Set regularities and feasibility problems
- Authors: Kruger, Alexander , Luke, Russell , Thao, Nguyen
- Date: 2018
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 168, no. 1-2 (2018), p. 279-311
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms
Quantitative characterizations of regularity properties of collections of sets
- Authors: Kruger, Alexander , Thao, Nguyen
- Date: 2015
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 164, no. 1 (2015), p. 41-67
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: Several primal and dual quantitative characterizations of regularity properties of collections of sets in normed linear spaces are discussed. Relationships between regularity properties of collections of sets and those of set-valued mappings are provided.
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.
Stationarity and regularity of infinite collections of sets
- Authors: Kruger, Alexander , López, Marco
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 154, no. 2 (2012), p. 339-369
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces. © 2012 Springer Science+Business Media, LLC.
Stationarity and Regularity of Infinite Collections of Sets. Applications to Infinitely Constrained Optimization
- Authors: Kruger, Alexander , López, Marco
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 155, no. 2 (2012), p. 390-416
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López (J. Optim. Theory Appl. 154(2), 2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements-normals and/or subdifferentials.