Benson-Subgradient And Set-Benson-Subgradient Of Set-valued Optimization

The scope of (weak) efficient solutions sets for set-valued optimization problems is bulky,so contraction of the scope of the set of solutions of optimization become to be an important work, and the concept of a variety of really effective solutions have been introduced, Benson true efficient solutions is a representative of the true effective solutions, and thus the study of Benson true efficient solutions become an important part of optimization theory. While the subderivative is an important method to depict the set-valued optimization, the operation of subderivative a very important part of analysis of non-smooth. For example, it play a very important role in studying the problem of optimal conjugate dual cone and the establishment of optimal conditions. The concept of subdifferential is first introduced by R.T.Rockafellar[19].In recent years, set-value optimization has become a concern, many scholars introduce subdifferential in set-valued optimization problems and study it.The sub-differential (sub-gradient) for describing the set-valued optimization problem are derivative or non-derivative, the existence conditions of non-derivative of the sub-differential (sub-gradient) are weaker than the derivative. X.Q. Yang [13] introduced a non-Derivative weak subdifferential; P.H. Sach [14] introduced the non-derivative-based Benson subdifferential to the set-valued optimization problems; Chen G.Y., Jahn J.[23] introduced a non-derivative-based Chen-weak subdifferential for set-valued optimization problems, which is the weak subdifferential about the image set on a point of the set-valued function avalued; S.J.Li, X.L.Guo[17] discussed the nature of these two subdifferential, and used them to describe the set-valued optimization problems. Li Taiyong[7] introduced the non-derivative-based subgradient under the strict efficient solutions, and discuss its existence, the conditions of its existence is weak, and which is used to characterize the strict efficient element. In this paper, Benson subgradient (non-derivative) for set-valued function is introduced in a weaker conditions,and its existence is proved, further more,several of its properties are discussed, and it is used to describe the Benson proper efficient solution of the set-valued optimization problems; at the same time the subdifferential is introduced under the Benson real efficent validity in the partial order complete the real linear topological space, its existence is proved by Hahn-Banach Theorem, and the Set-Benson-subdifferent is used to describe the set-valued subdifferential optimization problem.