Convexity and Optimization of Set-Valued Mapping in Vector Space

In order to discuss set-valued optimization under some weaker conditions in real linear topological space, the generalized subconvexlike set-valued mapping is defined in terms of cones and an alternative theorem is established for generalized subconvexlike set-valued mapping, which is used to discuss the optimal condition for set-valued optimization. As results, some optimal conditions are obtained in the form of Lagrange multiplier, a Lagrangian duality theorem is developed at the same time.