Abstract
A scalarization theorem and two Lagrange multiplier theorems are established for tightly proper efficiency in vector optimization involving nearly cone-subconvexlike set-valued maps. A dual is proposed, and some duality results are obtained in terms of tightly properly efficient solutions. A new type of saddle point, which is called tightly proper saddle point of an appropriate set-valued Lagrange map, is introduced and is used to characterize tightly proper efficiency.
Highlights
One important problem in vector optimization is to find efficient points of a set
In this paper, inspired by 10, 21–23, we extend the concept of tight properness from normed linear spaces to locally convex topological vector spaces, and study tightly proper efficiency for vector optimization problem involving nearly cone-subconvexlike set-valued maps and with nonempty interior of constraint cone in the framework of locally convex topological vector spaces
Suppose that S is a subset of Y and B C denotes the family of all bases of C. y is said to be a strictly efficient point with respect to Θ ∈ B C, written as y ∈ STE S, Θ, if there is a convex neighborhood U of 0Y such that cl cone S − y ∩ U − Θ ∅
Summary
One important problem in vector optimization is to find efficient points of a set. As observed by Kuhn, Tucker and later by Geoffrion, some efficient points exhibit certain abnormal properties. Weak efficiency and Benson proper efficiency in vector optimization problem involving ic-cone-convexlike set-valued maps. In this paper, inspired by 10, 21–23 , we extend the concept of tight properness from normed linear spaces to locally convex topological vector spaces, and study tightly proper efficiency for vector optimization problem involving nearly cone-subconvexlike set-valued maps and with nonempty interior of constraint cone in the framework of locally convex topological vector spaces.
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