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TRANSVERSALITY, REGULARITY AND ERROR BOUNDS IN VARIATIONAL ANALYSIS AND OPTIMISATION

Part of: Existence theories

Published online by Cambridge University Press:  09 June 2022

N. D. CUONG Open the ORCID record for N. D. CUONG [Opens in a new window]
N. D. CUONG*
Affiliation:
Centre for Informatics and Applied Optimization (CIAO), School of Engineering, Information Technology and Physical Sciences, Federation University Australia, POB 663, Ballarat, Victoria 3350, Australia and Department of Mathematics, College of Natural Sciences, Can Tho University, Can Tho, Vietnam
*
e-mail: [email protected]
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Abstract

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Keywords

variational analysistransversalitymetric regularityerror bounds

MSC classification

Primary: 49J53: Set-valued and variational analysis
Secondary: 65K10: Optimization and variational techniques 90C31: Sensitivity, stability, parametric optimization
Type
Abstracts of Australasian PhD Theses
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 167 - 169
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Transversality properties of collections of sets, regularity properties of set-valued mappings and error bounds of extended-real-valued functions lie at the core of variational analysis and optimisation because of their importance for stability analysis, constraint qualifications, qualification conditions in coderivative and subdifferential calculus, and convergence analysis of computational algorithms. The thesis is devoted to the investigation of several research questions related to the aforementioned properties and their applications in several optimisation problems. The main tools of the analysis are standard techniques of modern variational analysis (see [Reference Clarke2, Reference Ioffe10, Reference Mordukhovich11, Reference Rockafellar and Wets13]).

Quantitative analysis of transversality properties of collections of sets is investigated in the convex and nonconvex, linear and nonlinear settings by employing conventional tools of generalisation differentiation. The quantitative relations between transversality and 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 are discussed. These results are presented in recent publications [Reference Bui, Cuong and Kruger1, Reference Cuong and Kruger3–Reference Cuong and Kruger6, Reference Thao, Bui, Cuong and Verhaegen14].

We study theoretical and applied aspects of a new property called ‘linear semitransversality of collections of set-valued mappings’ in metric spaces [Reference Cuong and Kruger9]. The property can be seen as a generalisation of the conventional semitransversality of collections of sets and the negation of the corresponding stationarity which is a weaker property than the extremality of collections of set-valued mappings [Reference Mordukhovich, Treiman and Zhu12]. Necessary and sufficient conditions in terms of primal and dual objects and quantitative connections with the semiregularity of set-valued mappings are formulated. The results are applied to optimality conditions for a multiobjective optimisation problem with geometric constraints.

We examine a comprehensive (that is, not assuming the mapping to have any particular structure) view on the regularity theory of set-valued mappings and clarify the relationships between the existing primal and dual quantitative necessary and sufficient conditions including their hierarchy [Reference Cuong and Kruger7]. The typical sequence of regularity assertions, often hidden in the proofs, and the roles of the assumptions involved in the assertions, in particular, on the underlying space (general metric, normed, Banach or Asplund) are exposed. As a consequence, we formulate primal and dual conditions for the conventional metric regularity and subregularity properties as well as stability properties of solution mappings to inclusions.

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 [Reference Cuong and Kruger8]. 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. 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. As a consequence, the error bound theory is applied to characterise subregularity of set-valued mappings, and calmness of the solution mapping in convex semi-infinite optimisation problems for which linear perturbations of the objective function and continuous perturbations of the right-hand side of the constraint system are allowed.

Footnotes

Thesis submitted to Federation University in December 2020; degree approved on 24 May 2021; principal supervisor Alexander Kruger, associate supervisor David Yost.

References

Bui, H. T., Cuong, N. D. and Kruger, A. Y., ‘Transversality of collections of sets: geometric and metric characterizations’, Vietnam J. Math. 48(2) (2020), 277–297.Google Scholar
Clarke, F., Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264 (Springer, London, 2013).CrossRefGoogle Scholar
Cuong, N. D. and Kruger, A. Y., ‘Dual sufficient characterizations of transversality properties’, Positivity 24(5) (2020), 1313–1359.CrossRefGoogle Scholar
Cuong, N. D. and Kruger, A. Y., ‘Nonlinear transversality of collections of sets: dual space necessary characterizations’, J. Convex Anal. 27(1) (2020), 287–308.Google Scholar
Cuong, N. D. and Kruger, A. Y., ‘Primal necessary characterizations of transversality properties’, Positivity 25(2) (2021), 531–558.CrossRefGoogle Scholar
Cuong, N. D. and Kruger, A. Y., ‘Transversality properties: primal sufficient conditions’, Set-Valued Var. Anal. 29(2) (2021), 221–256.CrossRefGoogle Scholar
Cuong, N. D. and Kruger, A. Y., ‘Uniform regularity of set-valued mappings and stability of implicit multifunctions’, J. Nonsmooth Anal. Optim. 2 (2021), Article no. 6599, 24 pages.Google Scholar
Cuong, N. D. and Kruger, A. Y., ‘Error bounds revisited’, Optimization 71(4) (2022), 1021–1053.CrossRefGoogle Scholar
Cuong, N. D. and Kruger, A. Y., ‘Semitransversality of collections of set-valued mappings’. In preparation.Google Scholar
Ioffe, A. D., Variational Analysis of Regular Mappings. Theory and Applications, Springer Monographs in Mathematics (Springer, Cham, 2017).Google Scholar
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Mordukhovich, B. S., Treiman, J. S. and Zhu, Q. J., ‘An extended extremal principle with applications to multiobjective optimization’, SIAM J. Optim. 14(2) (2003), 359–379.CrossRefGoogle Scholar
Rockafellar, R. T. and Wets, R. J., Variational Analysis (Springer, Berlin, 1998).CrossRefGoogle Scholar
Thao, N. H., Bui, T. H., Cuong, N. D. and Verhaegen, M., ‘Some new characterizations of intrinsic transversality in Hilbert spaces’, Set-Valued Var. Anal. 28(1) (2020), 5–39.CrossRefGoogle Scholar