Stability Of Vector Variational Inequalities And Second Order Optimality Conditions Of Set-valued Optimization Problems

In this thesis, we study two problems: the lower semicontinuity and continuity of the solution map of parametric generalized vector variational inequality problems and second-order optimality conditions for strict local minimality in set-valued optimization. The detailed contents are listed below:On the one hand, we study the lower semicontinuity and continuity of the solution set map for parametric generalized vector variational inequality problems in Hausdorff topological vector spaces. We firstly define the notion of the nonemptyξ-efficient solutions of the vector variational inequality. Then, based on a scalarization representation of the solution mapping, a property for the union of a family of lower semicontinuous set-valued mappings and the models together with their properties studied in the paper [B.T. Kien, N.C. Wong and J.C. Yao, Generalized vector variational inequalities with star-pseudomonotone and discintinuous operators, 68(2008) 2859-2871], we establish the lower semicontinuity and continuity of the solution mapping to a parametric generalized vector variational inequality problem with set-valued mappings. These consequences partially improve and generalize the corresponding results in [C.R. Chen, S.J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, Journal of Global Optimization, 45 (2009) 309-318].On the another hand, we firstly introduce the second-order contingent set and the second-order adjacent set of set-valued mappings in real normed spaces and investigate their properties and relationships. Then, by virtue of the second-order contingent derivative and the second-order adjacent derivative for set-valued mappings, we obtain second order necessary optimality conditions and second order Fritz-John type necessary optimality conditions for a strict local minimizer of set-valued optimization problems whose constraint condition is determined by a set-valued map.