The computation of approximate feedback Stackelberg equilibria in multi-player nonlinear constrained dynamic games

Solving feedback Stackelberg games with nonlinear dynamics and coupled constraints, a common scenario in practice, presents significant challenges. This work introduces an efficient method for computing local feedback Stackelberg policies in multi-player general-sum dynamic games, with continuous state and action spaces. Different from existing (approximate) dynamic programming solutions that are primarily designed for unconstrained problems, our approach involves reformulating a feedback Stackelberg dynamic game into a sequence of nested optimization problems, enabling the derivation of Karush-Kuhn-Tucker (KKT) conditions and the establishment of a second-order sufficient condition for local feedback Stackelberg policies. We propose a Newton-style primal-dual interior point method for solving constrained linear quadratic (LQ) feedback Stackelberg games, offering provable convergence guarantees. Our method is further extended to compute local feedback Stackelberg policies for more general nonlinear games by iteratively approximating them using LQ games, ensuring that their KKT conditions are locally aligned with those of the original nonlinear games. We prove the exponential convergence of our algorithm in constrained nonlinear games. In a feedback Stackelberg game with nonlinear dynamics and (nonconvex) coupled costs and constraints, our experimental results reveal the algorithm's ability to handle infeasible initial conditions and achieve exponential convergence towards an approximate local feedback Stackelberg equilibrium.