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Dual sufficient characterizations of transversality properties

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Positivity Vol. 24, no. 5 (2020), p. 1313-1359
  • Relation: https://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. Our aim is to develop a general scheme for quantitative analysis of several transversality properties within the same framework. We consider a general nonlinear setting and establish dual (subdifferential and normal cone) sufficient characterizations of transversality properties of collections of sets in Banach/Asplund spaces. Besides quantitative estimates for the rates/moduli of the corresponding properties, we establish here also estimates for the other parameters involved in the definitions, particularly the size of the neighbourhood where a property holds. Interpretations of the main general nonlinear characterizations for the case of Hölder transversality are provided. Some characterizations are new even in the linear setting. As an application, we provide dual sufficient conditions for nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe. © 2020, Springer Nature Switzerland AG.
  • Description: The research was supported by the Australian Research Council, Project DP160100854, and the European Union’s Horizon 2020 research and innovation programme under the Marie Sk

Error bounds revisited

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2022
  • Type: Text , Journal article
  • Relation: Optimization Vol. 71, no. 4 (2022), p. 1021-1053
  • Relation: https://purl.org/au-research/grants/arc/DP160100854
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  • Description: We propose a unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings. The function is not assumed to possess any particular structure apart from the standard assumptions of lower semicontinuity in the case of sufficient conditions and (in some cases) convexity in the case of necessary conditions. We expose the roles of the assumptions involved in the error bound assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. Employing special collections of slope operators, we introduce a succinct form of sufficient error bound conditions, which allows one to combine in a single statement several different assertions: nonlocal and local primal space conditions in complete metric spaces, and subdifferential conditions in Banach and Asplund spaces. © 2022 Informa UK Limited, trading as Taylor & Francis Group.

Geometric and metric characterizations of transversality properties

  • Authors: Bui, Hoa , Cuong, Nguyen , Kruger, Alexander
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Vietnam Journal of Mathematics Vol. 48, no. 2 (2020), p. 277-297
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  • Description: This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. We clarify quantitative relations between several geometric and metric characterizations of the transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings. We expose all the parameters involved in the definitions and characterizations and establish relations between them. This allows us to classify the quantitative geometric and metric characterizations of transversality and regularity, and subdivide them into two groups with complete exact equivalences between the parameters within each group and clear relations between the values of the parameters in different groups. © 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

Nonlinear transversality of collections of sets : dual space necessary characterizations

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Journal of Convex Analysis Vol. 27, no. 1 (2020), p. 285-306
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of `good arrangements' of collections of sets in normed spaces near a point in their intersection. Our aim is to study general nonlinear transversality properties. We focus on dual space (subdifferential and normal cone) necessary characterizations of these properties. As an application, we provide dual necessary conditions for the nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe.
  • Description: The research was supported by the Australian Research Council, project DP160100854. The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.

Primal necessary characterizations of transversality properties

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Positivity Vol. 25, no. 2 (2021), p. 531-558
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Hölder case is given a special attention. Quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings as well as two nonlinear transversality properties of a convex set-valued mapping to a convex set in the range space are discussed. © 2020, Springer Nature Switzerland AG.

Some new characterizations of intrinsic transversality in hilbert spaces

  • Authors: Thao, Nguyen , Bui, Hoa , Cuong, Nguyen , Verhaegen, Michel
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 5-39
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  • Description: Motivated by a number of questions concerning transversality-type properties of pairs of sets recently raised by Ioffe and Kruger, this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. New results in terms of normal vectors clarify the picture of intrinsic transversality, its variants and sufficient conditions for subtransversality, and unify several of them. For the first time, intrinsic transversality is characterized by an equivalent condition which does not involve normal vectors. This characterization offers another perspective on intrinsic transversality. As a consequence, the obtained results allow us to answer a number of important questions about transversality-type properties. © 2020, The Author(s).

Transversality properties : primal sufficient conditions

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 29, no. 2 (2021), p. 221-256
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed. © 2020, Springer Nature B.V.

Transversality, regularity and error bounds in variational analysis and optimisation

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