Abstract
We study the notion of finitely determined functions defined on a topological vector space E equipped with a biorthogonal system. We prove that, for real-valued convex functions defined on a Banach space with a Schauder basis, the notion of finitely determined function coincides with the classical continuity but outside the convex case there are many finitely determined nowhere continuous functions. This notion will be used to obtain a necessary and sufficient condition for a convex function to attain a minimum at some point. An application to the Karush–Kuhn–Tucker theorem will be given.
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Acknowledgements
The third author was supported by the grants CONICYT-PFCHA/Doctorado Nacional/2018-21181905, Monge Invitation Programme of École Polytechnique, FONDECYT 1171854 and CMM Grant AFB170001.
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Communicated by Enrique A. Sanchez Perez.
The authors are grateful to the anonymous referee for these valuable remarks involving the following version of the paper.
This research has been conducted within the FP2M federation (CNRS FR 2036).
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Bachir, M., Fabre, A. & Tapia-García, S. Finitely determined functions. Adv. Oper. Theory 6, 28 (2021). https://doi.org/10.1007/s43036-020-00125-y
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DOI: https://doi.org/10.1007/s43036-020-00125-y
Keywords
- Schauder basis
- Convex optimization
- Finitely determined function
- Directional derivatives
- Karush–Kuhn–Tucker theorem