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.

中文翻译：

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Banach 空间中关于 Bregman 距离的正则化

Moreau 包络，也称为 Moreau-Yosida 正则化，以及相关的近端映射已广泛用于 Hilbert 和 Banach 空间。自从半个多世纪前它们的构想以来，它们一直是优化器非常感兴趣的对象。它们通过 D-Moreau 包络和 D-近端映射的概念通过用凸函数的 Bregman 距离的概念代替欧几里得距离的通常平方来推广。最近，D-Moreau 信封已在非常普遍的环境中开发。在本文中，我们提出了一种用于 Banach 空间中定义的凸函数的正则化和平滑技术。我们还研究了 D-Moreau 包络函数的几个性质及其在 Banach 空间中的相关 D-近端映射。由于技术原因，我们将注意力限制在 D-近端映射的 Lipschitz 连续性属性和 D-Moreau 包络函数的可微性属性上。特别地，我们证明了包络的 Fréchet 可微性和其导数的 Lipschitz 连续性。