Vector Variational Inequality And Vector Optimization In Locally Convex Topological Vector Spaces

In the framework of locally convex topological vector spaces, we introduce the concepts of radial epiderivative and contingent epiderivative, and discuss the relationship between them. For the vector variational inequality related to radial epiderivative (VVIP)R we proposed, various kinds of concepts of efficient solution pairs and proper efficient solution pairs are given. By using them, we obtain the necessary and sufficient conditions for various kinds of efficient pairs and proper efficient pairs of unconstrained set-valued vector optimization problem. Moreover, in light of the notion of f-contingent derivative in [17], we suggest a new derivative concept for set-valued map: f-radial derivative, and take use of it to derive a necessary and sufficient condition for f-efficient pair of set-valued vector optimization problem. Since radial epiderivative which we use to give the optimality conditions is very weak, we relax the requirement of objective set-valued maps greatly and extend the scope of set-valued maps we investigate. Based on the relationship between radial epiderivative and contingentepiderivative, we see that most of the results in [13], [17] have been generalized fromnormed spaces to locally convex topological vector spaces.