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 mappings provided that both primal and dual spaces are locally uniformly convex.
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References
Andrews, B., Hopper, C.: The Ricci Flow in Riemannian Geometry. Lecture Notes in Mathematics, vol. 2011. Springer, Berlin (2011)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Springer, Cham (2010)
Chen, Y., Kan, C., Song, W.: The Moreau envelope function and proximal mapping with respect to the Bregman distances in Banach spaces. Vietnam J. Math. 40, 181–199 (2012)
Cioranescu, I.: Geometry of Banach Spaces. Duality Mappings and Nonlinear Problems. Kluwer Acad. Publ, Dordrecht (1990)
Jourani, A., Thibault, L., Zagrodny, D.: Differential properties of the Moreau envelope. J. Funct. Anal. 266, 1185–1237 (2014)
Kartsatos, A.G.: A note on the duality mapping of a locally uniformly convex Banach space. Nonlinear Anal. 71, 5509–5512 (2009)
Kecis, I., Thibault, L.: Moreau envelopes of \(s\)-lower regular functions. Nonlinear Anal. 127, 157–181 (2015)
Laurent, P.: Approximation et Optimisation. Hermann, Paris (1972)
Moreau, J.-J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)
Nguyen, B.T., Khanh, P.D.: Lipschitz continuity of convex functions. Appl. Math. Optim. 84, 1623–1640 (2021)
Poliquin, R., Rockafellar, R.T.: Generalized Hessian properties of regularized nonsmooth functions. SIAM J. Optim. 6, 1121–1137 (1996)
Rockafellar, R.T.: Extension of Fenchel’s duality theorem for convex functions. Duke Math. J. 33, 81–89 (1966)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co. Inc, River Edge (2002)
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 mapping
- Convex function