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
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.
Fuzzy multiplier, sum and intersection rules in non-Lipschitzian settings : decoupling approach revisited
- Authors: Fabian, Marian , Kruger, Alexander , Mehlitz, Patrick
- Date: 2024
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 532, no. 2 (2024), p.
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: We revisit the decoupling approach widely used (often intuitively) in nonlinear analysis and optimization and initially formalized about a quarter of a century ago by Borwein & Zhu, Borwein & Ioffe and Lassonde. It allows one to streamline proofs of necessary optimality conditions and calculus relations, unify and simplify the respective statements, clarify and in many cases weaken the assumptions. In this paper we study weaker concepts of quasiuniform infimum, quasiuniform lower semicontinuity and quasiuniform minimum, putting them into the context of the general theory developed by the aforementioned authors. Along the way, we unify the terminology and notation and fill in some gaps in the general theory. We establish rather general primal and dual necessary conditions characterizing quasiuniform
Isolated calmness and sharp minima via Hölder Graphical Derivatives
- Authors: Kruger, Alexander , López, Marco , Yang, Xiaoqi , Zhu, Jiangxing
- Date: 2022
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 30, no. 4 (2022), p. 1423-1441
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper utilizes Hölder graphical derivatives for characterizing Hölder strong subregularity, isolated calmness and sharp minimum. As applications, we characterize Hölder isolated calmness in linear semi-infinite optimization and Hölder sharp minimizers of some penalty functions for constrained optimization. © 2022, The Author(s).
On the application of the SCD semismooth* newton method to variational inequalities of the second kind
- Authors: Gfrerer, Helmut , Outrata, Jiri , Valdman, Jan
- Date: 2022
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 30, no. 4 (2022), p. 1453-1484
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally, we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequality of the second kind, where one admits really large numbers of players (firms) and produced commodities. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.
On the isolated calmness property of implicitly defined multifunctions
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2023
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 30, no. 3 (2023), p. 1001-1023
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper deals with an extension of the available theory of SCD (subspace containing derivatives) mappings to mappings between spaces of different dimensions. This extension enables us to derive workable sufficient conditions for the isolated calmness of implicitly defined multifunctions around given reference points. This stability property differs substantially from isolated calmness at a point and, possibly in conjunction with the Aubin property, offers a new useful stability concept. The application area includes a broad class of parameterized generalized equations, where the respective conditions ensure a rather strong type of Lipschitzian behavior of their solution maps. © 2023 Heldermann Verlag. All rights reserved.
Radius theorems for subregularity in infinite dimensions
- Authors: Gfrerer, Helmut , Kruger, Alexander
- Date: 2023
- Type: Text , Journal article
- Relation: Computational Optimization and Applications Vol. 86, no. 3 (2023), p. 1117-1158
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper continues our previous work (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) on the radius of subregularity that was initiated by Asen Dontchev. We extend the results of (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) to general Banach/Asplund spaces and to other classes of perturbations, and sharpen the coderivative tools used in the analysis of the robustness of well-posedness of mathematical problems and related regularity properties of mappings involved in the statements. We also expand the selection of classes of perturbations, for which the formula for the radius of strong subregularity is valid. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Variational analysis of paraconvex multifunctions
- Authors: Van Ngai, Huynh , Tron, Nguyen , Van Vu, Nguyen , Théra, Michel
- Date: 2022
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 193, no. 1-3 (2022), p. 180-218
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: Our aim in this article is to study the class of so-called ρ- paraconvex multifunctions from a Banach space X into the subsets of another Banach space Y. These multifunctions are defined in relation with a modulus function ρ: X→ [0 , + ∞) satisfying some suitable conditions. This class of multifunctions generalizes the class of γ- paraconvex multifunctions with γ> 1 introduced and studied by Rolewicz, in the eighties and subsequently studied by A. Jourani and some others authors. We establish some regular properties of graphical tangent and normal cones to paraconvex multifunctions between Banach spaces as well as a sum rule for coderivatives for such class of multifunctions. The use of subdifferential properties of the lower semicontinuous envelope function of the distance function associated to a multifunction established in the present paper plays a key role in this study. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
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.