New Generalized Convexity Notion for Set-Valued Maps and Application to Vector Optimization

Theorems of the Alternative for Multivalued Mappings and Applications to Mixed Convex\Concave Systems of Inequalities

“…Since S i and M(C) are nonvoid, one has (M(C) + S) i = ∅. Picking a ∈ (M(C) + S) i we thus have (a, 0) ∈ E i .We conclude the proof by applying Theorem 3.1.Remark 3.5In the topological case, and for Z locally convex, intS = ∅, one can relax a bit the convexity of cone(M(C) + S) by assuming that clcone(M(C) + S) is convex[27, Corollary 2.2]. Then the conclusion of Theorem 3.5 holds, with λ continuous.In the single-valued case we getCorollary 3.3 Assume that S i = ∅ and let H : D → Z be a map such that cone(H(D) + S) is convex, D = ∅.…”

“…In the case when Problems like (σ a ) or (σ t ) are natural extensions of the systems of linear or convex inequalities ( [3, 12-14, 16, 19-21, 27],...). Statement (σ t ) has been considered in various situations ( [15,18,27],...). A typical example involving multivalued mappings in the statements (σ a ) or (σ t ) is presented in the proof of Proposition 3.1.…”

A note on “Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality” [Positivity. 18, 449–473(2014)]

“…Yang et al [1,2] introduced the concepts of generalized cone subconvexlike set-valued map and nearly cone-subconvexlike set-valued map. Sach [3] introduced a new convexity notion for set-valued maps, called ic-cone-convexlikeness. Xu and Song [4] obtained the following results: (a) when the ordering cone has nonempty interior, ic-cone-convexness is equivalent to near cone-subconvexlikeness; (b) when the ordering cone has empty interior, ic-cone-convexness implies near conesubconvexlikeness, a counter example is given to show that the converse implication is not true.…”

On super efficiency in set-valued optimization

“…Recently, Sach [13] introduced a new notion of generalized convexity for set-valued maps, called ic-cone-convexlikeness. It is a generalization of near-subconvexlike maps [6] , and can be used to investigate proper efficiency without requiring the cone to have nonempty interior.…”

“…Then y is called a super efficient point of M , written as y ∈ SE(M, D), if, for any neighbourhood V of 0 in Y , there exists a neighborhood U of 0 such that clcone(M − y) ∩ (U − D) ⊂ V. Remark 2.1 [2] . y ∈ SE(M, D) if and only if for each neighborhood V of 0 in Y , there exists a neighbourhood U of 0 such that cone(M − y) ∩ (U − D) ⊂ V. Definition 2.2 [13] . The set-valued map F : X → 2 Y is called ic-D-convexlike if intcone(F (X)+ D) is convex and F (X) + D ⊂ clintcone(F (X) + D).…”