  Motivated by the complex dynamics of cooperative and competitive interactions
within networked agent systems, multi-cluster games provide a framework for
modeling the interconnected goals of self-interested clusters of agents. For
this setup, the existing literature lacks comprehensive gradient-based
solutions that simultaneously consider constraint sets and directed
communication networks, both of which are crucial for many practical
applications. To address this gap, this paper proposes a distributed Nash
equilibrium seeking algorithm that integrates consensus-based methods and
gradient-tracking techniques, where inter-cluster and intra-cluster
communications only use row- and column-stochastic weight matrices,
respectively. To handle constraints, we introduce an averaging procedure, which
can effectively address the complications associated with projections. In turn,
we can show linear convergence of our algorithm, focusing on the contraction
property of the optimality gap. We demonstrate the efficacy of the proposed
algorithm through a microgrid energy management application.
