[Paper Review] Generalized Nash Equilibrium Problem by the Alternating Direction Method of Multipliers

Farzad Salehisadaghiani, Lacra Pavel|arXiv (Cornell University)|Mar 24, 2017

Distributed Control Multi-Agent Systems19 references3 citations

TL;DR

This paper proposes an inexact Alternating Direction Method of Multipliers (ADMM) algorithm for distributed generalized Nash equilibrium (GNE) computation in networked games with coupled linear constraints. Each player only knows their own cost function and private constraints, exchanging estimates of others' actions and local Lagrange multipliers over a communication graph; convergence to a GNE is proven under mild assumptions on cost functions, with simulation on a wireless ad-hoc network validating the approach.

ABSTRACT

In this paper, the problem of finding a generalized Nash equilibrium (GNE) of a networked game is studied. Players are only able to choose their decisions from a feasible action set. The feasible set is considered to be a private linear equality constraint that is coupled through decisions of the other players. We consider that each player has his own private constraint and it has not to be shared with the other players. This general case also embodies the one with shared constraints between players and it can be also simply extended to the case with inequality constraints. Since the players don't have access to other players' actions, they need to exchange estimates of others' actions and a local copy of the Lagrangian multiplier with their neighbors over a connected communication graph. We develop a relatively fast algorithm by reformulating the conservative GNE problem within the framework of inexact-ADMM. The convergence of the algorithm is guaranteed under a few mild assumptions on cost functions. Finally, the algorithm is simulated for a wireless ad-hoc network.

Motivation & Objective

• To develop a distributed algorithm for computing generalized Nash equilibria (GNE) in networked games where players have private linear coupled constraints.
• To enable players to compute a GNE using only local information—own cost functions, private constraints, and action sets—without direct knowledge of others’ actions.
• To design a scalable and robust optimization method that leverages communication graphs for exchanging action estimates and local Lagrange multipliers.
• To establish convergence guarantees for the proposed algorithm under mild assumptions on the players’ cost functions.
• To validate the algorithm’s performance through simulation in a wireless ad-hoc network (WANET) with data flow and capacity constraints.

Proposed method

• Reformulate the conservative GNE problem into its Lagrangian dual form and introduce local estimates of players’ actions and local copies of Lagrange multipliers to enable distributed computation.
• Use an augmented Lagrangian function per player to decouple the global optimization problem into local subproblems.
• Develop an inexact-ADMM algorithm that allows for inexact solutions in the subproblems, improving computational efficiency while preserving convergence.
• Model the communication network as a connected graph where each player exchanges action estimates and local dual variables with neighbors.
• Derive KKT conditions for the GNE and use them to establish convergence via a Lyapunov-like analysis of the algorithm’s iterates.
• Introduce auxiliary variables (e.g., u, w, v, z) to handle coupling constraints and dual variables in the ADMM update steps.

Experimental results

Research questions

• RQ1 Can a distributed algorithm be designed to compute a generalized Nash equilibrium in a networked game where players have private linear coupled constraints and limited information?
• RQ2 How can players achieve GNE computation using only local cost functions, private constraints, and neighbor-level communication of action estimates and dual variables?
• RQ3 What convergence guarantees can be established for an inexact-ADMM-based GNE seeking algorithm under mild assumptions on cost functions?
• RQ4 How does the proposed algorithm perform in a realistic networked system such as a wireless ad-hoc network with shared capacity constraints?
• RQ5 Can the algorithm handle both equality and inequality constraints, and how does it generalize from shared to private constraints?

Key findings

• The proposed inexact-ADMM algorithm converges to a generalized Nash equilibrium under mild assumptions, including strong convexity and Lipschitz continuity of the cost functions.
• The sequences of players’ actions, dual multipliers, and auxiliary variables converge to their optimal values, with the difference between consecutive iterates tending to zero.
• Simulation results on a 16-node wireless ad-hoc network with 15 users show convergence of flow rates to the GNE, marked by black stars in the figures.
• The dual Lagrange multipliers converge to zero, indicating that the coupling constraint is satisfied at equilibrium.
• The normalized error in the algorithm decreases over time, demonstrating robustness and convergence in practice.
• The algorithm is scalable and effective in a decentralized setting, requiring only local communication and no global coordination.