Robust Constrained Best Approximation in Banach Spaces

The theory of constrained best approximation in Banach spaces has been systematically developed well over a decade and effective characterizations of best approximations have been known under some qualifications on the constraints. The existing theory yet does not explain how to characterize a best...

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Bibliographic Details
Published in:Set-valued and variational analysis Vol. 33; no. 3; p. 38
Main Authors: Salkhordeh, S., Mohebi, H.
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.09.2025
Springer Nature B.V
Subjects:
Analysis
Approximation
Banach spaces
Constraints
Lagrange multiplier
Mathematics
Mathematics and Statistics
Optimization
Research Article
Robustness
Uncertainty
Perturbation property
Robust optimization
Lagrange multipliers
90C25
Robust sharpened nondegeneracy
Strong conical hull intersection property
90C22
41A65
41A50
Robust constrained best approximation
41A29
ISSN:1877-0533, 1877-0541
Online Access:Get full text
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Summary:The theory of constrained best approximation in Banach spaces has been systematically developed well over a decade and effective characterizations of best approximations have been known under some qualifications on the constraints. The existing theory yet does not explain how to characterize a best approximation in the face of data uncertainty in the constraints. In this paper, we explain when the best approximation over uncertain nonconvex constraints in Asplund aces is immunized against data uncertainty. This study is done by characterizing the best approximation of the robust counterpart of the uncertain constrained best approximation problem where the constraints are enforced for all possible uncertainties within the prescribed uncertainty sets. We show that under the robust nondegeneracy constraint qualification the same kind of effective characterizations of constrained best approximation hold for the robust best approximation that is immunized against data uncertainty. Moreover, we present Lagrange multipliers characterizations of the robust constrained best approximation that is immunized against data uncertainty under the robust nondegeneracy constraint qualification by following the robust optimization approach. Some examples are given to illustrate the obtained results.
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ISSN:1877-0533
1877-0541
DOI:10.1007/s11228-025-00776-0