Approximate Benson Proper Efficiency in Vector Optimization with Set-Valued Maps

In this paper, for vector optimization problems (VP) with the objective function and constraint function are set valued maps, the concepts of approximate Benson properly efficient solutions and approximate Benson properly efficient element are introduced, which extend ε-properly efficient solution introduced by Rong Weidong and Ma Yi, and an example is given to illustrate it. Then approximate Benson properly efficient solutions of vector optimization with set-valued maps are considered. Under the assumption of nearly cone subconvexlikeness, we obtain the conclusions about the approximate solutions of vector optimization problems through associated scalar optimization problems: x0,y0is approximate Benson properly efficient element of problem (VP) if and only if it is －εσ－C(μ)-suboptimal element for the scalar problem (Pμ) corresponds to (VP). Especially, the necessary and sufficient conditions have the same error, which extend and improve corresponding ones in the literature.