Contingent Derivatives Of Set-valued Maps With Applications To Parameterized Vector Optimization

By virtue of the concept of contingent derivatives of set-valued maps, sensitivity analysis for a family of parameterized vector optimization problems VOP(u) with u as the parameter is investigated first, and then the obtained results are applied to the study of the differential properties of gap functions for Minty vector variational inequalities (MVVI) and Minty weak vector variational inequalities (MWVVI), respectively.For parameterized vector optimization problems VOP(u), some chain rules are established first for calculating the contingent derivatives of a class of composition set-valued maps, each of which consists of a vector-valued function and a set-valued maps. Then, some explicit expressions for the contingent derivative of the feasible set map in the objective space are obtained. In the end, mainly under the compact and Lipschitz assumptions, some explicit expressions for the contingent derivatives of the perturbation maps and the weak perturbation maps are provided.In this thesis, two set-valued maps G^MVVI and G^MWVVI are introduced as the gap functions of MVVI and MWVVI, respectively. After constructing a special family of parameterized vector optimization problems VOP (u ) whose constraint sets do not depend on the parameter u, we find that -G^MVVI and -G^MWVVI are exactly a perturbation map and a weak perturbation map, respectively. Then, by applying the results obtained for VOP(u), we can easily obtain some explicit expressions for the contingent derivatives of the perturbation maps and the weak perturbation involving VOP (u ) and hence obtain some explicit expressions for the contingent derivatives of gap functions for MVVI and MWVVI. Furthermore, by virtue of the gap functions, we investigate optimality conditions for the solutions of MVVI and MWVVI, respectively.