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