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Distributed Generalized Nash Equilibria Seeking Algorithms Involving Synchronous and Asynchronous Schemes

Huaqing Li, Liang Ran, Lifeng Zheng, Zhe Li, Jinhui Hu, Jun Li, Tingwen Huang

TL;DR

This work addresses generalized Nash equilibria in noncooperative games with coupled inequality constraints by casting the problem as a variational inequality (VI) and introducing an edge-based local equilibrium condition. It develops a distributed primal-dual proximal DPDP_GNE algorithm and an asynchronous variant ASY_DPDP_GNE that operate without a central coordinator, using proximal updates on edge-variables, dual variables, and primal decisions. Theoretical results establish a $\gamma$-averaged operator framework for DPDP_GNE with a sublinear $o(1/k)$ convergence rate, and prove convergence in expectation and almost surely for ASY_DPDP_GNE under explicit step-size and delay bounds, with improved rates over prior work. Numerical experiments on a networked Cournot competition demonstrate fast convergence and robustness to communication delays, highlighting the practical impact of edge-based dual consensus for distributed GNE computation.

Abstract

This paper considers a class of noncooperative games in which the feasible decision sets of all players are coupled together by a coupled inequality constraint. Adopting the variational inequality formulation of the game, we first introduce a new local edge-based equilibrium condition and develop a distributed primal-dual proximal algorithm with full information. Considering challenges when communication delays occur, we devise an asynchronous distributed algorithm to seek a generalized Nash equilibrium. This asynchronous scheme arbitrarily activates one player to start new computations independently at different iteration instants, which means that the picked player can use the involved out-dated information from itself and its neighbors to perform new updates. A distinctive attribute is that the proposed algorithms enable the derivation of new distributed forward-backward-like extensions. In theoretical aspect, we provide explicit conditions on algorithm parameters, for instance, the step-sizes to establish a sublinear convergence rate for the proposed synchronous algorithm. Moreover, the asynchronous algorithm guarantees almost sure convergence in expectation under the same step-size conditions and some standard assumptions. An interesting observation is that our analysis approach improves the convergence rate of prior synchronous distributed forward-backward-based algorithms. Finally, the viability and performance of the proposed algorithms are demonstrated by numerical studies on the networked Cournot competition.

Distributed Generalized Nash Equilibria Seeking Algorithms Involving Synchronous and Asynchronous Schemes

TL;DR

This work addresses generalized Nash equilibria in noncooperative games with coupled inequality constraints by casting the problem as a variational inequality (VI) and introducing an edge-based local equilibrium condition. It develops a distributed primal-dual proximal DPDP_GNE algorithm and an asynchronous variant ASY_DPDP_GNE that operate without a central coordinator, using proximal updates on edge-variables, dual variables, and primal decisions. Theoretical results establish a -averaged operator framework for DPDP_GNE with a sublinear convergence rate, and prove convergence in expectation and almost surely for ASY_DPDP_GNE under explicit step-size and delay bounds, with improved rates over prior work. Numerical experiments on a networked Cournot competition demonstrate fast convergence and robustness to communication delays, highlighting the practical impact of edge-based dual consensus for distributed GNE computation.

Abstract

This paper considers a class of noncooperative games in which the feasible decision sets of all players are coupled together by a coupled inequality constraint. Adopting the variational inequality formulation of the game, we first introduce a new local edge-based equilibrium condition and develop a distributed primal-dual proximal algorithm with full information. Considering challenges when communication delays occur, we devise an asynchronous distributed algorithm to seek a generalized Nash equilibrium. This asynchronous scheme arbitrarily activates one player to start new computations independently at different iteration instants, which means that the picked player can use the involved out-dated information from itself and its neighbors to perform new updates. A distinctive attribute is that the proposed algorithms enable the derivation of new distributed forward-backward-like extensions. In theoretical aspect, we provide explicit conditions on algorithm parameters, for instance, the step-sizes to establish a sublinear convergence rate for the proposed synchronous algorithm. Moreover, the asynchronous algorithm guarantees almost sure convergence in expectation under the same step-size conditions and some standard assumptions. An interesting observation is that our analysis approach improves the convergence rate of prior synchronous distributed forward-backward-based algorithms. Finally, the viability and performance of the proposed algorithms are demonstrated by numerical studies on the networked Cournot competition.
Paper Structure (23 sections, 9 theorems, 54 equations, 2 figures, 2 algorithms)

This paper contains 23 sections, 9 theorems, 54 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Contribution Statements
  4. Organization
  5. Notations
  6. Game Formulation and Development
  7. Practical Scenarios
  8. Variational inequality formulation
  9. Local Equilibrium Condition
  10. Distributed Algorithms For Seeking GNE
  11. Synchronous Algorithm
  12. Asynchronous Algorithm
  13. Convergence Results
  14. Convergence of DPDP_GNE
  15. Convergence of ASY_DPDP_GNE
  16. ...and 8 more sections

Key Result

Lemma 1

Francisco2007generalized Let Assumption AssumSetconvex hold. Every solution ${x^ * }$ of VI is also a GNE of the game GENProblem, i.e., ${x^ * } \in {\rm{SOL}}\left( {\mathcal{X},F} \right) \Rightarrow {x^ * } \in {\rm{GNE}}\left({{\Xi}} \right)$.

Figures (2)

  • Figure 1: Comparison of different considerations.
  • Figure 2: Performance comparison with state-of-the-arts.

Theorems & Definitions (33)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Lemma 2
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 23 more