Proximal Subgradient Algorithm for a Class of Nonconvex Bilevel Equilibrium Problems

In this paper, we propose an algorithm for a bilevel problem of solving a monotone equilibrium problem over the solution set of a mixed equilibrium problem involving prox-convex functions in finite dimensional Euclidean space Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R<^>n$$\end{document}. The proposed algorithm is based on the proximal method for mixed variational inequalities by using proximal operators of prox-convex functions. The convergence of the sequences generated by the proposed algorithm is established. Furthermore, some consequences of the main result are given. Finally, we provide numerical examples to illustrate our algorithm;s convergence and compare it with others. As an application, we apply the proposed algorithm to solve a modified oligopolistic Nash-Cournot equilibrium model.