α-阶预凸集值映射弱有效解的最优性条件

The optimality conditions ofα-order set-valued optimization problem with weak effcient solution

设X,Y为实赋范线性空间,C为Y中的闭凸点锥,C诱导了Y中的偏序,F:X→2~Y为集值映射。本文新引入了α-阶C-预凸集值映射的概念,并介绍了集值映射α-阶伴随切导数的定义,给出了集值映射在以上两者假设下的一个引理和两个定理。定理1是关于集值映射F的弱有效解的导数型的充分必要条件,即(■,■)为F在S上的弱有效解■D^aF(■,■) (η(x,(■)))∩-intC=Φ,■x∈S.定理2说明了集值映射,的弱有效解即为F的局部弱有效解。

X, Y is real valued normed linear space, C is the closed conve X and pointed cone in Y, the partial order in Y is induced by C. F is set valued map from X into Y. In this paper, the concept of α - order preinvex set - valued map is introduced, At the same time, the definition of α - order contingent tangent derivative is given, Under the assumption of the above two, we obtain one lemma and two theorems, theorem one is about the sufficent and nessisary conditions of derivative for the weak effieient solution of set valued map, that is （x^-,y^-） is the weak efficient solution when and only when D＊F（x^-,y^-） （η（x,x^-）） ∩ - intC = Ф, arbitary x ∈ S. theorem two proves the weak efficient solution is the local weak efficent solution.