Abstract
Abstract A new class of generalized convex set-valued maps termed relatively solid generalized cone-subconvexlike maps is introduced in real linear spaces not equipped with any topology. This class is a generalization of generalized cone-subconvexlike maps and relatively solid cone-subconvexlike maps. Necessary and sufficient conditions for Benson proper efficiency of set-valued optimization problem are established by means of scalarization, Lagrange multipliers, saddle points and duality. The results generalize and improve some corresponding ones in the literature. Some examples are afforded to illustrate our results.
Highlights
It is well known that the notion of e ciency plays a very critical role in vector optimization problem
A new class of generalized convex set-valued maps termed relatively solid generalized conesubconvexlike maps is introduced in real linear spaces not equipped with any topology
Necessary and su cient conditions for Benson proper e ciency of set-valued optimization problem are established by means of scalarization, Lagrange multipliers, saddle points and duality
Summary
It is well known that the notion of (weak) e ciency plays a very critical role in vector optimization problem. [24] Let B ⊂ Y, y ∈ B be called a Benson proper minimal point of B (denoted by y ∈ PMin(B, C)) if vcl(cone(B − y + C)) ∩ (−C) = { }. Let B ⊂ Y, y ∈ B be called a Benson proper maximal point of B (denoted by y ∈ PMax(B, C)) if vcl(cone(B − y − C)) ∩ C = { }.
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