Continuous-Time Penalty Methods for Nash Equilibrium Seeking of a Nonsmooth Generalized Noncooperative Game

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 the₁penalty 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 exact₁penalty function and the continuously differentiable squared₂penalty function, respectively. The squared₂penalty function method works for games with smooth constraints and the exact₁penalty function works for certain scenarios. The proposed two distributed algorithms converge to anη-neighborhood of the unique normalized Nash equilibrium and anη-neighborhood of an approximated Nash equilibrium, respectively, withηbeing a positive constant. For eachη >0and each initial condition, there exists anε ^*such that for each0< ε < ε ^*, the convergence can be guaranteed whereεis a parameter in the algorithm.