About extensions of the extremal principle
- Authors: Bui, Hoa , Kruger, Alexander
- Date: 2018
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 215-242
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: In this paper, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets, and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions, and expand the applicability of the generalized separation results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity, and Lipschitz-like properties of set-valued mappings. © 2018, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
About intrinsic transversality of pairs of sets
- Authors: Kruger, Alexander
- Date: 2018
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 26, no. 1 (2018), p. 111-142
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The article continues the study of the ‘regular’ arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterizations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.
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.
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.
Necessary conditions for non-intersection of collections of sets
- Authors: Bui, Hoa , Kruger, Alexander
- Date: 2022
- Type: Text , Journal article
- Relation: Optimization Vol. 71, no. 1 (2022), p. 165-196
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections. © 2021 Informa UK Limited, trading as Taylor & Francis Group.