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

• 成果类型: Article
• 署名作者: Sun, Chao; Hu, Guoqiang
• 署名单位: Nanyang Technological University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2020.3040377
• 发表日期: 2021
• 页码: 4895-4902
• 关键词: Multiagent system; Nash equilibrium seeking; nonsmooth analysis; penalty methods

摘要

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 l(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 l(1) penalty function and the continuously differentiable squared l(2) penalty function, respectively. The squared l(2) penalty function method works for games with smooth constraints and the exact l(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 epsilon* such that for each 0 < epsilon < epsilon*, the convergence can be guaranteed where e is a parameter in the algorithm.