A globally convergent improved BFGS method for generalized Nash equilibrium problems

Abstract

In this article, we consider a class of Generalized Nash Equilibrium Problems (GNEPs) and solve it using one of the most effective quasi-Newton algorithms: the BFGS method. The considered GNEP is a player-convex GNEP. As the Armijo-type line search techniques are cost-effective in finding a step length, compared to Wolfe-type line search techniques, we use the Armijo–Goldstein line search technique in an improved BFGS method to solve GNEPs. In the BFGS method, the main drawback of using Armijo-type line search techniques is that it does not inherit the positive definiteness property of the generated Hessian approximation matrices. Therefore, we tactfully update approximate Hessian matrices so that the updated BFGS-matrices inherit the positive definiteness property. Accordingly, we prove its global convergence in the GNEP framework. The numerical performance of the proposed method is exhibited on three commonly used GNEPs and on two internet-switching GNEPs. © The Author(s), under exclusive licence to Sociedad Española de Matemática Aplicada 2023.