τw-contingent epiderivatives in reflexive spaces

In this paper we introduce a notion of the τw-contingent epiderivative of a set-valued map by considering the weak topology in the image space. We establish a chain rule formula for families of τw-contingent and contingent epiderivatives. When the image space is reflexive and the ordering cone a half-space, we give conditions for the existence of the family of τw-contingent epiderivatives of a stable set-valued map; moreover we obtain an interesting formula that connects this family with the contingent epiderivative of an associated scalar set-valued map. Finally we apply the previous results in order to establish optimality conditions for a set-valued optimization problem via epiderivatives with respect to half-spaces associated with the ordering cone. In particular we provide a scalarization method for computing these conditions in reflexive spaces.