On Lipschitz-like continuity of a class of set-valued mappings

Bednarczuk, Ewa M.; Minchenko, L. I.; Rutkowski, Krzysztof E.

doi:10.1080/02331934.2019.1696339

Cited by 16 publications

(38 citation statements)

References 29 publications

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“…By Remark 2.5 and Remark 4.1, there exist neighbourhoods U 2 (p) ⊂ U 1 (p) and V 1 (0), such that for all p ∈ U 2 (p) if (x, x ′ ) ∈ gphN (·; C(p))∩(x+V 1 (0),v −x+V 1 (0)), then I p (x) ⊂ Ip(x), x ∈ bd C(p) and x ′ = 0. Moreover, by [6,Theorem 6.40] 4 we have…”

Section: )mentioning

confidence: 99%

“…In view of Theorem 2.3 to prove (II) we need to show (I) and the condition (A). Condition (A) was investigated in details in [4,3] and it is discussed in Section 3. Condition (I) is proved in Proposition 6.3 in Section 6 with the help of a number of propositions proved in Section 4.…”

Section: Underlying Factsmentioning

confidence: 99%

“…3 We admit L = ∅. 4 Let us note that each equality constraint in the set C(·) can be represented as two inequalities,…”

Section: )mentioning

confidence: 99%

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On Lipschitz-Like Property for Polyhedral Moving Sets

Bednarczuk¹,

Rutkowski²

2019

SIAM J. Optim.

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In Hilbert space setting we prove local lipchitzness of projections onto parametric polyhedral sets represented as solutions to systems of inequalities and equations with parameters appearing both in left-hand-sides and right-hand-sides of the constraints. In deriving main results we assume that data are locally Lipschitz functions of parameter and the relaxed constant rank constraint qualification condition is satisfied.2010 Mathematics Subject Classification. 47N10,49J52,49J53,49K40,90C31.

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Section: )mentioning

confidence: 99%

Section: Underlying Factsmentioning

confidence: 99%

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On Lipschitz-Like Property for Polyhedral Moving Sets

Bednarczuk¹,

Rutkowski²

2019

SIAM J. Optim.

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“…In view of Theorem 1 to prove (II) we need to show (I) and the condition (A) . Condition (A) for C given by (2) was investigated in details in [3,5] and it is discussed in Sect. 3.…”

Section: Remarkmentioning

confidence: 99%

On Lipschitz Continuity of Projections onto Polyhedral Moving Sets

Bednarczuk

Rutkowski

2020

Appl Math Optim

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In Hilbert space setting we prove local lipchitzness of projections onto parametric polyhedral sets represented as solutions to systems of inequalities and equations with parameters appearing both in left- and right-hand sides of the constraints. In deriving main results we assume that data are locally Lipschitz functions of parameter and the relaxed constant rank constraint qualification condition is satisfied.

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“…In this paper, we focus on the derivation of sufficient criteria for the presence of so-called R-regularity of setvalued mappings, see Definition 2.2. This property, in turn, is beneficial in order to study Lipschitzian properties of marginal (or optimal value) functions and solution mappings in parametric optimization, see [6,32,36], and these features possess some extensions to bilevel optimization as well.…”

Section: Introductionmentioning

confidence: 99%

R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

Mehlitz

Minchenko

2021

Set-Valued Var. Anal

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The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

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On Lipschitz-like continuity of a class of set-valued mappings

Abstract: We study set-valued mappings defined by solution sets of parametric systems of equalities and inequalities. We prove Lipschitz-like continuity of these mappings under relaxed constant rank constraint qualification.2010 Mathematics Subject Classification. 41A50, 46C05, 49K27, 52A07, 90C31.

Cited by 16 publications

References 29 publications

On Lipschitz-Like Property for Polyhedral Moving Sets

On Lipschitz-Like Property for Polyhedral Moving Sets

On Lipschitz Continuity of Projections onto Polyhedral Moving Sets

R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

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