A unified scheme for scalarization in set optimization

Hā, Trúóng Xuân Dúc

doi:10.1080/02331934.2023.2171730

Cited by 4 publications

(3 citation statements)

References 42 publications

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“…It is obvious from Remarks 2.1 and 2.2 that the Bishop-Phelps functional φ ℓ and the Gerstewitz scalarizing functional ξ −C,k are −C representing functionals. In a recent paper by Ha [19], equality (3.1) is also called the cone representation property (P4), where ψ is named an abstract scalarizing function. + .…”

Section: Definitionmentioning

confidence: 99%

A unified approach to Bishop-Phelps and scalarizing functionals

2023

J. Appl. Numer. Optim.

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In this paper, functionals representing a negative convex cone as the solution set of an inequality are investigated. This general class of representing functionals includes various scalarizing functionals and Bishop-Phelps functionals. This unified approach extends some initial results to variable order structures, and additional properties of the representing functionals are given. The topics sublinearity, subdifferential, zeros, and separation are also treated in the context of representing functionals. Finally, the theory is applied to problems of vector and set optimization.

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Section: Definitionmentioning

confidence: 99%

A unified approach to Bishop-Phelps and scalarizing functionals

2023

J. Appl. Numer. Optim.

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“…It is known that Bishop-Phelps cones have a lot of useful properties and there are interesting applications in variational analysis and nonconvex optimization (see Phelps [32]) as well as in vector optimization (see, e.g., Eichfelder [33], Eichfelder and Kasimbeyli [17], Ha [34], Ha and Jahn [35], [31], Jahn [36], [3, p. 159-160], Kasimbeyli [10], Kasimbeyli and Kasimbeyli [37]).…”

Section: Introductionmentioning

confidence: 99%

Nonlinear strict cone separation theorems in real normed spaces

2024

J. Nonlinear Var. Anal.

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In this paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (reflexive) normed spaces. In essence, we follow the nonlinear and nonsymmetric separation approach developed by Kasimbeyli (2010, SIAM J. Optim. 20), which is based on augmented dual cones and Bishop-Phelps type (normlinear) separating functions. Compared to Kasimbeyli's separation theorem, we formulate our theorems for the separation of two cones under weaker conditions (concerning convexity and closedness requirements) with respect to the involved cones. By a new characterization of the algebraic interior of augmented dual cones in real normed spaces, we are able to establish relationships between our cone separation results and the results derived by Kasimbeyli (2010, SIAM

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“…[5] defined set order relations. One can see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] for further works based on these order relations, including existence theorems for minimal elements, scalarizations, derivatives and optimality conditions etc.…”

Section: Introductionmentioning

confidence: 99%

Relations Among Minimal Elements of a Family of Sets with Respect to Various Set Order Relations

ATASEVER GÜVENÇ

2023

Cumhuriyet Science Journal

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In this article, some set order relations given for set-optimization criterion which is one of the solution concept of set-valued optimization problems are considered. Minimal elements of a family of sets with respect to these set order relations are compared in detail. For this comparison relations between set order relations mentioned in this article are used. Also, in cases where a family of minimal sets is not a subset of the other one, examples are given.

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A unified scheme for scalarization in set optimization

Cited by 4 publications

References 42 publications

A unified approach to Bishop-Phelps and scalarizing functionals

A unified approach to Bishop-Phelps and scalarizing functionals

Nonlinear strict cone separation theorems in real normed spaces

Relations Among Minimal Elements of a Family of Sets with Respect to Various Set Order Relations

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