Continuity concepts for set-valued functions and a fundamental duality formula for set-valued optimization

Over the past few years a theory of conjugate duality for set-valued functions that map into the set of upper closed subsets of a preordered topological vector space has been developed. For scalar duality theory, continuity of convex functions plays an important role. For set-valued maps, different notions of continuity exist. We will compare the most prevalent ones for the special case where the image space is the set of upper closed subsets of a preordered topological vector space and analyze which of the results can be conveyed from the extended real-valued case.Moreover, we present a fundamental duality formula for set-valued optimization, using the weakest of the continuity concepts under consideration for a regularity condition.