Nonconvex separation theorems and some applications in vector optimization

Gerth, Christiane; Weidner, Petra

doi:10.1007/bf00940478

Cited by 327 publications

(71 citation statements)

References 7 publications

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“…But such methods rely heavily on some underlying convexity assumptions, which are hardly valid for many real problems. In this paper, by using Gerstewitz's function, which was used in Gerth and Weidner [19] to establish a useful nonconvex separation theorem, we develop another scalarization method for the vector-valued network equilibrium problem without any convexity assumptions. We denote…”

Section: Theorem 31 a Vector Flow V ∈ D Is An Equilibrium Pattern Flmentioning

confidence: 99%

Super Efficiency of Multicriterion Network Equilibrium Model and Vector Variational Inequality

Peng

et al. 2011

J Optim Theory Appl

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The super efficiency of a vector variational inequality is considered in this paper. We show that for both the single and multiple criteria cases, a network equilibrium model can be recast as super efficient solutions to a kind of variational inequality. For the network equilibrium model with a vector-valued cost function, we derive the necessary and sufficient condition in terms of the super efficiency of a vector variational inequality by using the Gerstewitz's function without any convex assumptions.

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Section: Theorem 31 a Vector Flow V ∈ D Is An Equilibrium Pattern Flmentioning

confidence: 99%

Super Efficiency of Multicriterion Network Equilibrium Model and Vector Variational Inequality

Peng

et al. 2011

J Optim Theory Appl

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“…Proposition 3.1 (Gerth and Weidner 1990). Let e ∈ int K , v ∈ Y and r ∈ R. The following statements hold:…”

Section: A New Nonlinear Scalarization Function and Its Propertiesmentioning

confidence: 99%

“…They introduced three kinds of well-posedness for set optimization problems. Using a generalization of the Gerstewitz function (Gerth and Weidner 1990;Hernández and Rodríguez-Marín 2007), they established some equivalence relationships between the well-posedness of a set optimization problem with cone-bounded objective values and the well-posedness of some scalar optimization problems. Also, they provided some criteria to the wellposedness for set optimization problems via a generalized forcing function.…”

Section: Introductionmentioning

confidence: 99%

Pointwise well-posedness and scalarization in set optimization

Khoshkhabar-amiranloo

Khorram

2015

Math Meth Oper Res

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In this paper, some notions of pointwise well-posedness for set optimization problems are introduced. Some relationships among these notions are established. Using a new nonlinear scalarization function, pointwise well-posed set optimization problems are characterized by means of a family of Tykhonov well-posed scalar optimization problems. Also, three classes of well-posed set optimization problems are identified.

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“…The next definition was firstly introduced in [5] (for more details, see [6]) in order to apply in optimization theory and then was used for equilibrium problems in [1]. Definition 1.9.…”

Section: Introductionmentioning

confidence: 99%

On the Relationship Between Topological Boundedness and the Ordered Boundedness for Solid Cones via Scalarizing

Karamian¹,

Lashkaripour²

2016

Int. J. of Pure and Appl. Math.

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In this paper, first we prove some lemmas, then by using the nonlinear scalarization function, we prove that there is no difference between the topological boundedness and the ordered boundedness. As an application, a fixed point theorem which is a new version of the main result obtained by Zangenehmehr et al (Positivity, 19, 333-340, 2015), by relaxing or weakening some assumptions, is established.

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Nonconvex separation theorems and some applications in vector optimization

Cited by 327 publications

References 7 publications

Super Efficiency of Multicriterion Network Equilibrium Model and Vector Variational Inequality

Super Efficiency of Multicriterion Network Equilibrium Model and Vector Variational Inequality

Pointwise well-posedness and scalarization in set optimization

On the Relationship Between Topological Boundedness and the Ordered Boundedness for Solid Cones via Scalarizing

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