Smoothness properties of a regularized gap function for quasi-variational inequalities

This article studies continuity and differentiability properties for a reformulation of a finite-dimensional quasi-variational inequality (QVI) problem using a regularized gap function approach. For a special class of QVIs, this gap function is continuously differentiable everywhere, in general, however, it has nondifferentiability points. We therefore take a closer look at these nondifferentiability points and show, in particular, that under mild assumptions all locally minimal points of the reformulation are differentiability points of the regularized gap function. The results are specialized to generalized Nash equilibrium problems. Numerical results are also included and show that the regularized gap function provides a valuable approach for the solution of QVIs.