AI 中文总结
针对动态博弈中广义纳什均衡求解，提出惯性修正牛顿法，通过KKT矩阵惯性验证二阶充分条件，并引入修正步提升收敛性，实现带最优性检查的快速求解器，经数值与实物实验验证。
AI 中文摘要
牛顿法通过求解KKT必要条件，能高效地找到动态博弈中的广义纳什均衡（GNE）。这些方法速度快，可支持高度动态机器人的多智能体模型预测控制（MPC）。然而，仅凭较小的KKT残差并不能证明返回的解满足局部GNE的二阶充分条件。本文提出一种高效的数值方法，用于验证局部GNE的二阶充分条件（SOSC）。我们将智能体KKT矩阵的惯性与其约化Hessian在约束零空间上的正定性联系起来。此外，我们引入一种惯性修正的更新步，通过去稳定具有弱跨智能体耦合的严格鞍点，提高向局部GNE的收敛性。我们的主要贡献是一个用于约束动态博弈的快速牛顿求解器，该求解器提供高效的最优性检查。通过数值基准测试，我们展示了该求解器在实际多智能体规划问题中的运行时间和收敛性能。我们还利用微型自动驾驶赛车平台进行了物理实验，验证了该求解器的实时能力。
英文摘要
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.
