Distributed Nash Equilibrium Seeking with Linear Convergence Rate for Constrained Multi-Cluster Aggregative Games over Digraphs

This paper studies the distributed Nash equilibrium (NE) seeking problem for constrained multi-cluster aggregative games over digraphs, where the closed convex set constraints are involved. In the considered game, the players within the same cluster collaboratively minimize the collective cost within locally closed convex sets regardless of the interests of other clusters, where the local objective functions depend not only on local decisions but also on a global aggregative variable. A distributed algorithm is designed to seek the NE based on distributed aggregate estimation and gradient tracking. Specially, the eigenvector learning method is employed to address the information asymmetry caused by unbalanced digraphs. Moreover, the method of feasible direction is incorporated into the proposed algorithm to handle the involved closed convex set constraints. Under certain conditions, it is proved that the proposed algorithm achieves a linear convergence rate. Finally, the proposed algorithm is applied in a numerical simulation associated with the Energy Internet System to verify its effectiveness.