Existence of weak efficient solutions of set-valued optimization problems

Abstract

In this paper, we introduce a novel scalarization function for set-valued maps and establish several significant properties that highlight its utility. We extend the concept of regularglobal- inf functions, initially studied in the context of single-valued maps, to the realm of set-valued maps. As the first main result, we derive a new Weierstrass-type theorem for set-valued maps satisfying the coercivity condition. Furthermore, utilizing tools from asymptotic analysis, we present an existence theorem for strict weakly l-efficient solutions of set optimization problems under certain non-coercive conditions. To underscore the relevance and applicability of our findings, we provide several corollaries and illustrative examples.