Abstract
It is shown that the Moreau envelope of a convex lower semicontinuous function on a real Banach space with strictly convex dual is Fréchet differentiable at every its minimizer, and continuously Fréchet differentiable at every its non-minimizer satisfying that the dual space is uniformly convex at every norm one element around its normalized gradient vector at those points. As an application, we obtain the continuous Fréchet differentiability of the Moreau envelope functions on Banach spaces with locally uniformly duals and the continuity of the corresponding proximal map**s provided that both primal and dual spaces are locally uniformly convex.
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Acknowledgements
The authors are grateful to the editor and two anonymous referees for constructive comments and suggestions, which greatly improved the paper. The research of the first author is funded by Ho Chi Minh City University of Education Foundation for Science and Technology under grant number CS.2021.19.01TD. The research of the second author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2020.21.
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Communicated by Nguyen Dong Yen.
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Khanh, P.D., Nguyen, B.T. Continuous Fréchet Differentiability of the Moreau Envelope of Convex Functions on Banach Spaces. J Optim Theory Appl 195, 1007–1018 (2022). https://doi.org/10.1007/s10957-022-02126-8
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DOI: https://doi.org/10.1007/s10957-022-02126-8
Keywords
- Strict convexity
- Local uniform convexity
- Fréchet differentiability
- Moreau envelope
- Proximal map**
- Convex function