# A distributed primal-dual algorithm for computation of generalized Nash equilibria with shared affine coupling constraints via operator splitting methods

## TL;DR

This paper introduces a distributed primal-dual algorithm based on operator splitting methods for computing generalized Nash equilibria with shared affine constraints in networked noncooperative games, ensuring convergence under mild conditions.

## Contribution

It develops a novel distributed primal-dual algorithm utilizing operator splitting for variational GNE computation with shared constraints, allowing decentralized implementation and convergence guarantees.

## Key findings

- Algorithm converges to variational GNE under fixed step-sizes.
- Distributed approach requires only local information and neighbor communication.
- Numerical simulations demonstrate the algorithm's efficiency in network Cournot competition.

## Abstract

In this paper, we propose a distributed primal-dual algorithm for computation of a generalized Nash equilibrium (GNE) in noncooperative games over network systems. In the considered game, not only each player's local objective function depends on other players' decisions, but also the feasible decision sets of all the players are coupled together with a globally shared affine inequality constraint. Adopting the variational GNE, that is the solution of a variational inequality, as a refinement of GNE, we introduce a primal-dual algorithm that players can use to seek it in a distributed manner. Each player only needs to know its local objective function, local feasible set, and a local block of the affine constraint. Meanwhile, each player only needs to observe the decisions on which its local objective function explicitly depends through the interference graph and share information related to multipliers with its neighbors through a multiplier graph. Through a primal-dual analysis and an augmentation of variables, we reformulate the problem as finding the zeros of a sum of monotone operators. Our distributed primal-dual algorithm is based on forward-backward operator splitting methods. We prove its convergence to the variational GNE for fixed step-sizes under some mild assumptions. Then a distributed algorithm with inertia is also introduced and analyzed for variational GNE seeking. Finally, numerical simulations for network Cournot competition are given to illustrate the algorithm efficiency and performance.