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Error bounds revisited

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2022
  • Type: Text , Journal article
  • Relation: Optimization Vol. 71, no. 4 (2022), p. 1021-1053
  • Relation: https://purl.org/au-research/grants/arc/DP160100854
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  • Description: We propose a unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings. The function is not assumed to possess any particular structure apart from the standard assumptions of lower semicontinuity in the case of sufficient conditions and (in some cases) convexity in the case of necessary conditions. We expose the roles of the assumptions involved in the error bound assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. Employing special collections of slope operators, we introduce a succinct form of sufficient error bound conditions, which allows one to combine in a single statement several different assertions: nonlocal and local primal space conditions in complete metric spaces, and subdifferential conditions in Banach and Asplund spaces. © 2022 Informa UK Limited, trading as Taylor & Francis Group.

Isolated calmness and sharp minima via Hölder Graphical Derivatives

Necessary conditions for non-intersection of collections of sets

  • Authors: Bui, Hoa , Kruger, Alexander
  • Date: 2022
  • Type: Text , Journal article
  • Relation: Optimization Vol. 71, no. 1 (2022), p. 165-196
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

Optimality conditions, approximate stationarity, and applications 'a story beyond lipschitzness

  • Authors: Kruger, Alexander , Mehlitz, Patrick
  • Date: 2022
  • Type: Text , Journal article
  • Relation: ESAIM - Control, Optimisation and Calculus of Variations Vol. 28, no. (2022), p.
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: Approximate necessary optimality conditions in terms of Frechet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's variational principle, the fuzzy Frechet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established. © The authors.

Zero duality gap conditions via abstract convexity

  • Authors: Bui, Hoa , Burachik, Regina , Kruger, Alexander , Yost, David
  • Date: 2022
  • Type: Journal article
  • Relation: Optimization Vol. 71, no. 4 (2022), p. 811-847
  • Relation: https://purl.org/au-research/grants/arc/DP160100854
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  • Description: Using tools provided by the theory of abstract convexity, we extend conditions for zero duality gap to the context of non-convex and nonsmooth optimization. Mimicking the classical setting, an abstract convex function is the upper envelope of a family of abstract affine functions (being conventional vertical translations of the abstract linear functions). We establish new conditions for zero duality gap under no topological assumptions on the space of abstract linear functions. In particular, we prove that the zero duality gap property can be fully characterized in terms of an inclusion involving (abstract) (Formula presented.) -subdifferentials. This result is new even for the classical convex setting. Endowing the space of abstract linear functions with the topology of pointwise convergence, we extend several fundamental facts of functional/convex analysis. This includes (i) the classical Banach–Alaoglu–Bourbaki theorem (ii) the subdifferential sum rule, and (iii) a constraint qualification for zero duality gap which extends a fact established by Borwein, Burachik and Yao (2014) for the conventional convex case. As an application, we show with a specific example how our results can be exploited to show zero duality for a family of non-convex, non-differentiable problems. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

Enlargements of the moreau–rockafellar subdifferential

Gateaux differentiability revisited

Primal necessary characterizations of transversality properties

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Positivity Vol. 25, no. 2 (2021), p. 531-558
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Hölder case is given a special attention. Quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings as well as two nonlinear transversality properties of a convex set-valued mapping to a convex set in the range space are discussed. © 2020, Springer Nature Switzerland AG.

Transversality properties : primal sufficient conditions

  • 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.

Dual sufficient characterizations of transversality properties

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Positivity Vol. 24, no. 5 (2020), p. 1313-1359
  • Relation: https://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. Our aim is to develop a general scheme for quantitative analysis of several transversality properties within the same framework. We consider a general nonlinear setting and establish dual (subdifferential and normal cone) sufficient characterizations of transversality properties of collections of sets in Banach/Asplund spaces. Besides quantitative estimates for the rates/moduli of the corresponding properties, we establish here also estimates for the other parameters involved in the definitions, particularly the size of the neighbourhood where a property holds. Interpretations of the main general nonlinear characterizations for the case of Hölder transversality are provided. Some characterizations are new even in the linear setting. As an application, we provide dual sufficient conditions for nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe. © 2020, Springer Nature Switzerland AG.
  • Description: The research was supported by the Australian Research Council, Project DP160100854, and the European Union’s Horizon 2020 research and innovation programme under the Marie Sk

Geometric and metric characterizations of transversality properties

  • Authors: Bui, Hoa , Cuong, Nguyen , Kruger, Alexander
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Vietnam Journal of Mathematics Vol. 48, no. 2 (2020), p. 277-297
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  • Description: This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. We clarify quantitative relations between several geometric and metric characterizations of the transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings. We expose all the parameters involved in the definitions and characterizations and establish relations between them. This allows us to classify the quantitative geometric and metric characterizations of transversality and regularity, and subdivide them into two groups with complete exact equivalences between the parameters within each group and clear relations between the values of the parameters in different groups. © 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

Nonlinear transversality of collections of sets : dual space necessary characterizations

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Journal of Convex Analysis Vol. 27, no. 1 (2020), p. 285-306
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of `good arrangements' of collections of sets in normed spaces near a point in their intersection. Our aim is to study general nonlinear transversality properties. We focus on dual space (subdifferential and normal cone) necessary characterizations of these properties. As an application, we provide dual necessary conditions for the nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe.
  • Description: The research was supported by the Australian Research Council, project DP160100854. The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.

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

Extremality, stationarity and generalized separation of collections of sets

  • Authors: Bui, Hoa , Kruger, Alexander
  • Date: 2019
  • Type: Text , Journal article
  • Relation: Journal of Optimization Theory and Applications Vol. 182, no. 1 (2019), p. 211-264
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  • Description: The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analysed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining ‘extremal’ statements. For that, we examine and clarify quantitative relationships between the parameters involved in the respective definitions and statements. Some new characterizations of extremality properties are obtained.

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.

On semiregularity of mappings

  • 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
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  • 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.

About extensions of the extremal principle

  • Authors: Bui, Hoa , Kruger, Alexander
  • Date: 2018
  • Type: Text , Journal article
  • Relation: Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 215-242
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: In this paper, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets, and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions, and expand the applicability of the generalized separation results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity, and Lipschitz-like properties of set-valued mappings. © 2018, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

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.

Perturbation of error bounds

  • 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
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  • 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.

Set regularities and feasibility problems