Newton Methods in Generalized Nash Equilibrium Problems with Applications to Game-Theoretic Model Predictive Control

Abstract

We prove input-to-state stability (ISS) of perturbed Newton-type methods for generalized equations arising from Nash equilibrium (NE) and generalized NE (GNE) problems. This ISS property allows the use of inexact computations in equilibrium-seeking to enable fast solution tracking in dynamic systems such as in model predictive control (MPC). For NE problems, we address the local convergence of perturbed Josephy-Newton methods from the variational inequality (VI) stability analysis, and establish the ISS result under less restrictive regularity conditions compared to the existing results established for nonlinear optimization. Agent-distributed algorithms are also developed. For GNE problems, since they cannot be reduced to VI problems in general, we use semismooth Newton methods to solve the semismooth equations arising from the Karush-Kuhn-Tucker (KKT) systems of the GNE problem and establish the ISS result under a quasi-regularity condition. To illustrate the use of the ISS in dynamic systems, applications to constrained game-theoretic MPC (CG-MPC) are studied with time-distributed solution-tracking for real-time implementation. Boundness of tracking errors is proven. Numerical examples are reported.