Abstract convexity of set-valued topical functions with application in DC-type optimization

In the previous work, we established a conjugate duality for the constrained vector optimization, in which the concept of a set-valued topical function introduced there played a key role. This is a map with good features and can be interpreted as an abstract convex function in some sense. This paper is devoted to the further study for the properties and characterizations of the set-valued topical function, within the framework of abstract convexity, especially about the subdifferentials. As applications, we also investigate some set-valued DC-type optimization problems. Some optimal conditions and dual results are obtained by virtue of the conjugation and subdifferentials.