Lipschitz properties of the scalarization function and applications

Tammer, Christiane; Zălinescu, Constantin

doi:10.1080/02331930801951033

Cited by 62 publications

(39 citation statements)

References 16 publications

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“…PROOF. For x * ∈ A + (14) follows immediately from (13). If (2) we may take x, x * = 1 in the second term of (13), getting so (15).…”

Section: Proposition 4 Let a ⊂ X And Setmentioning

confidence: 98%

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Duality results involving functions associated to nonempty subsets of locally convex spaces

Zălinescu

2009

Rev. R. Acad. Cien. Serie A. Mat.

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In many papers on consumer theory and production analysis duality results between profit, revenue, cost, input, output and shortage functions are established. This functions are associated to certain subsets of R n . The aim of this paper is to study in a systematic way such duality results in locally convex spaces and to derive them under minimal hypotheses. Resultados sobre dualidad mediante funciones asociadas a subconjuntos no vacios de espacios localmente convexosResumen. En muchos artículos sobre teoría del consumo y análisis de la producción, se establecen resultados de dualidad entre beneficios y costes, e inversiones y rendimientos, proponiéndose diversas funciones de insuficiencia asociadas a ciertos subconjuntos de R n . El objeto de este trabajo es el estudio sistemático de dichos resultados de dualidad en espacios localmente convexos, y su obtención bajo condiciones mínimas.Moreover, for s ∈ R we set sA := {s}A and for x ∈ X we set x + A := {x} + A. We denote by i A or icr A, int A, cl A or A, conv A and aff A the intrinsic core, the interior, the closure, the convex hull and the affine hull of A ⊂ X, respectively; moreover, convA := cl(conv A) and affA := cl(aff A). To A ⊂ X we associate the sets

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“…PROOF. For x * ∈ A + (14) follows immediately from (13). If (2) we may take x, x * = 1 in the second term of (13), getting so (15).…”

Section: Proposition 4 Let a ⊂ X And Setmentioning

confidence: 98%

“…For this reason it is sufficient to study ϕ A,k or ψ A,k . A detailed study of the function ϕ A,k in the case A closed and A = A + R + k is performed in [9, Section 2.3]; other properties of ϕ A,k are established in [13].…”

Section: Gauges and Scalarization Functionsmentioning

confidence: 99%

Duality results involving functions associated to nonempty subsets of locally convex spaces

Zălinescu

2009

Rev. R. Acad. Cien. Serie A. Mat.

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“…See [11,22,23,27] for basic definitions and concepts for vector optimization. Also, see [19,20,34,41] for some scalarization methods for solving vector optimization problems with respect to fixed order structures and some properties of these scalarization methods.…”

Section: Preliminariesmentioning

confidence: 99%

Concepts for Approximate Solutions of Vector Optimization Problems with Variable Order Structures

Soleimani

Tammer

2014

Vietnam J. Math.

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In this paper, we introduce the concepts for approximately minimal, approximately nondominated solutions, and approximate minimizers of vector optimization problems with respect to variable order structures. In order to describe solution concepts, we use a set-valued map and this map is not necessarily a (pointed, convex) cone-valued map. We illustrate the different concepts for approximate solutions by several examples. Important properties of these three different kinds of approximate solutions of vector optimization problems with respect to variable order structures are discussed. Finally, we give some necessary conditions for approximate solutions of vector optimization problems with variable order structures using a vector-valued variant of Ekeland's variational principle.

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“…and p k defined in (2.3). Hence, we minimize here the well-known Tammer-Weidner-functional [11,14] which has many important properties, see for instance [17], and which turns out to be very useful in many theoretical and numerical approaches to multi-objective optimization. It is also the base of a very general scalarization in multiobjective optimization and covers several other scalarizations as special cases, see [5] and the recent review [1].…”

Section: Basic Definitions and Algorithm Mhtmentioning

confidence: 99%

Representation of the Pareto front for heterogeneous multi-objective optimization

2019

J. Appl. Numer. Optim.

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Optimization problems with multiple objectives which are expensive, i. e., where function evaluations are time consuming, are difficult to solve. Finding at least one locally optimal solution is already a difficult task. In case only one of the objective functions is expensive while the others are cheap, for instance, analytically given, this can be used in the optimization procedure. Using a trust-region approach and the Tammer-Weidner-functional for finding descent directions, in [19] an algorithm was proposed which makes use of the heterogeneity of the objective functions. In this paper, we present three heuristic approaches, which allow to find additional optimal solutions of the multiobjective optimization problem and by that representations at least of parts of the Pareto front. We present the related theoretical results as well as numerical results on some test instances.

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Lipschitz properties of the scalarization function and applications

Cited by 62 publications

References 16 publications

Duality results involving functions associated to nonempty subsets of locally convex spaces

Duality results involving functions associated to nonempty subsets of locally convex spaces

Concepts for Approximate Solutions of Vector Optimization Problems with Variable Order Structures

Representation of the Pareto front for heterogeneous multi-objective optimization

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