aiqbench analysis
This paper presents a novel numerical method for verifying second-order sufficient conditions in dynamic games, addressing a critical gap where standard Newton solvers fail to certify local optimality despite low residuals. By integrating inertia correction into the update step, the approach accelerates convergence and effectively destabilizes strict saddle points in multi-agent systems with weak coupling. The technique is validated through both numerical benchmarks and physical experiments using autonomous race cars, demonstrating its viability for real-time Model Predictive Control applications.
Inertia-Corrected Newton Method For Generalized Nash Equilibria in Dynamic Games with Optimality Verification
arXiv:2609.31929v1 Announce Type: new 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.
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.
Robotics (cs.RO); Computer Science and Game Theory (cs.GT)
arXiv:2609.31929 [cs.RO]
(or arXiv:2609.31929v1 [cs.RO] for this version)
https://doi.org/10.48550/arXiv.2609.31929 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 278 words
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