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Continuous-Time Online Distributed Seeking for Generalized Nash Equilibrium of Nonmonotone Online Game

Jianing Chen, Sichen Qian, Chuangyin Dang, Sitian Qin

TL;DR

This work addresses distributed GNE seeking for online nonmonotone games with time-varying coupling constraints by introducing a continuous-time algorithm with a time-varying control gain, achieving a constant regret bound $\mathcal{R}^{\top}=O(1)$ and a sublinear fit bound $\mathcal{F}^{\top}=O(\sqrt{T})$. It tackles communication costs with a dynamic event-triggered mechanism, while preserving the same performance guarantees and ensuring no Zeno behavior. The approach combines a passivity-based estimation of other players’ actions with projection-based updates, and is validated on a five-player numerical example showing robust performance and reduced communication, underscoring its practicality for continuous-time, online, nonmonotone games.

Abstract

This paper mainly investigates a class of distributed generalized Nash equilibrium (GNE) seeking problems for online nonmonotone game with time-varying coupling inequality constraints. Based on a time-varying control gain, a novel continuous-time distributed GNE seeking algorithm is proposed, which realizes the constant regret bound and sublinear fit bound, matching those of the criteria for online optimization problems. Furthermore, to reduce unnecessary communication among players, a dynamic event-triggered mechanism involving internal variables is introduced into the distributed GNE seeking algorithm, while the constant regret bound and sublinear fit bound are still achieved. Also, the Zeno behavior is strictly prohibited. Finally, a numerical example is given to demonstrate the validity of the theoretical results.

Continuous-Time Online Distributed Seeking for Generalized Nash Equilibrium of Nonmonotone Online Game

TL;DR

This work addresses distributed GNE seeking for online nonmonotone games with time-varying coupling constraints by introducing a continuous-time algorithm with a time-varying control gain, achieving a constant regret bound and a sublinear fit bound . It tackles communication costs with a dynamic event-triggered mechanism, while preserving the same performance guarantees and ensuring no Zeno behavior. The approach combines a passivity-based estimation of other players’ actions with projection-based updates, and is validated on a five-player numerical example showing robust performance and reduced communication, underscoring its practicality for continuous-time, online, nonmonotone games.

Abstract

This paper mainly investigates a class of distributed generalized Nash equilibrium (GNE) seeking problems for online nonmonotone game with time-varying coupling inequality constraints. Based on a time-varying control gain, a novel continuous-time distributed GNE seeking algorithm is proposed, which realizes the constant regret bound and sublinear fit bound, matching those of the criteria for online optimization problems. Furthermore, to reduce unnecessary communication among players, a dynamic event-triggered mechanism involving internal variables is introduced into the distributed GNE seeking algorithm, while the constant regret bound and sublinear fit bound are still achieved. Also, the Zeno behavior is strictly prohibited. Finally, a numerical example is given to demonstrate the validity of the theoretical results.
Paper Structure (12 sections, 5 theorems, 51 equations, 6 figures, 1 table)

This paper contains 12 sections, 5 theorems, 51 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Description
  3. Graph Theory
  4. Operator Theory
  5. Online Noncooperative Game
  6. Performance Metrics
  7. Main Results
  8. Continuous-time distributed GNE seeking algorithm
  9. GNE seeking algorithm with dynamic event-triggered mechanism
  10. Numerical Example
  11. Conclusion
  12. References

Key Result

Lemma II.1

guo2020predefined For a connected undirected graph $\mathcal{G}$, $\mathcal{L}$ is positive semidefinite, whose eigenvalues have order as $\lambda_1=0<\lambda_2 \leq \cdots \leq \lambda_N$. Also, $\mathcal{L} \mathbf{1}_N=\mathbf{0}_N$ and $\mathbf{1}_N^{\top} \mathcal{L}=\mathbf{0}_N^{\top}$.

Figures (6)

  • Figure 1: The communication topology among five players.
  • Figure 2: Evolutions of $\mathcal{R}^\top$ and $\frac{\mathcal{F}^\top}{\sqrt{T}}$ based on algorithm \ref{['algo1']}.
  • Figure 3: Evolutions of $\mathcal{R}^\top$ and $\frac{\mathcal{F}^\top}{\sqrt{T}}$ based on algorithm \ref{['algo2']} with dynamic event-triggered mechanism \ref{['triggerc']}.
  • Figure 4: Evolutions of $\mathcal{F}^\top$ based on algorithms \ref{['algo1']} (Left) and \ref{['algo2']} (Right).
  • Figure 5: The event time sequence for $\hat{\Upsilon}^i$.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Lemma II.1
  • Definition II.1
  • Definition II.2
  • Lemma II.2
  • Definition II.3
  • Remark II.1
  • Theorem III.1
  • proof
  • Remark III.1
  • Theorem III.2
  • ...and 3 more