Inertia-Corrected Newton Method For Generalized Nash Equilibria in Dynamic Games with Optimality Verification

2026-09-25

AI Summary

This study addresses the limitations of Newton's method in dynamic games, specifically its inability to verify second-order sufficient conditions and its susceptibility to weakly coupled saddle points. To overcome these issues, this work proposes an inertia-corrected Newton method that establishes an equivalence between the inertia of the KKT matrix and the positive definiteness of the projected Hessian. By introducing an inertia correction step, the proposed approach effectively disrupts saddle-point structures, thereby rigorously verifying the optimality of local Nash equilibria and significantly enhancing convergence performance in multi-agent planning. The research is formulated within the frameworks of constrained dynamic games and model predictive control. Benchmark evaluations and micro-car racing experiments validate the algorithm's real-time solving capability and superior convergence properties.

Abstract

Newton methods efficiently find Generalized Nash Equilibria (GNE) in dynamic games by solving for the KKT necessary conditions. These methods are fast and can support multi-agent Model Predictive Control (MPC) for highly dynamic robots. However, a small KKT residual alone does not certify that the returned solution satisfies the second-order sufficient conditions for a local GNE. In this paper, we propose an efficient numerical method to verify the second-order sufficient conditions (SOSC) for a local GNE. We connect the inertia of the agent KKT matrix with the positive definiteness of the reduced Hessian of the cost function, projected onto the null space of the constraints. Furthermore, we introduce an inertia-corrected update step that improves convergence to local GNEs by destabilizing strict saddle points with weak cross-agent coupling. Our main contribution is a fast Newton solver for Constrained Dynamic Games that provides efficient optimality checking. Through numerical benchmarks, we demonstrate the proposed solver's runtime and convergence performance in practical multi-agent planning problems. We also validate the solver's real-time capabilities in physical experiments using a platform of miniature autonomous race cars.

Problem

• Generalized Nash Equilibria
• Dynamic Games
• Second-Order Sufficient Conditions
• Optimality Verification
• Saddle Points

Innovation

• Generalized Nash Equilibria
• Inertia-Corrected Newton Method
• Second-Order Sufficient Conditions
• Dynamic Games
• Model Predictive Control