Nash Equilibrium Seeking Algorithm Design for Distributed Nonsmooth Multicluster Games Over Weight-Balanced Digraphs

In this article, we study the multicluster games over weight-balanced digraphs, where the cost functions of all players are nonsmooth. Besides, in the problem, not only are the decisions of all players constrained by heterogeneous local constraints but also the decisions of players in the same cluster are constrained by coupling constraints. Due to the nonsmooth cost functions, the coupling constraints, the general local convex constraints, and the weight-balanced digraphs, existing Nash equilibrium seeking algorithms cannot solve our problem. In order to seek the Nash equilibrium of the game, we design a distributed algorithm based on subgradient descent, differential inclusions, and projection operations. In the algorithm, a distributed learning strategy is embedded for the players to estimate the decisions of other players. Moreover, we analyze the asymptotical convergence of the algorithm via set-valued LaSalle invariance principle. Finally, a numerical simulation about electricity market games is presented to illustrate the effectiveness of our result.