A symmetric Gauss-Seidel based alternating proximal ALM for generalized Nash Equilibrium problems in Banach spaces

Abstract

In this paper, we study a class of monotone generalized Nash equilibrium problems (GNEPs) with jointly linear constraints. The players' strategy spaces are real Hilbert spaces, while the joint constraint is formulated in a Banach space. To solve such problems, we propose a novel symmetric Gauss-Seidel (sGS) based alternating proximal augmented Lagrangian method (sGS-APALM) which incorporates newly designed quadratic surrogates. In contrast to existing regularization and ALM-type methods, the proposed method avoids solving coupled Nash equilibrium subproblems at each iteration and instead updates the players' strategies alternately by solving a sequence of unconstrained quadratic programs. Moreover, unlike many existing splitting-based methods, our global convergence analysis and convergence rate estimation require only monotonicity and Lipschitz continuity of the pseudo-gradient mapping, without imposing stronger assumptions such as strong monotonicity or co-coercivity. Finally, we apply the method to a class of risk-neutral PDE-constrained GNEPs with joint state constraints, and preliminary numerical results demonstrate its efficiency and effectiveness.