Beyond the Fermat Optimality Rules
Abstract.
This work proposes a general framework for analyzing the behavior at its extrema of an extended real-valued function assumed neither convex nor differentiable and for which the classical Fermat rules of optimality do not apply. The tools used for building this framework are the notions of sup-subdifferential, recently introduced by two of the authors together with Kruger, and partial sup-subdifferentials. The sup-subdifferential is a nonempty enlargement of the Moreau–Rockafellar subdifferential that satisfies most of its fundamental properties and enjoys certain calculus rules. The partial sup-subdifferentials are obtained by breaking down the sup-subdifferential into one-dimensional components through basis elements and play the same role as the partial derivatives in the Fermat optimality rules.
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Acknowledgments.
We would like to express our gratitude to Constantin Zălinescu who adopted a much more rigorous approach to [1] and for suggestions leading to improving some statements, Jérôme Bolte for some well-pointed observations and questions regarding the sup-subdifferential, and Felipe Lara for bringing to our attention a possible connection between the sup-subdifferential and the asymptotic functions considered in [10]. We would also like to thank Alexander Kruger for his help in promoting this work. Comments and suggestions of four anonymous referees that contributed to improving the presentation (in particular to simplifying some proofs and correcting some errors in a previous version of the manuscript), and clarifying some issues are gratefully acknowledged.
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© 2025 Society for Industrial and Applied Mathematics.
History
Submitted: 8 June 2023
Accepted: 9 January 2025
Published online: 24 April 2025
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Funding Information
French-Australian Science and Innovation Collaboration (FASIC): 49763ZL
Agence Nationale de la Recherche (ANR): ANR-11-LABX-0056-LMH
Funding: This research was partially supported by the FASIC project 49763ZL and a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH (Grad and Théra).
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