Zhi Ang ZHOU, Shuang YANG
In this paper, we study $(C,\varepsilon)$-super subdifferentials of set-valued maps. First, we introduce a notion of $(C,\varepsilon)$-super efficient point of a set. Some properties and equivalent characterizations of the $(C,\varepsilon)$-super efficient points are presented. Scalarization theorems of the set-valued optimization problem are obtained in the sense of $(C,\varepsilon)$-super efficiency. Second, we define $(C,\varepsilon)$-subdifferentials of set-valued maps and research the existence conditions of $(C,\varepsilon)$-subdifferentials. Moreau-Rockafellar type theorems characterized by $(C,\varepsilon)$-subdifferentials are also established. Finally, as the applications, we establish some optimality conditions of the set-valued optimization problem involving the $(C,\varepsilon)$-super subdifferentials. The results obtained in this paper unify and generalize some results characterized by the super subdifferentials or $\varepsilon$-super subdifferentials of the set-valued maps in the literature.