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.
Error bounds and Hölder metric subregularity
- Authors: Kruger, Alexander
- Date: 2015
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 705-736
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- Description: The Holder setting of the metric subregularity property of set-valued mappings between general metric or Banach/Asplund spaces is investigated in the framework of the theory of error bounds for extended real-valued functions of two variables. A classification scheme for the general Holder metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.
About subtransversality of collections of sets
- Authors: Kruger, Alexander , Luke, Russell , Thao, Nguyen
- Date: 2017
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 25, no. 4 (2017), p. 701-729
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: We provide dual sufficient conditions for subtransversality of collections of sets in an Asplund space setting. For the convex case, we formulate a necessary and sufficient dual criterion of subtransversality in general Banach spaces. Our more general results suggest an intermediate notion of subtransversality, what we call weak intrinsic subtransversality, which lies between intrinsic transversality and subtransversality in Asplund spaces.
Holder error bounds and holder calmness with applications to convex semi-infinite optimization
- Authors: Kruger, Alexander , Lopez, Marco , Yang, Xiaoqi , Zhu, Jiangxing
- Date: 2019
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 27, no. 4 (Dec 2019), p. 995-1023
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- Description: Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Holder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Holder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Holder calmness modulus of the argmin mapping in the framework of linear programming.
Directional metric pseudo subregularity of set-valued mappings: a general model
- Authors: Van Ngai, Huynh , Tron, Nguyen , Van Vu, Nguyen , Théra, Michel
- Date: 2020
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 61-87
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- Description: This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established. © 2019, Springer Nature B.V.
The radius of metric subregularity
- Authors: Dontchev, Asen , Gfrerer, Helmut , Kruger, Alexander , Outrata, Jiri
- Date: 2020
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 28, no. 3 (2020), p. 451-473, http://purl.org/au-research/grants/arc/DP160100854
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- Description: There is a basic paradigm, called here the radius of well-posedness, which quantifies the “distance” from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings. © 2019, Springer Nature B.V.
- Description: Funding details: Austrian Science Fund, FWF, P26132-N25, P26640-N25, P29190-N32 Funding details: National Science Foundation, NSF Funding details: Australian Research Council, ARC Funding details: Australian Research Council, ARC, DP160100854 Funding details: Austrian Science Fund, FWF Funding details: Universiteit Stellenbosch, US, P26640-N25 P26132-N25, BodyRef/PDF/11228_2019_Article_523.pdf Funding details: Grantová Agentura