Solving Multiparametric Generalized Nash Equilibrium Problems and Explicit Game-Theoretic Model Predictive Control

Abstract

We present a method for computing explicit solutions to parametric generalized Nash equilibrium (GNE) problems with convex quadratic cost functions and linear coupling and local constraints. Assuming that the parameters enter only the linear terms of the cost functions and the constraint right-hand sides, we provide the exact multiparametric solution of the GNE problem. Such a solution enables: (i) minimal real-time computation; (ii) inherent interpretability and explainability, as well as exact enumeration of all multiple equilibria; (iii) selection of desired GNE solution types in the case of infinitely many equilibria; and (iv) zero-shot updates of the GNE solution in response to changes in constraint right-hand sides and/or linear costs. In line with explicit model predictive control (MPC) approaches, we apply our method to solve game-theoretic MPC problems, also known as receding horizon games, explicitly. We compare its performance against centralized solvers in a battery charging game and a toy two-mass-spring-damper system control problem. A Python implementation of the algorithms presented in this paper is available at https://github.com/bemporad/nash_mpqp.