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

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