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.
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.
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
Holder error bounds and holder calmness with applications to convex semi-infinite optimization
- Authors: Kruger, Alexander , Lopez, Marco , Yang, Xiaoqi , Zhu, Jiangxing
- Date: 2019
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 27, no. 4 (Dec 2019), p. 995-1023
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- Description: Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Holder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Holder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Holder calmness modulus of the argmin mapping in the framework of linear programming.
Metric regularity and systems of generalized equations
- Authors: Dmitruk, Andrei , Kruger, Alexander
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 342, no. 2 (2008), p. 864-873
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- Description: The paper is devoted to a revision of the metric regularity property for mappings between metric or Banach spaces. Some new concepts are introduced: uniform metric regularity and metric multi-regularity for mappings into product spaces, when each component is perturbed independently. Regularity criteria are established based on a nonlocal version of Lyusternik-Graves theorem due to Milyutin. The criteria are applied to systems of generalized equations producing some "error bound" type estimates. © 2007 Elsevier Inc. All rights reserved.