Asymptotic Behavior of Regularized OptimizationProblems with Quasi-variational Inequality Constraints

The great interest into hierarchical optimization problems and the increasing use of game theory in many economic or engineering applications led to investigate optimization problems with constraints described by the solutions to a quasi-variational inequality (variational problems having constraint sets depending on their own solutions, present in many applications as social and economic networks, financial derivative models, transportation network congestion and traffic equilibrium). These problems are bilevel problems such that at the lower level a parametric quasi-variational inequality is solved (by one or more followers) meanwhile at the upper level the leader solves a scalar optimization problem with constraints determined by the solutions set to the lower level problem. In this paper, mainly motivated by the use of approximation methods in infinite dimensional spaces (penalization, discretization, Moreau-Yosida regularization ...), we are interested in the asymptotic behavior of the sequence of the infimal values and of the sequence of the minimum points of the upper level when a general scheme of perturbations is considered. Unfortunately, we show that the global convergence of exact values and exact solutions of the perturbed bilevel problems cannot generally be achieved. Thus, we introduce suitable concepts of regularized optimization problems with quasi-variational inequality constraints and we investigate, in Banach spaces, the behavior of the approximate infimal values and of the approximate solutions under and without perturbations.