Online distributed seeking for first-order Nash equilibria of nonconvex noncooperative games with multiple clusters

In this paper, we study the problem of online distributed seeking for first-order Nash equilibria of nonconvex noncooperative games with multiple clusters. In this game, each cluster is composed of multiple players, whose goals are to cooperatively minimize the sum of time-varying cost functions in their own cluster. Each player can only have access to its own cost function and action set, and can only communicate with its immediate neighbors in the same cluster through a time-varying graph. Moreover, cost functions cannot be known by players in advance. Unlike existing studies on noncooperative games, the cost function we consider is nonconvex. To address this challenge, an online distributed dual averaging algorithm is proposed. Interestingly enough, the performance of the algorithm is measured by regrets involving the first-order Nash equilibrium condition, and the sublinearity of the regrets is achieved under the proposed algorithm. The validity of the results is illustrated by a numerical example.