Further on Inner Regularizations in Bilevel Optimization

A crucial difficulty in pessimistic bilevel optimization is the possible lack of existence of exact solutions since marginal functions of the sup-type may fail to be lower semicontinuous. So, to overcome this drawback, we have introduced, in Lignola and Morgan, J. Optim. Theory Appl. 173, 2017, suitable inner regularizations of the lower level optimization problem together with relative viscosity solutions for the pessimistic bilevel problem. Here, we continue this research by considering new inner regularizations of the lower level optimization problem, which not necessarily satisfy the constraints but that are close to them, and by deriving an existence result of related viscosity solutions to the pessimistic bilevel problem.