Proper Efficiency in Vector Optimization on Real Linear Spaces

Adán, Miguel; Novo, Vicente

doi:10.1023/b:jota.0000037602.13941.ed

Cited by 43 publications

(36 citation statements)

References 11 publications

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“…Moreover, assume that Y is a locally convex space in Corollary 4.2. Then the result of Corollary 4.2 remains true if the condition (b) is replaced by the following condition: When cor(K) = ∅, where cor(K) (see [1,17]) denotes the algebraic interior of K, we can easily obtain the following result. …”

Section: Remark 43mentioning

confidence: 94%

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Vectorial variational principle with variable set-valued perturbation

Zhang

Qiu

2015

Acta. Math. Sin.-English Ser.

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We give a general vectorial Ekeland's variational principle, where the objective function is defined on an F-type topological space and taking values in a pre-ordered real linear space. Being quite different from the previous versions of vectorial Ekeland's variational principle, the perturbation in our version is no longer only dependent on a fixed positive vector or a fixed family of positive vectors. It contains a family of set-valued functions taking values in the positive cone and a family of subadditive functions of topology generating quasi-metrics. Hence, the direction of the perturbation in the new version is a family of variable subsets which are dependent on the objective function values. The general version includes and improves a number of known versions of vectorial Ekeland's variational principle. From the general Ekeland's principle, we deduce the corresponding versions of Caristi-Kirk's fixed point theorem and Takahashi's nonconvex minimization theorem. Finally, we prove that all the three theorems are equivalent to each other.

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Section: Remark 43mentioning

confidence: 94%

“…For subsets of real linear spaces, we recall some concepts of vectorial closedness (see, e.g., [1,17,27]). Let A be a nonempty subset of a real linear space Y .…”

Section: Definition 22mentioning

confidence: 99%

Vectorial variational principle with variable set-valued perturbation

Zhang

Qiu

2015

Acta. Math. Sin.-English Ser.

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“…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 )). Lemma 1.2 [8] Let A be convex in X and cor(A) = φ. Then (i) vcl(A) is convex;…”

Section: Preliminariesmentioning

confidence: 99%

“…Lemma 1.3 [8] If K is a convex cone in Y , then cor(K) = φ if and only if cor(K + ) = φ. Lemma 1.4 [5] Let K be a nontrivial convex cone in Y and cor(K) = φ. If y ∈ cor(K) and y * ∈ K + \ {o}, then ≺ y, y * > 0.…”

Section: Preliminariesmentioning

confidence: 99%

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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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Nonlinear Multiobjective Programming

Engau

2011

Wiley Encyclopedia of Operations Research and Management Science

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This introductory treatment of the theory and methodology of nonlinear multiobjective programming offers an annotated bibliographic overview of various concepts and approaches to solve continuous deterministic multicriteria optimization problems. First, the notions of Pareto optimality, efficiency, and nondominance are discussed and then related to approximate, epsilon, and proper efficiency. Second, after highlighting similarities and pointing out differences to first‐ and second‐order optimality conditions in classical nonlinear programming, several other topics including duality and sensitivity are also mentioned. Third, a number of solution approaches and generating methods are described with a particular focus on scalarizations using objective aggregation, prioritization, and norm or distance minimization to some utopian reference point.

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Proper Efficiency in Vector Optimization on Real Linear Spaces

Cited by 43 publications

References 11 publications

Vectorial variational principle with variable set-valued perturbation

Vectorial variational principle with variable set-valued perturbation

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

Nonlinear Multiobjective Programming

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