Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming

In this paper we adapt the main results from Amahroq and Gadhi (J Glob Optim 21:435-443, 2001) for a general set-valued optimization problem to an optimistic bilevel programming problem as an optimization problem with implicitly given set-valued constraint. Since this constraint is assumed to be upper but not lower semicontinuous in the sense of Berge, we need to deal with a lower semicontinuous distance function to this mapping. In order to approximate the gradient of the distance function, we introduce a new concept for a directional convexificator. Some calculus rules for this new tool are adapted and several properties are characterized. The main result presents optimality conditions for an optimistic bilevel programming problem using a convexificator constructed with the aid of the directional convexificator.