Study On Optimality Conditions Of Set-valued Optimization Problems

The optimality conditions for set-valued optimization problems in the sense of various solutions are the important components of set-valued optimization problems and are the important bases of developing modern algorithms. This paper is mainly devoted to discuss the optimality conditions of set-valued optimization problems by virtue of the higher-order tangent sets and the higher-order generalized derivative and generalized higher-order epiderivative.The main results are as follows:1. The relationships of Clarke tangent epiderivative, contingent epiderivative, Y- epiderivative, radial epiderivative and generalized first-order epiderivative were obtained.2. Some properties of generalized higher-order tangent sets were discussed. Then, by virtue of the generalized higher-order tangent sets and the Gerstewitz's nonconvex separation functional, necessary and sufficient optimality conditions were obtained for weakly Benson proper efficient solutions of set-valued optimization problems without any convexity assumption on objective and constraint mappings.3. By using the generalized higher-order epiderivative, the optimality conditions of the weakly efficient solution and different kinds of properly efficient solutions for set-valued vector equilibrium problems with functional constraints were obtained.4. By virtue of the higher-order generalized derivative and the generalized higher-order epiderivative, the optimality conditions of the weakly efficient solution for nonconvex set-valued vector equilibrium problems with functional constraints were obtained.