ADMM-Type Methods for Generalized Nash Equilibrium Problems in Hilbert Spaces

Abstract

We consider the generalized Nash equilibrium problem (GNEP) with $ N $ players in a Hilbert space setting. The joint constraints are eliminated by an augmented Lagrangian-type approach, leading to an ADMM (alternating direction method of multipliers) algorithm. In contrast to standard optimization problems, however, the direct extension of ADMM to GNEPs is not necessarily convergent even for $N = 2$ players. We therefore use a regularized version of ADMM and present a global convergence result for $ N \geq 2 $ players under a partial strong monotonicity and a partial Lipschitz condition. Furthermore, also different from the optimization context, it turns out that the corresponding regularization parameters have to be sufficiently large in order to guarantee global convergence. We therefore also discuss a second ADMM-type method with an adaptive choice of the regularization parameters, with the aim of keeping the regularization parameters smaller and, hence, getting faster convergence. Numerical results are presented for some examples arising in infinite-dimensional Hilbert spaces.