Continuous-Time Penalty Methods for Nash Equilibrium Seeking of a Nonsmooth Generalized Noncooperative Game
Chao Sun, Guoqiang Hu · IEEE Transactions on Automatic Control · 2020
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$\ell _{1}$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$\ell _{1}$penalty function and the continuously differentiable squared$\ell _{2}$penalty function, respectively. The squared$\ell _{2}$penalty function method works for games with smooth constraints and the exact$\ell _{1}$penalty function works for certain scenarios. The proposed two distributed algorithms converge to an$\eta$-neighborhood of the unique normalized Nash equilibrium and an$\eta$-neighborhood of an approximated Nash equilibrium, respectively, with$\eta$being a positive constant. For each$\eta >0$and each initial condition, there exists an$\varepsilon ^*$such that for each$0< \varepsilon < \varepsilon ^*$, the convergence can be guaranteed where$\varepsilon$is a parameter in the algorithm.