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

In this paper, the notion of weak sharp minima is employed to the investigation of set-valued vector variational inequalities. The gap function φTfor set-valued strong vector variational inequalities (for short, SVVI) is proved to be less than the gap function φ{symbol}Tfor 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 sqrt(min1 ≤ i ≤ npTi) and for gap functions sqrt(φT) and sqrt(φ{symbol}T) under the assumption of strong pseudomonotonicity, where pTiis 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 sqrt(min1 ≤ i ≤ np∇ gi) when each component giof objective function is strongly convex.