Efficient Solution of DC-Type Vector Optimization via Abstract Convex Analysis

The concept of vector topical functions, which take values in a partially ordered Banach space endowed with a complete lattice structure, was introduced in our preceding study. This structure enabled the development of an abstract convexity theory for vector topical functions, utilizing the notion of vector support. In this paper, applying these abstract convex theories, a DC-type vector optimization is investigated. Using the idea of a slack, the support set of a vector-valued map can be fully characterized by the subdifferential in abstract convex sense. Then, with the aid of this result, a sufficient condition to detect the efficient solutions for the DC-type vector optimization is obtained. In addition, a dual problem for the DC optimization is proposed, for which some strong dual results are established.