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Prescribed-Time Distributed Generalized Nash Equilibrium Seeking

Liraz Mudrik, Isaac Kaminer, Sean Kragelund, Abram H. Clark

Abstract

This paper proposes the first fully distributed algorithm for finding the Generalized Nash Equilibrium (GNE) of a non-cooperative game with shared coupling constraints and general cost coupling at a user-prescribed finite time T. As a foundation, a centralized gradient-based prescribed-time convergence result is established for the GNE problem, extending the optimization Lyapunov function framework to gradient dynamics, the only known realization among existing alternatives that naturally decomposes into per-agent computations. Building on this, a fully distributed architecture is designed in which each agent concurrently runs three coupled dynamics: a prescribed-time distributed state observer, a gradient-based optimization law, and a dual consensus mechanism that enforces the shared-multiplier requirement of the variational GNE, thus guaranteeing convergence to the same solution as the centralized case. The simultaneous operation of these layers creates bidirectional perturbations between consensus and optimization, which are resolved through gain synchronization that matches the temporal singularities of the optimization and consensus layers, ensuring all error components vanish exactly at T. The Fischer-Burmeister reformulation renders the algorithm projection-free and guarantees constraint satisfaction at the deadline. Numerical simulations on a Nash-Cournot game and a time-critical sensor coverage problem validate the approach.

Prescribed-Time Distributed Generalized Nash Equilibrium Seeking

Abstract

This paper proposes the first fully distributed algorithm for finding the Generalized Nash Equilibrium (GNE) of a non-cooperative game with shared coupling constraints and general cost coupling at a user-prescribed finite time T. As a foundation, a centralized gradient-based prescribed-time convergence result is established for the GNE problem, extending the optimization Lyapunov function framework to gradient dynamics, the only known realization among existing alternatives that naturally decomposes into per-agent computations. Building on this, a fully distributed architecture is designed in which each agent concurrently runs three coupled dynamics: a prescribed-time distributed state observer, a gradient-based optimization law, and a dual consensus mechanism that enforces the shared-multiplier requirement of the variational GNE, thus guaranteeing convergence to the same solution as the centralized case. The simultaneous operation of these layers creates bidirectional perturbations between consensus and optimization, which are resolved through gain synchronization that matches the temporal singularities of the optimization and consensus layers, ensuring all error components vanish exactly at T. The Fischer-Burmeister reformulation renders the algorithm projection-free and guarantees constraint satisfaction at the deadline. Numerical simulations on a Nash-Cournot game and a time-critical sensor coverage problem validate the approach.
Paper Structure (22 sections, 4 theorems, 56 equations, 5 figures, 1 table)

This paper contains 22 sections, 4 theorems, 56 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Centralized Gradient-Based GNE Seeking
  4. Stationarity Mapping and the Optimization Lyapunov Function
  5. Sublevel Set Compactness
  6. Centralized Prescribed-Time Convergence
  7. Distributed Control Architecture
  8. Prescribed-Time Distributed State Observer
  9. Optimization Feedback Law with Dual Consensus
  10. Error Coordinates and Closed-Loop Dynamics
  11. Distributed Convergence Analysis
  12. Main Convergence Result
  13. Proof of Theorem \ref{['thm:main']}
  14. Numerical Simulations
  15. Generalized Nash-Cournot Competition
  16. ...and 7 more sections

Key Result

Proposition III.1

Under Assumptions ass:monotonicity--ass:slater, define the compactness threshold Then, $c^* > 0$ and for all $0 < c < c^*$, the sublevel set is compact. Moreover, if $\mathbf{g}(\mathbf{x}) = \mathbf{C}\mathbf{x} - \mathbf{d}$ where $\mathbf{C} \in \mathbb{R}^{p \times n}$ has full row rank, then $c^* = +\infty$ and $\Omega_c$ is compact for all $c > 0$.

Figures (5)

  • Figure 1: Communication topology: A tree graph with $N=20$ agents. This minimal connectivity represents a challenging scenario for distributed consensus.
  • Figure 2: Trajectories of decision variables $x_i(t)$ for $N=20$ agents. All agents converge to their optimal production levels exactly at $T=10$ s.
  • Figure 3: Stationarity residual $\|\mathbf{S}(\mathbf{z}(t))\|$ over time. The residual decays to zero, indicating that the KKT conditions for the v-GNE are satisfied at $T$.
  • Figure 4: Convergence of the Stationarity vector for the Sensor Network. Note the vertical drop below $\approx 10^{-8}$ at exactly $T=0.5$ s, validating the prescribed-time property.
  • Figure 5: Sensor Deployment at $t=T$. The sensors are distributed along the boundary of the active power constraint, balancing target tracking (dotted lines).

Theorems & Definitions (7)

  • Remark II.6: Special constraint structures
  • Proposition III.1: Sublevel Set Compactness
  • Lemma III.2: Non-Singularity of $\nabla\mathbf{S}$
  • proof
  • Proposition III.3: Centralized Prescribed-Time GNEPs
  • Remark IV.1: Dual scaling factor
  • Theorem V.1: Prescribed-Time Convergence to v-GNE