Efficient and weakly efficient solutions for the vector topical optimization

In this paper, we consider some scalarization functions, which consist of the generalized min-type function, the so-called plus-Minkowski function and their convex combinations. We investigate the abstract convexity properties of these scalarization functions and use them to identify the maximal points of a set in an ordered vector space. Then, we establish some versions of Farkas type results for the infinite inequality system involving vector topical functions. As applications, we obtain the necessary and sufficient conditions of efficient solutions and weakly efficient solutions for a vector topical optimization problem, respectively.