Proper Efficiency in Vector Optimization on Real Linear Spaces

Adán, Miguel; Novo, Vicente

doi:10.1007/s10957-004-1184-x

Cited by 7 publications

(7 citation statements)

References 1 publication

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“…In the following, we always assume that Y is a real vector space. For a nonempty subset A ⊂ Y , the vector closure of A is defined as follows (refer to [1,41,43]):…”

Section: Generalized Gerstewitz's Functions and Their Propertiesmentioning

confidence: 99%

Generalized Gerstewitz’s Functions and Vector Variational Principle for ϵ-Efficient Solutions in the Sense of Németh

Qiu

2018

Acta. Math. Sin.-English Ser.

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In this paper, we establish a partial order principle, which is useful to deriving vector Ekeland variational principle (denoted by EVP). By using the partial order principle and extending Gerstewitz's functions, we obtain a vector EVP for ǫ-efficient solutions in the sense of Németh, which essentially improves the earlier results by removing a usual assumption for boundedness of range of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.

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“…In the following, we always assume that Y is a real vector space. For a nonempty subset A ⊂ Y , the vector closure of A is defined as follows (refer to [1,41,43]):…”

Section: Generalized Gerstewitz's Functions and Their Propertiesmentioning

confidence: 99%

Generalized Gerstewitz’s Functions and Vector Variational Principle for ϵ-Efficient Solutions in the Sense of Németh

Qiu

2018

Acta. Math. Sin.-English Ser.

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“…→ , Definition 5 reduces to Definition 4.1 in [13]. When the linear spaces becomes a topological space, Definition 5 becomes Definition 2.2 in [4].…”

Section: Letmentioning

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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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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“…It is known that in vector optimization (see Jahn [34]) as well as in Image Space Analysis (ISA) in infinite dimensional linear spaces (see Giannessi [19,20] and references therein) difficulties may arise because of the possible non-solidness of ordering cones (for instance in the fields of optimal control, approximation theory, duality theory). Thus, it is of increasing interest to derive optimality conditions and duality results for such vector optimization problems using generalized interiority conditions (see, e.g., Adán and Novo [1][2][3][4], Bagdasar and Popovici [6], Bao and Mordukhovich [7], Borwein and Goebel [10], Borwein and Lewis [11], Grad [23,24], Grad and Pop [25], Khazayel et al [36], Zȃlinescu [43,44], and Cuong et al [14]). Such conditions can be formulated using the well-established generalized interiority notions given by quasi-interior, quasi-relative interior, algebraic interior (also known as core), relative algebraic interior (also known as intrinsic core, pseudo-relative interior or intrinsic relative interior).…”

Section: Introductionmentioning

confidence: 99%

Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

Günther

Khazayel

Tammer

2021

J Optim Theory Appl

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In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological/algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper efficiency concept based on generalized dilating cones which are relatively solid (i.e., their intrinsic cores are nonempty). Using linear functionals from the dual cone of the ordering cone, we are able to characterize the sets of (weakly, properly) efficient solutions under certain generalized convexity assumptions. Toward this end, we employ separation theorems that are working in the considered setting.

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Proper Efficiency in Vector Optimization on Real Linear Spaces

Cited by 7 publications

References 1 publication

Generalized Gerstewitz’s Functions and Vector Variational Principle for ϵ-Efficient Solutions in the Sense of Németh

Generalized Gerstewitz’s Functions and Vector Variational Principle for ϵ-Efficient Solutions in the Sense of Németh

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

Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

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