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
- Full Text: false
- Reviewed:
- 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.
Regularity of collections of sets and convergence of inexact alternating projections
- Authors: Kruger, Alexander , Thao, Nguyen
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 23, no. 3 (2016), p. 823-847
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text: false
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- Description: We study the usage of regularity properties of collections of sets in convergence analysis of alternating projection methods for solving feasibility problems. Several equivalent characterizations of these properties are provided. Two settings of inexact alternating projections are considered and the corresponding convergence estimates are established and discussed.
About [q]-regularity properties of collections of sets
- Authors: Kruger, Alexander , Thao, Nguyen
- Date: 2014
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 416, no. 2 (2014), p. 471-496
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: We examine three primal space local Holder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed.
- Description: We examine three primal space local Holder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed. (C) 2014 Elsevier Inc. All rights reserved.
Error bounds : Necessary and sufficient conditions
- Authors: Fabian, Marian , Henrion, René , Kruger, Alexander , Outrata, Jiri
- Date: 2010
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 18, no. 2 (2010), p. 121-149
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- Description: The paper presents a general classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several derivative-like objects both from the primal as well as from the dual space are used to characterize the error bound property of extended-real-valued functions on a Banach space. © 2010 Springer Science+Business Media B.V.