Vector Topical Function,Abstract Convexity and Image Space Analysis

In this paper, we introduce a new type of vector topical function. It contains some other categories of topical functions as special cases and can be interpreted as weak separation functions in image space analysis. We establish its envelope result and investigate its properties in the frame of abstract convexity. Then, we present the corresponding conjugation and subdifferential, and observe the relationships among these concepts. Finally, as applications, we obtain some dual results for some vector optimization, where the object is expressed as the difference of vector topical functions.