Regularization in Banach Spaces with Respect to the Bregman Distance

Mohamed Soueycatt (), Yara Mohammad () and Yamar Hamwi ()
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Mohamed Soueycatt: AL-Andalus University for Medical Sciences
Yara Mohammad: AL-Andalus University for Medical Sciences
Yamar Hamwi: Tishreen University

Journal of Optimization Theory and Applications, 2020, vol. 185, issue 2, No 2, 327-342

Abstract: Abstract The Moreau envelope, also known as Moreau–Yosida regularization, and the associated proximal mapping have been widely used in Hilbert and Banach spaces. They have been objects of great interest for optimizers since their conception more than half a century ago. They were generalized by the notion of the D-Moreau envelope and D-proximal mapping by replacing the usual square of the Euclidean distance with the conception of Bregman distance for a convex function. Recently, the D-Moreau envelope has been developed in a very general setting. In this article, we present a regularizing and smoothing technique for convex functions defined in Banach spaces. We also investigate several properties of the D-Moreau envelope function and its related D-proximal mapping in Banach spaces. For technical reasons, we restrict our attention to the Lipschitz continuity property of the D-proximal mapping and differentiability properties of the D-Moreau envelope function. In particular, we prove the Fréchet differentiability property of the envelope and the Lipschitz continuity property of its derivative.

Keywords: Moreau envelope; Proximal mapping; D-Moreau envelope; Bregman distance; Totally convex; Legendre function; 90C25; 65K10; 47H05 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10957-020-01655-4

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