Efficient and weak efficient points in vector optimization with generalized cone convexity

Adán, Miguel; Novo, Vicente

doi:10.1016/s0893-9659(03)80035-6

Cited by 11 publications

(9 citation statements)

References 6 publications

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“…Many works have been done with vector optimization problems under real linear spaces without any particular topology [1,2,[4][5][6][7][8][9][10][11][12][13][14]. However, only a few authors focus on nonconvex vector optimization problems [5,7,[13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Nonconvex Vector Optimization and Optimality Conditions for Proper Efficiency

Vaezpour¹,

Elham²,

Tavakoli³

2022

Int. J. Anal. Appl.

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In this paper, we consider, a new nonlinear scalarization function in vector spaces which is a generalization of the oriented distance function. Using the algebraic type of closure, which is called vector closure, we introduce the algebraic boundary of a set, without assuming any topology, in our context. Furthermore, some properties of this algebraic boundary set are given and present the concept of the oriented distance function via this set in the concept of vector optimization. We further investigate Q-proper efficiency in a real vector space, where Q is some nonempty (not necessarily convex) set. The necessary and sufficient conditions for Q-proper efficient solutions of nonconvex optimization problems are obtained via the scalarization technique. The scalarization technique relies on the use of two different scalarization functions, the oriented distance function and nonconvex separation function, which allow us to characterize the Q-proper efficiency in vector optimization with and without constraints.

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Section: Introductionmentioning

confidence: 99%

Nonconvex Vector Optimization and Optimality Conditions for Proper Efficiency

Vaezpour¹,

Elham²,

Tavakoli³

2022

Int. J. Anal. Appl.

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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%

“…In recent works related to vector optimization in real linear spaces (see, e.g., Adán and Novo [1][2][3][4], Bao and Mordukhovich [7], Khazayel et al [36], Novo and Zȃlinescu [40], Popovici [41], and Zhou, Yang and Peng [45]), the intrinsic core notion is studied in more detail. Having two real linear spaces X and E, a vector-valued objective function f : X → E, a certain set of constraints Ω ⊆ X , a convex (ordering) cone K ⊆ E (with possibly empty algebraic interior), a vector optimization problem is defined by f (x) → min w.r.t.…”

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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“…Convex Analysis and Vector Optimization under real vector spaces, without any topology, have been studied by various scholars in recent years [1,2,3,8,16,18,19,20,29,30]. Studying these problems opens new connections between Optimization, Functional Analysis, and Convex analysis.…”

Section: Introductionmentioning

confidence: 99%

“…To this end, the concepts of algebraic (relative) interior and vectorial closure have been investigated in the literature, and many results have been provided invoking these algebraic concepts; see e.g. [1,2,3,11,16,18,20,23,24,29,30] and the references therein. The main aim of this paper is to unifying vector optimization in real vector spaces with vector optimization in topological vector spaces.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of the core convex topology and its applications in vector optimization and convex analysis

Mohammadi¹,

Soleimani-damaneh²

2017

Preprint

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In this paper, the core convex topology on a real vector space X, which is constructed just by X operators, is investigated. This topology, denoted by τc, is the strongest topology which makes X into a locally convex space. It is shown that some algebraic notions (closure and interior) existing in the literature come from this topology. In fact, it is proved that algebraic interior and vectorial closure notions, considered in the literature as replacements of topological interior and topological closure, respectively, in vector spaces not necessarily equipped with a topology, are actually nothing else than the interior and closure with the respect to the core convex topology. We reconstruct the core convex topology using an appropriate topological basis which enables us to characterize its open sets.Furthermore, it is proved that (X, τc) is not metrizable when X is infinite-dimensional, and also it enjoys the Hine-Borel property. Using these properties, τc-compact sets are characterized and a characterization of finite-dimensionality is provided. Finally, it is shown that the properties of the core convex topology lead to directly extending various important results in convex analysis and vector optimization from topological vector spaces to real vector spaces.

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Efficient and weak efficient points in vector optimization with generalized cone convexity

Cited by 11 publications

References 6 publications

Nonconvex Vector Optimization and Optimality Conditions for Proper Efficiency

Nonconvex Vector Optimization and Optimality Conditions for Proper Efficiency

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

Reconstruction of the core convex topology and its applications in vector optimization and convex analysis

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