Continuous-time penalty methods for Nash equilibrium seeking of a nonsmooth generalized noncooperative game

Abstract

In this article, we propose centralized and distributed continuous-time penalty methods to find a Nash equilibrium for a generalized noncooperative game with shared inequality and equality constraints and private inequality constraints that depend on the player itself. By using the ℓ1 penalty function, we prove that the equilibrium of a differential inclusion is a normalized Nash equilibrium of the original generalized noncooperative game, and the centralized differential inclusion exponentially converges to the unique normalized Nash equilibrium of a strongly monotone game. Suppose that the players can communicate with their neighboring players only and the communication topology can be represented by a connected undirected graph. Based on a leader-following consensus scheme and singular perturbation techniques, we propose distributed algorithms by using the exact ℓ1 penalty function and the continuously differentiable squared ℓ2 penalty function, respectively. The squared ℓ2 penalty function method works for games with smooth constraints and the exact ℓ1 penalty function works for certain scenarios. The proposed two distributed algorithms converge to an η-neighborhood of the unique normalized Nash equilibrium and an -neighborhood of an approximated Nash equilibrium, respectively, with being a positive constant. For each 0 and each initial condition, there exists an such that for each 0, the convergence can be guaranteed where is a parameter in the algorithm.