[Paper Review] Generalized Nash Equilibrium Seeking Algorithm Design for Distributed Constrained Multi-Cluster Games

Zhenhua Deng, Yan Zhao|arXiv (Cornell University)|Jun 24, 2021

Distributed Control Multi-Agent Systems39 references4 citations

TL;DR

This paper proposes a distributed gradient-based algorithm to seek the variational generalized Nash equilibrium (GNE) in distributed constrained multi-cluster games, where players cooperate within clusters and compete across clusters under coupled, local, and set constraints. The algorithm ensures asymptotic convergence to the GNE using projection methods and Lyapunov stability analysis, validated through simulations.

ABSTRACT

The multi-cluster games are addressed in this paper, where all players team up with the players in the cluster that they belong to, and compete against the players in other clusters to minimize the cost function of their own cluster. The decision of every player is constrained by coupling inequality constraints, local inequality constraints and local convex set constraints. Our problem extends well-known noncooperative game problems and resource allocation problems by considering the competition between clusters and the cooperation within clusters at the same time. Besides, without involving the resource allocation within clusters, the noncooperative game between clusters, and the aforementioned constraints, existing game algorithms as well as resource allocation algorithms cannot solve the problem. In order to seek the variational generalized Nash equilibrium (GNE) of the multi-cluster games, we design a distributed algorithm via gradient descent and projections. Moreover, we analyze the convergence of the algorithm with the help of variational analysis and Lyapunov stability theory. Under the algorithm, all players asymptotically converge to the variational GNE of the multi-cluster game. Simulation examples are presented to verify the effectiveness of the algorithm.

Motivation & Objective

• Address the gap in existing game and resource allocation algorithms that fail to handle both inter-cluster competition and intra-cluster cooperation under complex constraints.
• Model a distributed multi-cluster game where players minimize their cluster’s cost function subject to coupling, local inequality, and convex set constraints.
• Design a distributed algorithm that enables players to autonomously converge to the variational GNE without centralized coordination.
• Establish theoretical convergence guarantees for the proposed algorithm using variational analysis and Lyapunov stability theory.

Proposed method

• Formulate the multi-cluster game as a generalized Nash equilibrium problem with coupling constraints across clusters and local constraints per player.
• Design a distributed algorithm using gradient descent updates for cost function minimization and projection operations to enforce constraint satisfaction.
• Integrate projection operators to handle local inequality and convex set constraints at each player’s decision variable.
• Apply variational analysis to characterize the variational GNE as a solution to a variational inequality problem.
• Use Lyapunov stability theory to prove that all players asymptotically converge to the variational GNE under the proposed algorithm.
• Ensure distributed implementation by allowing each player to update based only on local information and local communication with cluster members.

Experimental results

Research questions

• RQ1 How can a distributed algorithm be designed to compute the variational GNE in multi-cluster games with coupled, local, and set constraints?
• RQ2 What conditions ensure the asymptotic convergence of players to the variational GNE in such a constrained multi-cluster setting?
• RQ3 In what way does the proposed algorithm outperform existing noncooperative game and resource allocation algorithms in handling inter-cluster competition and intra-cluster cooperation?
• RQ4 How can the convergence of the algorithm be rigorously proven using variational analysis and Lyapunov stability theory?
• RQ5 What role do projections and gradient updates play in maintaining constraint feasibility and driving convergence?

Key findings

• The proposed distributed algorithm ensures that all players asymptotically converge to the variational generalized Nash equilibrium of the multi-cluster game.
• The convergence is established through rigorous analysis using variational inequality theory and Lyapunov stability, confirming global convergence under the algorithm.
• The algorithm effectively handles the coexistence of inter-cluster competition and intra-cluster cooperation, which existing methods fail to address.
• The method maintains feasibility of all constraints—coupled, local inequality, and convex set constraints—through projection-based updates.
• Simulation results demonstrate the effectiveness of the algorithm in achieving equilibrium under various constraint configurations.
• The algorithm operates without requiring centralized coordination, enabling scalable and robust deployment in distributed systems.