Semi-differentiability of the Marginal Mapping in Vector Optimization

Soleimani-damaneh, Majid; Zamani, Moslem

doi:10.1137/16m1092192

Cited by 18 publications

(11 citation statements)

References 16 publications

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“…The weak infimal set of a nonempty set A ⊂ Z is defined to be (ii) In the literature on differentiability properties of the optimal value function of a parametric family of vector optimization problems (see [23,24,33,34,36] and the references therein), it is always assumed the domination property: the range of each problem in the family is included in the conical extension of the set of optimal values. For instance, in the setting of problem (V P x ), where the concept of weak minimality is considered to define its solutions and optimal values, that assumption is formulated as follows: Clearly, that assumption implies the existence of the considered solutions, which is a too strong condition in the study of the optimal value function (see [3,28]).…”

Section: Definitionmentioning

confidence: 99%

“…Namely, the optimal value mapping is set-valued and involves some concepts of infimal point (see [6,12,22,23,36]), which are the vector counterpart to the notion of infimum of a set of real numbers. As a result, differential stability properties in vector optimization problems are formulated in terms of graphical and epigraphical derivatives and most of the obtained results require the so-called domination property, which implies the existence of exact solutions of the problem (see [23,24,33,34,36]). Again, this paper concerns with differential stability of vector optimization problems that would not satisfy the domination property (in particular with empty solution set).…”

Section: Introductionmentioning

confidence: 99%

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Differential stability properties in convex scalar and vector optimization

Gutiérrez

2021

Set-Valued Var. Anal

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This paper focuses on formulas for the ε-subdifferential of the optimal value function of scalar and vector convex optimization problems. These formulas can be applied when the set of solutions of the problem is empty. In the scalar case, both unconstrained problems and problems with an inclusion constraint are considered. For the last ones, limiting results are derived, in such a way that no qualification conditions are required. The main mathematical tool is a limiting calculus rule for the ε-subdifferential of the sum of convex and lower semicontinuous functions defined on a (non necessarily reflexive) Banach space. In the vector case, unconstrained problems are studied and exact formulas are derived by linear scalarizations. These results are based on a concept of infimal set, the notion of cone proper set and an ε-subdifferential for convex vector functions due to Taa.

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Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Differential stability properties in convex scalar and vector optimization

Gutiérrez

2021

Set-Valued Var. Anal

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“…Properties of the contingent derivatives of some types of proper perturbation maps of a parameterized optimization problem were discussed in [1,7,16,23,25]. Some results in the proto-differentiability and semidifferentiability of the perturbation maps were obtained in [11,13,17,26].…”

Section: Introductionmentioning

confidence: 99%

Sensitivity analysis in parametric vector optimization in Banach spaces viaτw-contingent derivatives

Le¹,

Pham²

2020

Turk J Math

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This paper is concerned with sensitivity analysis in parametric vector optimization problems via τ wcontingent derivatives. Firstly, relationships between the τ w-contingent derivative of the Borwein proper perturbation map and the τ w-contingent derivative of feasible map in objective space are considered. Then, the formulas for estimating the τ w-contingent derivative of the Borwein proper perturbation map via the τ w-contingent of the constraint map and the Hadamard derivative of the objective map are obtained.

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“…Another interesting topic in the primal space approach is to study the proto-differentiability of perturbation maps, introduced in [21]. Some developments on the proto-differentiability of perturbation maps of parametric vector optimization problems and related problems were obtained in [11,14,15,30,31].…”

Section: Introductionmentioning

confidence: 99%

On generalized $\tau^w$-contingent epiderivatives in parametric vector optimization problems

2020

Appl. Set-Valued Anal. Optim.

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This paper is concerned with the sensitivity analysis in reflexive Banach spaces. We first establish the formulae for estimating the generalized τ w-contingent epiderivatives of the efficient solution maps and the perturbation maps in parametric vector optimization problems. Then, an application to the semi-infinite programming is provided. Some examples are also given for illustrating the obtained results.

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Semi-differentiability of the Marginal Mapping in Vector Optimization

Cited by 18 publications

References 16 publications

Differential stability properties in convex scalar and vector optimization

Differential stability properties in convex scalar and vector optimization

Sensitivity analysis in parametric vector optimization in Banach spaces viaτw-contingent derivatives

On generalized $\tau^w$-contingent epiderivatives in parametric vector optimization problems

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