Error Bounds and Implicit Multifunction Theorem in Smooth Banach Spaces and Applications to Optimization

Ngai, Huynh Van; Théra, Michel

doi:10.1023/b:svan.0000023396.58424.98

Cited by 58 publications

(38 citation statements)

References 42 publications

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“…In recent years, error bound properties have been extensively studied (cf. [3,7,12,20,21,22,28,30,34] and references therein). In particular, studies on error bounds have been well carried out in terms of subdifferentials; these studies are mainly carried out in two directions of approach.…”

Section: D(x M (B)) ≤ L Y − B ∀(Y X) ∈ Gr(m ) Close To (B A)mentioning

confidence: 99%

Metric Subregularity and Calmness for Nonconvex Generalized Equations in Banach Spaces

Zheng¹,

Ng²

2010

SIAM J. Optim.

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Abstract. This paper concerns a generalized equation defined by a closed multifunction between Banach spaces, and we employ variational analysis techniques to provide sufficient and/or necessary conditions for a generalized equation to have the metric subregularity (i.e., local error bounds for the concerned multifunction) in general Banach spaces. Following the approach of Ioffe [Trans. Amer. Math. Soc., 251 (1979), pp. 61-69] who studied the numerical function case, our conditions are described in terms of coderivatives of the concerned multifunction at points outside the solution set. Motivated by the existing modulus representation and point-based criteria for the metric regularity, we establish the corresponding results for the metric subregularity. In the Asplund space case, sharper results are obtained.

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Section: D(x M (B)) ≤ L Y − B ∀(Y X) ∈ Gr(m ) Close To (B A)mentioning

confidence: 99%

Metric Subregularity and Calmness for Nonconvex Generalized Equations in Banach Spaces

Zheng¹,

Ng²

2010

SIAM J. Optim.

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“…The pioneering works of Robinson [12][13][14][15] gave good samples for implicit function theorems and their applications. Later, Ledyaev and Zhu [7], Ngai and Théra [11] established sufficient conditions for the metric regularity of the implicit multifunction (1.2) in terms of Fréchet coderivatives in Banach spaces with Fréchet-smooth Lipschitz bump functions. By using the scheme given by Yen [19], Lee et al [8] showed some sufficient conditions for the nonemptiness, the lower semicontinuity, the metric regularity and the Lipschitz-like property of the implicit multifunction (1.2) in terms of normal coderivatives in Asplund spaces.…”

Section: Introductionmentioning

confidence: 99%

Stability of random implicit multifunctions in separable Asplund spaces

Yang¹,

Xu²

2017

J. Nonlinear Sci. Appl.

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This paper is mainly devoted to present new sufficient conditions in terms of Fréchet coderivatives for the local metric regularity, the metric regularity, the Lipschitz-like property, the nonemptiness and the lower semicontinuity of random implicit multifunctions in separable Asplund spaces. An example is given to illustrate the above random implicit multifunction results. Some applications to stability analysis of solution maps for random parametric generalized equations are also given.

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“…This concept goes back to the surjectivity of a linear continuous mapping in the Banach Open Mapping Theorem and to its extension to nonlinear operators known as the Lyusternik & Graves Theorem ( [40], [27], see also [15]) and [21]). For a detailed account the reader is referred to the books or works of many researchers, [3], [5], [9], [10], [11], [12], [13], [17], [18], [20], [22], [29], [30], [32], [34], [36], [37], [40], [42], [43], [41], [44], [45], [46], [50], [51], [52], [57] and the references given therein for many theoretical results on the metric regularity as well as its various applications. Metric regularity or its equivalent notions (covering at a linear rate) [38] or Aubin property of the inverse [1] is now considered as a central concept in modern variational analysis (see the survey paper by Ioffe [34]).…”

Section: Introductionmentioning

confidence: 99%

Implicit multifunction theorems in complete metric spaces

2013

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In this paper, we establish some new characterizations of the metric regularity of implicit multifunctions in complete metric spaces by using the lower semicontinuous envelopes of the distance functions for set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the perturbation stability of implicit multifunctions.Mathematics Subject Classification: 49J52, 49J53, 90C30.

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Error Bounds and Implicit Multifunction Theorem in Smooth Banach Spaces and Applications to Optimization

Cited by 58 publications

References 42 publications

Metric Subregularity and Calmness for Nonconvex Generalized Equations in Banach Spaces

Metric Subregularity and Calmness for Nonconvex Generalized Equations in Banach Spaces

Stability of random implicit multifunctions in separable Asplund spaces

Implicit multifunction theorems in complete metric spaces

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