$ V $-Moreau envelope of nonconvex functions on smooth Banach spaces

Messaoud Bounkhel; Messaoud Bounkhel

doi:10.3934/math.20241387

AIMS Mathematics

2024, Volume 9, Issue 10: 28589-28610. doi: 10.3934/math.20241387

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Research article

$ V $-Moreau envelope of nonconvex functions on smooth Banach spaces

Messaoud Bounkhel ^,

Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 24 August 2024 Revised: 15 September 2024 Accepted: 19 September 2024 Published: 09 October 2024
MSC : 34A60, 49J53

We continue the study of the properties of the $ V $-Moreau envelope and generalized $ (f, \lambda) $-projection that we started in ^[5]. In this paper, we study the differentiability and the regularity of the $ V $-Moreau envelope and the Hölder continuity of the generalized $ (f, \lambda) $-projection. Our results extend many existing results from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. Even on Hilbert spaces and for convex functions and sets, we derived new results.
Citation: Messaoud Bounkhel. $ V $-Moreau envelope of nonconvex functions on smooth Banach spaces[J]. AIMS Mathematics, 2024, 9(10): 28589-28610. doi: 10.3934/math.20241387

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Abstract

We continue the study of the properties of the $ V $-Moreau envelope and generalized $ (f, \lambda) $-projection that we started in ^[5]. In this paper, we study the differentiability and the regularity of the $ V $-Moreau envelope and the Hölder continuity of the generalized $ (f, \lambda) $-projection. Our results extend many existing results from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. Even on Hilbert spaces and for convex functions and sets, we derived new results.

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