Two-Timescale Distributed Approach to Nash Equilibrium Seeking of Euler–Lagrange Systems in Multicluster Nonconvex Games

In this article, we investigate a distributed generalized Nash equilibrium (GNE) seeking problem for multicluster games, where each cluster consists of multiple players. Different from most of the existing multicluster game studies, the multicluster games considered nonconvex cost functions, nonconvex (coupled) inequality constraints, and nonaffine (coupled) equality constraints. Each player in a cluster communicates and cooperates with other players based on a directed connected network to make optimal actions that minimize the cost function of their cluster. We propose a two-timescale multiagent system (MAS) for multicluster nonconvex games. We further propose a two-timescale MAS in which players are formulated by Euler–Lagrange dynamics. We prove the convergence of the MAS with derived conditions to one of the local GNEs. We elaborate on a numerical example, a price-bidding problem in an electricity market, and a Nash–Cournot game to illustrate the characteristics of the proposed approach.