Local Generalized Nash Equilibria With Nonconvex Coupling Constraints

In this article, we address a class of Nash games with nonconvex coupling constraints for which we define a novel notion of local equilibrium, here named local generalized Nash equilibrium (LGNE). Our first technical contribution is to show the stability in the game theoretic sense of these equilibria on a specific local subset of the original feasible set. Remarkably, we show that the proposed notion of local equilibrium can be equivalently formulated as the solution of a quasi-variational inequality with equal Lagrange multipliers. Next, under the additional proximal smoothness assumption of the coupled feasible set, we define conditions for the existence and local uniqueness of an LGNE. To compute such an equilibrium, we propose two discrete-time dynamics, or fixed-point iterations implemented in a centralized fashion. Our third technical contribution is to prove convergence under (strongly) monotone assumptions on the pseudogradient mapping of the game and proximal smoothness of the coupled feasible set. Finally, we apply our theoretical results to a noncooperative version of the optimal power flow control problem.