Multi-Cluster Aggregative Games: A Linearly Convergent Nash Equilibrium Seeking Algorithm and Its Applications in Energy Management

We propose a class of non-cooperative games, termed multi-cluster aggregative games. In this framework, clusters serve as non-cooperative players, with each cluster comprising collaborative agents whose cost functions depend on both their individual decisions and the aggregate decisions of all clusters. This game model is motivated by decision-making problems in competitive-cooperative networked systems with a large number of participants, such as the Energy Internet. To address challenges in seeking Nash equilibrium for such networked systems, we develop a distributed algorithm under a hierarchical communication scheme which is hybrid with semi-decentralized and distributed protocols. The cluster aggregate decisions are acquired through a semi-decentralized structure, whereas the estimations of averaged cluster gradients and the aggregate decisions for other clusters are obtained by distributed structures. In particular, the algorithm employs an aggregate estimator instead of an all-decision estimator. Under strongly monotone and Lipschitz continuous assumptions, we prove that the algorithm linearly converges to a Nash equilibrium with a fixed step size. We present the applications in the context of the Energy Internet, and the numerical results verify the effectiveness of the algorithm.