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Finite alternation theorems and a constructive approach to piecewise polynomial approximation in chebyshev norm

- Crouzeix, Jean-Pierre, Sukhorukova, Nadezda, Ugon, Julien

  • Authors: Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 123-147. http://purl.org/au-research/grants/arc/DP180100602
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  • Description: One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then propose an original approach to obtain new proofs of the well known necessary and sufficient optimality conditions. There are two main advantages of this approach. First of all, the proofs are intuitive and easy to understand. Second, these proofs are constructive and therefore they lead to new alternation-based algorithms. In the second part of this paper, we develop new local optimality conditions for free knot polynomial spline approximation. The proofs for free knot approximation are relying on the techniques developed in the first part of this paper. The piecewise polynomials are required to be continuous on the approximation segment. © 2020, Springer Nature B.V.

Enlargements of the moreau–rockafellar subdifferential

- Abbasi, Malek, Kruger, Alexander, Théra, Michel

The radius of metric subregularity

- Dontchev, Asen, Gfrerer, Helmut, Kruger, Alexander, Outrata, Jiri

  • 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

Alternative representations of the normal cone to the domain of supremum functions and subdifferential calculus

- Correa, Rafael, Hantoute, Abderrahim, Lopez, Marco

  • Authors: Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 683-699
  • Relation: http://purl.org/au-research/grants/arc/DP180100602
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  • Description: The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.

About intrinsic transversality of pairs of sets

- Kruger, Alexander

  • 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.

Transversality properties : primal sufficient conditions

- Cuong, Nguyen, Kruger, Alexander

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 29, no. 2 (2021), p. 221-256
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed. © 2020, Springer Nature B.V.

Stability analysis for parameterized variational systems with implicit constraints

- Benko, Matúš, Gfrerer, Helmut, Outrata, Jiri

  • Authors: Benko, Matúš , Gfrerer, Helmut , Outrata, Jiri
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 167-193
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  • Description: In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non- restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples. © 2019, The Author(s).

Directional metric pseudo subregularity of set-valued mappings: a general model

- Van Ngai, Huynh, Tron, Nguyen, Van Vu, Nguyen, Théra, Michel

Some new characterizations of intrinsic transversality in hilbert spaces

- Thao, Nguyen, Bui, Hoa, Cuong, Nguyen, Verhaegen, Michel

  • Authors: Thao, Nguyen , Bui, Hoa , Cuong, Nguyen , Verhaegen, Michel
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 5-39
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  • Description: Motivated by a number of questions concerning transversality-type properties of pairs of sets recently raised by Ioffe and Kruger, this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. New results in terms of normal vectors clarify the picture of intrinsic transversality, its variants and sufficient conditions for subtransversality, and unify several of them. For the first time, intrinsic transversality is characterized by an equivalent condition which does not involve normal vectors. This characterization offers another perspective on intrinsic transversality. As a consequence, the obtained results allow us to answer a number of important questions about transversality-type properties. © 2020, The Author(s).
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