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  • 0906 Electrical and Electronic Engineering
  • 0101 Pure Mathematics
  • Kruger, Alexander
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1Cibulka, Radek 1Fabian, Marian 1Plubtieng, Somyot 1Seangwattana, Thidaporn 1Thao, Nguyen
Subject
1Borwein-Preiss variational principle 1Gauge-type function 1Linear openness 1Metric regularity 1Metric semiregularity 1Normal cone 1Open mapping theorem 1Perturbation 1Set-valued perturbation 1Smooth variational principle 1Subdifferential 1Uniform regularity
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Full Text
2Yes 1No
Creator
1Cibulka, Radek 1Fabian, Marian 1Plubtieng, Somyot 1Seangwattana, Thidaporn 1Thao, Nguyen
Subject
1Borwein-Preiss variational principle 1Gauge-type function 1Linear openness 1Metric regularity 1Metric semiregularity 1Normal cone 1Open mapping theorem 1Perturbation 1Set-valued perturbation 1Smooth variational principle 1Subdifferential 1Uniform regularity
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About [q]-regularity properties of collections of sets

- Kruger, Alexander, Thao, Nguyen


  • 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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  • Reviewed:
  • 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.

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
  • Full Text:
  • Reviewed:
  • 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.
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Borwein-Preiss variational principle revisited

- Kruger, Alexander, Plubtieng, Somyot, Seangwattana, Thidaporn


  • Authors: Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn
  • Date: 2016
  • Type: Text , Journal article
  • Relation: Journal of Mathematical Analysis and Applications Vol. 435, no. 2 (2016), p. 1183-1193
  • Relation: http://purl.org/au-research/grants/arc/DP110102011
  • Full Text:
  • Reviewed:
  • Description: In this article, we refine and slightly strengthen the metric space version of the Borwein-Preiss variational principle due to Li and Shi (2000) [12], clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein and Zhu (2005) [4] and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein and Preiss (1987) [3] along several directions. (C) 2015 Elsevier Inc. All rights reserved.

Borwein-Preiss variational principle revisited

  • Authors: Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn
  • Date: 2016
  • Type: Text , Journal article
  • Relation: Journal of Mathematical Analysis and Applications Vol. 435, no. 2 (2016), p. 1183-1193
  • Relation: http://purl.org/au-research/grants/arc/DP110102011
  • Full Text:
  • Reviewed:
  • Description: In this article, we refine and slightly strengthen the metric space version of the Borwein-Preiss variational principle due to Li and Shi (2000) [12], clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein and Zhu (2005) [4] and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein and Preiss (1987) [3] along several directions. (C) 2015 Elsevier Inc. All rights reserved.

On semiregularity of mappings

- Cibulka, Radek, Fabian, Marian, Kruger, Alexander

  • Authors: Cibulka, Radek , Fabian, Marian , Kruger, Alexander
  • Date: 2019
  • Type: Text , Journal article
  • Relation: Journal of Mathematical Analysis and Applications Vol. 473, no. 2 (2019), p. 811-836
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
  • Full Text: false
  • Reviewed:
  • Description: There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part. © 2019 Elsevier Inc.

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