraction mappings. The main objective is to find a common solution to the pseudomonotone variational inequality and the fixed point problem of the pseudocontraction mapping using the resolvent of the pseudocontraction mapping. We establish the strong convergence of the iterative sequence generated by the proposed algorithm under mild conditions. Numerical experiments demonstrate the efficiency and applicability of our method.