On the Quantitative Solution Stability of Parameterized Set-Valued Inclusions

Uderzo, Amos

doi:10.1007/s11228-020-00571-z

Cited by 9 publications

(19 citation statements)

References 26 publications

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“…From the proof of Proposition 1 it should be evident that such a result embeds Theorem 3.1 in [29], which provides a sufficient condition for Lipschitz lower semicontinuity in the special case with R being given by R(p) = X, for every p ∈ P. Notice that, in such an event, Liplsc R( p, x) = 0 while, for every p ∈ P, the function x → dist (x, R(p)) vanishes. The condition in hypothesis (iii) can therefore be replaced with the mere positivity of |∇ x ν F | > ( p, x), as ν 1 reduces to ν F .…”

Section: Remarkmentioning

confidence: 62%

“…Discussions about this property and its relationships with other quantitative semicontinuity properties for set-valued mappings can be found, for instance, in [15,29]. For the purpose of the present analysis, it is relevant to observe that the requirement in (12) entails local solvability for problems (PSV) and nearness to the reference value x of at least some among the solutions to the perturbed problems.…”

Section: Parameterized Set-valued Inclusions With Moving Feasible Regionmentioning

confidence: 99%

“…It is plain to see that the search for ideal efficient solutions to problems (VOPp ) can be regarded in fact as a specialization of a more general class of problems involving set-valued mappings and cones, a kind of parameterized generalized equations which are referred to as set-valued inclusions in [29]. More precisely, given set-valued mappings R : P ⇒ X, F : P × X ⇒ Y and a nontrivial cone C ⊆ Y, these problems require to…”

Section: Introduction and Problem Statementmentioning

confidence: 99%

“…Set-valued inclusions, in simple as well as in parameterized form, have been recently studied from several viewpoints in [3,27,28,29,30]. The idea underlying the research exposed in the present paper is that useful insights into the stability behaviour of ideal efficient solutions can be obtained by refining in a proper way the study of solution stability of parameterized set-valued inclusions.…”

Section: Introduction and Problem Statementmentioning

confidence: 99%

See 3 more Smart Citations

A condition for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

Uderzo¹

2021

Preprint

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In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result specific for ideal efficient solutions to unperturbed problem is also discussed.

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Section: Remarkmentioning

confidence: 62%

Section: Parameterized Set-valued Inclusions With Moving Feasible Regionmentioning

confidence: 99%

Section: Introduction and Problem Statementmentioning

confidence: 99%

Section: Introduction and Problem Statementmentioning

confidence: 99%

See 2 more Smart Citations

A condition for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

Uderzo¹

2021

Preprint

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“…Conditions for solution existence, global error bounds and characterizations of the contingent cone to the solution set are also investigated in [30], following the convex analysis approach initiated in [6]. Besides, a perturbation analysis of the solution set to parameterized set-valued inclusions has been started in [31]. More precisely, given a set-valued mapping F : P × X ⇒ Y and a nonempty closed set C ⊂ Y , the following set-valued inclusion problem is considered there: find x ∈ X such that…”

Section: Introduction and Problem Statementmentioning

confidence: 99%

On Differential Properties of Multifunctions Defined Implicitly by Set-Valued Inclusions

Uderzo¹

2020

Preprint

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In the present paper, several properties concerning generalized derivatives of multifunctions implicitly defined by set-valued inclusions are studied by techniques of variational analysis. Set-valued inclusions are problems formalizing the robust fulfilment of cone constraint systems, whose data are affected by a "crude knowledge" of uncertain elements, so they can not be casted in traditional generalized equations.The focus of this study in on the first-order behaviour of the solution mapping associated with a parameterized set-valued inclusion, starting with Lipschitzian properties and then considering its graphical derivative. In particular, a condition for the Aubin continuity of the solution mapping is established in terms of outer prederivative of the set-valued mapping defining the inclusion. A large class of parameterized set-valued inculsions is singled out, whose solution mapping turns out to be convex. Some relevant consequences on the graphical derivative are explored. In the absence of that, formulae for the inner and outer approximation of the graphical derivative are provided by means of prederivatives of the problem data. A representation useful to calculate the coderivative of the solution mapping is also obtained via the subdifferential of a merit function.

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Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

Uderzo

2022

J Glob Optim

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In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result, specific for ideal efficient solutions to unperturbed problem and enhanced by related error bounds, is discussed. Some connections with the concept of robustness in multi-objective optimization are also sketched.

show abstract

On the Quantitative Solution Stability of Parameterized Set-Valued Inclusions

Cited by 9 publications

References 26 publications

A condition for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

A condition for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

On Differential Properties of Multifunctions Defined Implicitly by Set-Valued Inclusions

Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

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