Weak sharp minima for set-valued vector variational inequalities with an application

Abstract

In this paper, the notion of weak sharp minima is employed to the investigation of set-valued vector variational inequalities. The gap function φ T for set-valued strong vector variational inequalities (for short, SVVI) is proved to be less than the gap function ϕ T for set-valued weak vector variational inequalities (for short, WVVI) under certain conditions, which implies that the solution set of SVVI is equivalent to the solution set of WVVI. Moreover, it is shown that weak sharp minima for the solution sets of SVVI and WVVI hold for min 1 ⩽ i ⩽ n p T i and for gap functions φ T and ϕ T under the assumption of strong pseudomonotonicity, where p T i is a gap function for i-th component of SVVI and WVVI. As an application, the weak Pareto solution set of vector optimization problems (for short, VOP) is proved to be weak sharp minimum for min 1 ⩽ i ⩽ n p ∇ g i when each component g i of objective function is strongly convex.