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90103 Numerical and Computational Mathematics
50906 Electrical and Electronic Engineering
30802 Computation Theory and Mathematics
3Error bound
3Metric subregularity
3Normal cone
3Subdifferential
2Calmness
2Ekeland variational principle
2Slope
10101 Pure Mathematics
1Alternating projections
1Asplund spaces
1CHIP
1Clarke regularity
1Coderivative
1Collections of sets
1Directional metric regularity

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Error bounds and metric subregularity

**Authors:**Kruger, Alexander**Date:**2015**Type:**Text , Journal article**Relation:**Optimization Vol. 64, no. 1 (2015), p. 49-79**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**Necessary and sufficient criteria for metric subregularity (or calmness) of set-valued mappings between general metric or Banach spaces are treated in the framework of the theory of error bounds for a special family of extended real-valued functions of two variables. A classification scheme for the general error bound and metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.

**Authors:**Kruger, Alexander**Date:**2015**Type:**Text , Journal article**Relation:**Optimization Vol. 64, no. 1 (2015), p. 49-79**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**Necessary and sufficient criteria for metric subregularity (or calmness) of set-valued mappings between general metric or Banach spaces are treated in the framework of the theory of error bounds for a special family of extended real-valued functions of two variables. A classification scheme for the general error bound and metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.

Metric Regularity of the Sum of Multifunctions and Applications

- Van Ngai, Huynh, Tron, Nguyen Tron, Thera, Michel

**Authors:**Van Ngai, Huynh , Tron, Nguyen Tron , Thera, Michel**Date:**2014**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 160, no. 2 (2014), p. 355-390**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:**false**Reviewed:****Description:**The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported. © 2013 Springer Science+Business Media New York.

Perturbation of error bounds

- Kruger, Alexander, López, Marco, Théra, Michel

**Authors:**Kruger, Alexander , López, Marco , Théra, Michel**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

**Authors:**Kruger, Alexander , López, Marco , Théra, Michel**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Quantitative characterizations of regularity properties of collections of sets

- Kruger, Alexander, Thao, Nguyen

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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

- 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**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.

**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.

An induction theorem and nonlinear regularity models

- Khanh, Phan, Kruger, Alexander, Thao, Nguyen

**Authors:**Khanh, Phan , Kruger, Alexander , Thao, Nguyen**Date:**2015**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 25, no. 4 (2015), p. 2561-2588**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**A general nonlinear regularity model for a set-valued mapping F : X x R+ paired right arrows Y, where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik, and Graves. Namely, we revise the induction theorem from Khanh [J. Math. Anal. Appl., 118 (1986), pp. 519-534] and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution for the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping F : X paired right arrows Y. An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.

**Authors:**Khanh, Phan , Kruger, Alexander , Thao, Nguyen**Date:**2015**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 25, no. 4 (2015), p. 2561-2588**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**A general nonlinear regularity model for a set-valued mapping F : X x R+ paired right arrows Y, where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik, and Graves. Namely, we revise the induction theorem from Khanh [J. Math. Anal. Appl., 118 (1986), pp. 519-534] and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution for the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping F : X paired right arrows Y. An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.

Directional Holder metric regularity

- Ngai, Huynh Van, Tron, Nguyen Huu, Thera, Michel

**Authors:**Ngai, Huynh Van , Tron, Nguyen Huu , Thera, Michel**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 785-819**Full Text:****Reviewed:****Description:**This paper sheds new light on regularity of multifunctions through various characterizations of directional Holder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Holder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Holder/Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.

**Authors:**Ngai, Huynh Van , Tron, Nguyen Huu , Thera, Michel**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 785-819**Full Text:****Reviewed:****Description:**This paper sheds new light on regularity of multifunctions through various characterizations of directional Holder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Holder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Holder/Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.

Nonlinear metric subregularity

**Authors:**Kruger, Alexander**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 820-855**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this article, we investigate nonlinear metric subregularity properties of set-valued mappings between general metric or Banach spaces. We demonstrate that these properties can be treated in the framework of the theory of (linear) error bounds for extended real-valued functions of two variables developed in Kruger (Error bounds and metric subregularity. Optimization 64(1):49-79, 2015). Several primal and dual space local quantitative and qualitative criteria of nonlinear metric subregularity are formulated. The relationships between the criteria are established and illustrated.

**Authors:**Kruger, Alexander**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 820-855**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this article, we investigate nonlinear metric subregularity properties of set-valued mappings between general metric or Banach spaces. We demonstrate that these properties can be treated in the framework of the theory of (linear) error bounds for extended real-valued functions of two variables developed in Kruger (Error bounds and metric subregularity. Optimization 64(1):49-79, 2015). Several primal and dual space local quantitative and qualitative criteria of nonlinear metric subregularity are formulated. The relationships between the criteria are established and illustrated.

Directional metric regularity of multifunctions

- Ngai, Huynh Van, Thera, Michel

**Authors:**Ngai, Huynh Van , Thera, Michel**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics of Operations Research Vol. 40, no. 4 (2015), p. 969-991**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.**Description:**In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity. © 2015 INFORMS.

**Authors:**Ngai, Huynh Van , Thera, Michel**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics of Operations Research Vol. 40, no. 4 (2015), p. 969-991**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.**Description:**In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity. © 2015 INFORMS.

Set regularities and feasibility problems

- Kruger, Alexander, Luke, Russell, Thao, Nguyen

**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**Full Text:****Reviewed:****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

**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**Full Text:****Reviewed:****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

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