On a Smooth Dual Gap Function for a Class of Player Convex Generalized Nash Equilibrium Problems

We consider a class of generalized Nash equilibrium problems (GNEPs) where both the objective functions and the constraints are allowed to depend on the decision variables of the other players. It is well-known that this problem can be reformulated as a constrained optimization problem via the (regularized) Nikaido-Isoda-function, but this reformulation is usually nonsmooth. Here we observe that, under suitable conditions, this reformulation turns out to be the difference of two convex functions. This allows the application of the Toland-Singer duality theory in order to obtain a dual formulation which provides an unconstrained and continuously differentiable optimization reformulation of the GNEP. Moreover, based on a result from parametric optimization, the gradient of the unconstrained objective function is shown to be piecewise smooth. Some numerical results are presented to illustrate the theory.