Weak subdifferential for set-valued mappings and its applications. (English) Zbl 1171.49013
Summary: The existence theorems of two kinds of weak subgradients for set-valued mappings, which are the generalizations of Theorem 7 in [G. A. Chen and J. Jahn, Math. Methods Oper. Res. 48, No. 2, 187–200 (1998; Zbl 0927.90095)] and Theorem 4.1 in [J. W. Peng, H. W. J. Lee, W. D. Rong and X. M. Yang, Math. Methods Oper. Res. 61, No. 2, 281–297 (2005; Zbl 1079.49016)], respectively, are proved by virtue of a Hahn-Banach extension theorem. Moreover, some properties of the weak subdifferential for set-valued mappings are obtained by using a so-called Sandwich theorem. Finally, necessary and sufficient optimality conditions are discussed for set-valued optimization problems, whose constraint sets are determined by a fixed set and a set-valued mapping, respectively.
MSC:
| 49J52 | Nonsmooth analysis |
References:
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