Constrained multi-cluster game: Distributed Nash equilibrium seeking over directed graphs

TL;DR

This paper introduces a distributed algorithm for finding Nash equilibria in multi-cluster games with constraints over directed graphs, combining consensus, gradient tracking, and averaging to ensure convergence.

Contribution

It presents a novel gradient-based method that handles constraints and directed communication, achieving linear convergence in multi-cluster networked games.

Findings

Algorithm converges linearly under certain conditions

Effective handling of constraints via averaging procedure

Successful application demonstrated in microgrid energy management

Abstract

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.