On the Existence of Projected Solutions of Quasi-Variational Inequalities and Generalized Nash Equilibrium Problems

A quasi-variational inequality is a variational inequality, in which the constraint set is depending on the variable. However, as shown by a motivating example in electricity market, the constraint map may not be a self-map, and then, there is usually no solution. Thus, we define the concept of projected solution and, based on a fixed point theorem, we establish some results on existence of projected solution for quasi-variational inequality problem in a finite-dimensional space where the constraint map is not necessarily self-map. As an application of our results, we obtain an existence theorem for quasi-optimization problems. Finally, we introduce the concept of projected Nash equilibrium and study the existence of such equilibrium for noncooperative games as another application.