Generalized Subdiffernetials Of Set-valued Maps With Applications To Vecter Optimization

In this thesis, we present a new concept of subdifferential for a set-valued map and study the optimality conditions for set-valued optimization problems in trems of this subdifferential.We first introduce the basic definition of generalized subdifferential for a setvalued map and give a simple calculation rule of this subdifferential, and discuss the relationships between the generalized subdifferential and several kinds of notions of weak subdifferential for set-valued maps. Subsequently, an existence theorem of generalized subgradient for a set-valued map is established by using the generalized Hahn-Banach theorem. Then, we study some basic properties of the generalized subdifferential, and discuss the relation between the generalized subdifferential and directional derivative when the set-valued map reduces to single-valued case and give a calculation formula of the subdifferential in this case. Simultaneously, using the semidifferentiablity of set-valued maps, a sum rule of the generalized subdifferential is obtained.As applications, in terms of the generalized subdifferential we also study the optimality conditions of set-valued optimization problems. Sufficient and necessary optimality conditions are, respectively, derived for weak minimizer of a set-valued unconstrained optimization problem and a set-valued optimization problem with generalized inequation constraints.In the end, the main works in this thesis are briefly summed up. Some problems which are remained and are worth thoughting in the future are put forward.