Approximating Security Values of MinSup Problems with Quasi-variational Inequality Constraints

We consider a pessimistic two-stage model where a leader, according to its risk-averse prone- ness, solves a MinSup problem with constraints, corresponding to the reaction set of a follower, defined by the solutions of a (quasi)variational inequality. We show that, in general, the security value of a MinSup problem with (quasi)variational in- equality constraints is not stable under perturbations, in the sense that the sequence of the security values for the perturbed problems may not converge to the security value of the orig- inal problem even in presence of nice data. So, we introduce different types of approximate security values, close to the exact value also in the case of possibly discontinuous functions, and we investigate their asymptotic behavior in the presence of perturbations and under suitable assumptions of minimal character.