Local Generalized Nash Equilibria with Nonconvex Coupling Constraints

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 a 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 pseudo- gradient 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.