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

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.

💡 Research Summary

This paper establishes input‑to‑state stability (ISS) for perturbed Newton‑type methods applied to generalized equations that arise from Nash equilibrium (NE) and generalized Nash equilibrium (GNE) problems, and demonstrates how this property enables fast solution tracking in dynamic systems such as model predictive control (MPC).

NE Problems
The authors first formulate NE problems as a generalized equation f(z)+F(z)∋0, where f is a continuously Fréchet‑differentiable mapping and F is a normal‑cone set‑valued mapping. The classical Josephy‑Newton method linearizes f and retains the set‑valued term, yielding the iteration (4). Existing convergence analyses rely on strong regularity or strong subregularity of f+F. By leveraging variational inequality (VI) stability theory, the paper shows that a much weaker condition—strict semicopositivity of the game Hessian H on the critical cone C(a*;A,F)—is sufficient to guarantee local uniqueness (isolatedness) and stability of a Nash equilibrium a*. Under this condition, the Josephy‑Newton iteration converges Q‑quadratically to a* (Theorem 4).

ISS for Perturbed Josephy‑Newton
A perturbed version of the method is introduced, allowing inexact gradient evaluations or residual errors, modeled by a disturbance sequence v_k. Assuming bounded disturbances and Lipschitz continuity of f and its Jacobian at (a*,0), the perturbed iteration is shown to be locally ISS (Theorem 5). Specifically, the error dynamics satisfy ‖a_{k+1}−a*‖ ≤ L_a‖a_k−a*‖ + L_v‖v_k‖, and a refined bound with a quadratic term is given (Corollary 1). This result provides a quantitative link between computational inaccuracy and tracking error, which is essential for real‑time implementations.

Distributed Josephy‑Newton
The paper further develops a distributed scheme where each agent updates only its own decision variable using local cost, gradient, and Hessian information, while receiving other agents’ previous iterates. By solving a proximal‑response subproblem via a Newton step, the method achieves Q‑quadratic convergence to the proximal response, and, under monotonicity of the pseudogradient F and a sufficiently large proximal parameter τ, the collective proximal responses converge to a Nash equilibrium (Theorem 6). The ISS property extends to this distributed setting by the same analysis as in the centralized case.

GNE Problems
For GNE problems, where agents’ feasible sets depend on the decisions of others, the KKT conditions lead to a quasi‑variational inequality (QVI) that cannot be reduced to a standard VI. The authors reformulate the KKT system as a nonsmooth equation using complementarity functions and apply a semismooth Newton method. Under a quasi‑regularity (nonsingularity) condition, they prove ISS and Q‑quadratic local convergence for the perturbed semismooth Newton scheme. This extends the ISS framework to GNE settings without requiring strong regularity.

Application to Constrained Game‑theoretic MPC (CG‑MPC)
The theoretical developments are applied to CG‑MPC, where at each sampling instant a constrained NE or GNE problem must be solved. To reduce computational burden, a time‑distributed optimization (TDO) strategy is employed: Newton iterations are spread over successive sampling periods, so the optimizer evolves in parallel with the plant. Because the Newton solver is ISS, the plant‑optimizer closed loop inherits stability, robustness, and constraint satisfaction properties. The authors prove boundedness of the tracking error between the true optimal control trajectory and the trajectory generated by the TDO‑based CG‑MPC. Numerical simulations illustrate the effectiveness of the approach, showing reduced computation time and satisfactory tracking performance compared with conventional MPC solvers.

Key Contributions

1. Demonstrates that strict semicopositivity of the game Hessian, a weaker condition than strong regularity, suffices for local uniqueness, stability, and ISS of Newton methods for NE problems.
2. Extends ISS analysis to perturbed semismooth Newton methods for GNE problems under a quasi‑regularity condition.
3. Provides distributed Josephy‑Newton algorithms with provable convergence and ISS.
4. Integrates the ISS‑based Newton solvers into CG‑MPC via time‑distributed optimization, establishing closed‑loop stability and bounded tracking error.

Overall, the paper bridges the gap between advanced Newton‑type equilibrium‑seeking algorithms and real‑time control applications, offering a rigorous stability foundation that permits inexact computations and distributed implementations while guaranteeing performance in dynamic environments.