In this work, we study stationarity conditions and constraint qualifications (CQs) tailored to Generalized Nash Equilibrium Problems (GNEPs) and analyze their relationships and implications for the global convergence of algorithms. We recall that GNEPs generalize Nash Equilibrium Problems (NEPs) in that the feasible strategy set of each player depends on the strategies chosen by the other players, thereby introducing additional difficulties in both theoretical analysis and algorithm design.
Our stationary concepts provide a theoretical framework for analyzing the global convergence of several numerical methods. In particular, we establish new convergence results for the safeguarded augmented Lagrangian method and propose a new adaptation of the Hyperbolic Augmented Lagrangian Algorithm (HALA) tailored to GNEPs.
More specifically, we investigate the convergence properties of these methods with respect to both feasibility and optimality of the limit points. Furthermore, we prove global convergence to a Karush–Kuhn–Tucker (KKT) point under a weak CQ and establish boundedness of the associated multipliers under a strong quasinormality-type CQ adapted to GNEPs.
Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed methods.
Keywords: Stationary conditions; convergence; Hyperbolic augmented Lagrangian; generalized Nash equilibrium; numerical experiments