Stability of Error Bounds for Convex Constraint Systems in Banach Spaces

Kruger, Alexander Y.; Ngai, Huynh Van; Théra, Michel

doi:10.1137/100782206

Cited by 52 publications

(54 citation statements)

References 23 publications

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“…The reader is addressed to the monographs [6,12,16,18] for details and references about calmness and Aubin properties. The existing relationship between the calmness property and local error bounds is well known (see, e.g., [1,15]). As far as calmness plays an important role in relation to issues from optimization (theory and algorithms), one can find in the literature deep contributions to the analysis of this property in different linear and nonlinear frameworks, mainly devoted to the calmness of feasible set mappings under right-hand-side perturbations; see, e.g., [7,10,13,14].…”

Section: Introductionmentioning

confidence: 99%

Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization

et al. 2014

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Artículo de publicación ISIThis paper was originally motivated by the problem of providing a point-based formula (only involving the nominal data, and not data in a neighborhood) for estimating the calmness modulus of the optimal set mapping in linear semi-infinite optimization under perturbations of all coefficients. With this aim in mind, the paper establishes as a key tool a basic result on finite-valued convex functions in the -dimensional Euclidean space. Specifically, this result provides an upper limit characterization of the boundary of the subdifferential of such a convex function. When applied to the supremum function associated with our constraint system, this characterization allows us to derive an upper estimate for the aimed calmness modulus in linear semi-infinite optimization under the uniqueness of nominal optimal solution.Fondecyt 111001

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Section: Introductionmentioning

confidence: 99%

Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization

et al. 2014

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“…Statement (i) in this theorem comes from [16, Propositions 1, 11 and 5(ii)], whereas (ii) follows directly from [17,Theorem 1]. In it, we have taken into account the well-known relationship between clmF(b, x) and the error bound modulus of g at x, specifically…”

Section: Preliminaries On the Feasible Set Mappingmentioning

confidence: 99%

“…3), and then, taking also (6) and (7) into account, we obtain inequality (8). (ii) Equality (9) is held under convexity, even without differentiability assumptions on the f i 's (see again [17,Theorem 1]), in which case, ∂g stands for the usual subdifferential of convex analysis.…”

Section: Preliminaries On the Feasible Set Mappingmentioning

confidence: 99%

“…Comprehensive studies on calmness and other variational properties for generic multifunctions can be traced out from the monographs [7][8][9][10]; see also [11][12][13][14] in relation to the calmness of constraint systems in the context of RHS perturbations; where calmness translates into the existence of a local error bound for the corresponding supremum function (see [15][16][17]). Other subdifferential approaches to calmness/local error bounds can be found in [18,19].…”

Section: Introductionmentioning

confidence: 99%

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Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming

Cánovas

Henrion

López

et al. 2015

J Optim Theory Appl

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With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions, which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying special attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a Karush-Kuhn-Tucker index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems, without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given. B M. A. López marco.antonio@ua.es M. J. Cánovas canovas@umh.es R. Henrion henrion@wias-berlin.de

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“…[22,23]. Stability and some other properties of error bounds are examined in [27,32,33,34,46]. Several conditions using subdifferential operators or directional derivatives and ensuring the error bound property in Banach spaces have been established, for example, in [10,41,31].…”

mentioning

confidence: 99%

Directional Metric Regularity of Multifunctions

Ngai

Théra

2015

Mathematics of OR

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In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.

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Stability of Error Bounds for Convex Constraint Systems in Banach Spaces

Cited by 52 publications

References 23 publications

Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization

Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization

Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming

Directional Metric Regularity of Multifunctions

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