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Distributed GNE seeking under partial-decision information over networks via a doubly-augmented operator splitting approach

Lacra Pavel

TL;DR

A single-layer algorithm is introduced, fully distributed with respect to both primal and dual variables, and its convergence to a variational GNE with fixed step sizes is shown, by reformulating it as a forward–backward iteration for a pair of doubly-augmented monotone operators.

Abstract

We consider distributed computation of generalized Nash equilibrium (GNE) over networks, in games with shared coupling constraints. Existing methods require that each player has full access to opponents' decisions. In this paper, we assume that players have only partial-decision information, and can communicate with their neighbours over an arbitrary undirected graph. We recast the problem as that of finding a zero of a sum of monotone operators through primal-dual analysis. To distribute the problem, we doubly augment variables, so that each player has local decision estimates and local copies of Lagrangian multipliers. We introduce a single-layer algorithm, fully distributed with respect to both primal and dual variables. We show its convergence to a variational GNE with fixed step-sizes, by reformulating it as a forward-backward iteration for a pair of doubly-augmented monotone operators.

Distributed GNE seeking under partial-decision information over networks via a doubly-augmented operator splitting approach

TL;DR

A single-layer algorithm is introduced, fully distributed with respect to both primal and dual variables, and its convergence to a variational GNE with fixed step sizes is shown, by reformulating it as a forward–backward iteration for a pair of doubly-augmented monotone operators.

Abstract

We consider distributed computation of generalized Nash equilibrium (GNE) over networks, in games with shared coupling constraints. Existing methods require that each player has full access to opponents' decisions. In this paper, we assume that players have only partial-decision information, and can communicate with their neighbours over an arbitrary undirected graph. We recast the problem as that of finding a zero of a sum of monotone operators through primal-dual analysis. To distribute the problem, we doubly augment variables, so that each player has local decision estimates and local copies of Lagrangian multipliers. We introduce a single-layer algorithm, fully distributed with respect to both primal and dual variables. We show its convergence to a variational GNE with fixed step-sizes, by reformulating it as a forward-backward iteration for a pair of doubly-augmented monotone operators.

Paper Structure

This paper contains 12 sections, 8 theorems, 68 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary background
  3. Monotone operators
  4. Graph theory
  5. Game formulation
  6. Iterative Algorithm under Full-Decision Information
  7. Distributed Algorithm under Partial-Decision Information
  8. Convergence analysis
  9. Nash-Cournot game over a network
  10. Assumptions
  11. Numerical results
  12. Conclusions

Key Result

Lemma 1

Let $x_k\!\!= \!\!col(x_{i,k})_{i \in \mathcal{N}}$, $\mathbf{x}_k\!\!=\!\! col(\mathbf{x}^i_{k})_{i \in \mathcal{N}}$, $\bm{\lambda}_k\!\!=\!\!col(\lambda_{i,k})_{i \in \mathcal{N}}$, $\bm{z}_k\!\!=\!\! col(z_{i,k})_{i \in \mathcal{N}}$. Then, Algorithm dal_1 is equivalently written in stacked nota where $\mathbf{F}$ is defined as $\bm{A}\! \! = \!diag(\!(A_i)_{i \in \mathcal{N}}\!$, $\!\!\bm{b}

Figures (5)

  • Figure 1: Network Nash-Cournot game: An edge from $i$ to $M_k$ on this graph implies that agent/firm $i$ participates in Market $M_k$.
  • Figure 2: Communication graph $\mathcal{G}_c$: Firms $i$ and $j$ are able to exchange their local $\mathbf{x}^i$ and $\mathbf{x}^j$ if there exists an edge between them on this graph.
  • Figure 3: The trajectories of local decisions $x_{i,k}$ for selected players ($c=100$)
  • Figure 4: The trajectories of dual variables $\lambda_{i,k}$ for selected players ($c=100$)
  • Figure 5: The trajectories of local decisions $x_{i,k}$ for selected players ($c=10$)

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Lemma 2
  • Remark 4
  • Remark 5
  • Theorem 1
  • Remark 6
  • Lemma 3
  • ...and 6 more