Asymptotic behavior of semi-quasivariational optimistic bilevel problems in Banach spaces

The great interest into hierarchical optimization problems and the increasing use of game theory in many economic or engineering applications led to investigate scalar bilevel problems in which the upper level is an optimization problem and the lower level is a parametrized quasi-variational inequality. In this paper, we analyze the convergence of the sequences of infima and minima for the upper level when the data of the problem are perturbed. First, we show that general results on the convergence of the infima and minima may not be possible. Thus, we introduce suitable concepts of regularized semi-quasivariational optimistic bilevel problems and we study, in Banach spaces, the convergence properties of the infima and minima to these regularized problems in the presence or not of perturbations.