Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness

Adán, Miguel; Novo, Vicente

doi:10.1016/s0377-2217(02)00444-7

Cited by 45 publications

(24 citation statements)

References 9 publications

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“…Lemma 10 (see [11]). Let be a nonempty subset of , and let be a nontrivial and convex cone with cor( ) ̸ = 0.…”

Section: Definition 7 a Set-valued Mapmentioning

confidence: 99%

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The Relationship between Two Kinds of Generalized Convex Set-Valued Maps in Real Ordered Linear Spaces

Zhou¹

2013

Abstract and Applied Analysis

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A new notion of the ic-cone convexlike set-valued map characterized by the algebraic interior and the vector closure is introduced in real ordered linear spaces. The relationship between the ic-cone convexlike set-valued map and the nearly cone subconvexlike set-valued map is established. The results in this paper generalize some known results in the literature from locally convex spaces to linear spaces.

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“…Lemma 10 (see [11]). Let be a nonempty subset of , and let be a nontrivial and convex cone with cor( ) ̸ = 0.…”

Section: Definition 7 a Set-valued Mapmentioning

confidence: 99%

“…Lemma 9 (see [11]). If is a nonempty convex set in and icr( ) ̸ = 0, then (a) vcl( ) is a convex set in ;…”

Section: Definition 7 a Set-valued Mapmentioning

confidence: 99%

The Relationship between Two Kinds of Generalized Convex Set-Valued Maps in Real Ordered Linear Spaces

Zhou¹

2013

Abstract and Applied Analysis

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“…Following Adán and Novo [1], we say that a set A is solid if corA = ∅, and relatively solid if icr A = ∅, respectively. [28, p. 3]).…”

Section: Generalized Interiority Conceptsmentioning

confidence: 99%

Unifying local–global type properties in vector optimization

Bagdasar

Popovici

2018

J Glob Optim

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It is well-known that all local minimum points of a semistrictly quasiconvex real-valued function are global minimum points. Also, any local maximum point of an explicitly quasiconvex real-valued function is a global minimum point, provided that it belongs to the intrinsic core of the function's domain. The aim of this paper is to show that these "local min -global min" and "local maxglobal min" type properties can be extended and unified by a single general localglobal extremality principle for certain generalized convex vector-valued functions with respect to two proper subsets of the outcome space. For particular choices of these two sets, we recover and refine several local-global properties known in the literature, concerning unified vector optimization (where optimality is defined with respect to an arbitrary set, not necessarily a convex cone) and, in particular, classical vector/multicriteria optimization.

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“…Lemma 1.1 [7] Let A be a nonempty subset in X, and P be a nontrivial convex cone in X. If cor(P ) = φ, then (i) A+cor(P )=cor(vcl(A + P )); (ii) vcl(cone(A) + P )=vcl(cone(A + P )); (iii) vcl(A + P )=vcl(A+cor(P )).…”

Section: Preliminariesmentioning

confidence: 99%

“…In Section 3, under the assumption of generalized subconvexlikeness, we establish optimality conditions in the sense of Benson proper efficiency, including scalarization, multiplier and saddle point theorems. Thus, some results of [7][8] are extended.…”

Section: Introductionmentioning

confidence: 97%

Benson proper efficiency for vector optimization of generalized subconvexlike set-valued maps in ordered linear spaces

Zhou

2010

J. Shanghai Univ.(Engl. Ed.)

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Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness

Cited by 45 publications

References 9 publications

The Relationship between Two Kinds of Generalized Convex Set-Valued Maps in Real Ordered Linear Spaces

The Relationship between Two Kinds of Generalized Convex Set-Valued Maps in Real Ordered Linear Spaces

Unifying local–global type properties in vector optimization

Benson proper efficiency for vector optimization of generalized subconvexlike set-valued maps in ordered linear spaces

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