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Tracking-based distributed equilibrium seeking for aggregative games

Guido Carnevale, Filippo Fabiani, Filiberto Fele, Kostas Margellos, Giuseppe Notarstefano

TL;DR

The paper addresses the challenge of computing exact (generalized) Nash equilibria in aggregative games over networks with partial information. It introduces two fully distributed TRADES schemes: Primal TRADES for problems with local constraints and Primal-Dual TRADES for problems with affine coupling constraints, each employing tracking mechanisms to estimate the aggregative variable and, in the constrained case, augmented primal-dual updates. Using a singular perturbation framework, the authors prove linear (global exponential) convergence to the (G)NE under suitable step-size and network conditions, without requiring compactness of local sets or global information. Numerical experiments on demand-response-like scenarios corroborate the theoretical results, showing favorable convergence behavior and competitive efficiency compared to state-of-the-art distributed methods. The work provides scalable, distributed tools for networked decision-making in smart grids and related applications, with a clear pathway for extensions to time-varying graphs and richer constraint structures.

Abstract

We propose fully-distributed algorithms for Nash equilibrium seeking in aggregative games over networks. We first consider the case where local constraints are present and we design an algorithm combining, for each agent, (i) the projected pseudo-gradient descent and (ii) a tracking mechanism to locally reconstruct the aggregative variable. To handle coupling constraints arising in generalized settings, we propose another distributed algorithm based on (i) a recently emerged augmented primal-dual scheme and (ii) two tracking mechanisms to reconstruct, for each agent, both the aggregative variable and the coupling constraint satisfaction. Leveraging tools from singular perturbations analysis, we prove linear convergence to the Nash equilibrium for both schemes. Finally, we run extensive numerical simulations to confirm the effectiveness of our methods and compare them with state-of-the-art distributed equilibrium-seeking algorithms.

Tracking-based distributed equilibrium seeking for aggregative games

TL;DR

The paper addresses the challenge of computing exact (generalized) Nash equilibria in aggregative games over networks with partial information. It introduces two fully distributed TRADES schemes: Primal TRADES for problems with local constraints and Primal-Dual TRADES for problems with affine coupling constraints, each employing tracking mechanisms to estimate the aggregative variable and, in the constrained case, augmented primal-dual updates. Using a singular perturbation framework, the authors prove linear (global exponential) convergence to the (G)NE under suitable step-size and network conditions, without requiring compactness of local sets or global information. Numerical experiments on demand-response-like scenarios corroborate the theoretical results, showing favorable convergence behavior and competitive efficiency compared to state-of-the-art distributed methods. The work provides scalable, distributed tools for networked decision-making in smart grids and related applications, with a clear pathway for extensions to time-varying graphs and richer constraint structures.

Abstract

We propose fully-distributed algorithms for Nash equilibrium seeking in aggregative games over networks. We first consider the case where local constraints are present and we design an algorithm combining, for each agent, (i) the projected pseudo-gradient descent and (ii) a tracking mechanism to locally reconstruct the aggregative variable. To handle coupling constraints arising in generalized settings, we propose another distributed algorithm based on (i) a recently emerged augmented primal-dual scheme and (ii) two tracking mechanisms to reconstruct, for each agent, both the aggregative variable and the coupling constraint satisfaction. Leveraging tools from singular perturbations analysis, we prove linear convergence to the Nash equilibrium for both schemes. Finally, we run extensive numerical simulations to confirm the effectiveness of our methods and compare them with state-of-the-art distributed equilibrium-seeking algorithms.
Paper Structure (21 sections, 8 theorems, 113 equations, 6 figures, 2 tables, 2 algorithms)

This paper contains 21 sections, 8 theorems, 113 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Paper organization
  5. Mathematical preliminaries
  6. Problem definition and main assumptions
  7. A key result on singularly perturbed systems
  8. Aggregative games over networks without coupling constraints
  9. Primal TRADES
  10. Proof of Theorem \ref{['th:convergence']}
  11. Generalized Nash equilibrium problems in aggregative form
  12. Primal-Dual TRADES
  13. Proof of Theorem \ref{['th:convergence_primal_dual']}
  14. Numerical Examples
  15. Example without coupling constraints
  16. ...and 6 more sections

Key Result

Theorem 2.5

Consider the system with $x^{t} \in \mathcal{D} \subseteq \mathbb{R}^n$, $w^{t} \in \mathbb{R}^m$, $f: \mathcal{D} \times \mathbb{R}^m \rightarrow \mathbb{R}^n$, $g: \mathbb{R}^m \times \mathbb{R}^n \times\mathbb{R} \rightarrow \mathbb{R}^m$, $\delta > 0$. Let $f$ and $g$ be Lipschitz continuous with respect to both $x and further assume that there exists $x^\star \in \mathbb{R}^n$ such that The

Figures (6)

  • Figure 1: Block diagram of the original interconnected system \ref{['eq:interconnected_system_generic']}.
  • Figure 2: Block diagram of the boundary layer system \ref{['eq:boundary_layer_system_generic']}.
  • Figure 3: Block diagram of the reduced system \ref{['eq:reduced_system_generic']}.
  • Figure 4: Mean and $1$-standard deviation band (based on Monte Carlo simulations) of the normalized distance of the iterates from the NE achieved by Primal TRADES (Algorithm \ref{['algo:unconstrained']}), the algorithm by parise2020distributed, and the algorithm by bianchi2022fast on a case study introduced in parise2020distributed.
  • Figure 5: Execution time of a single iterate with Primal TRADES (Algorithm \ref{['algo:unconstrained']}) and the considered algorithms in parise2020distributed and bianchi2022fast. Mean and $1$-standard deviation are based on Monte Carlo simulations of the case study in parise2020distributed.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 2.1: Generalized Nash equilibrium FacchineiKanzowGNE2010
  • Theorem 2.5: Global exponential stability for singularly perturbed systems
  • Remark 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Definition A.1
  • ...and 2 more