We systematically study the local single-valuedness of the Bregman proximal mapping and local smoothness of the Bregman–Moreau envelope of a nonconvex function under relative prox-regularity—an extension of prox-regularity—which was originally introduced by Poliquin and Rockafellar. As Bregman distances are asymmetric in general, in accordance with Bauschke et al., it is natural to consider two variants of the Bregman proximal mapping, which, depending on the order of the arguments, are called left and right Bregman proximal mapping. We consider the left Bregman proximal mapping first. Then, via translation result, we obtain analogue (and partially sharp) results for the right Bregman proximal mapping. The class of relatively prox-regular functions significantly extends the recently considered class of relatively hypoconvex functions. In particular, relative prox-regularity allows for functions with a possibly nonconvex domain. Moreover, as a main source of examples and analogously to the classical setting, we introduce relatively amenable functions, i.e. convexly composite functions, for which the inner nonlinear mapping is component-wise smooth adaptable, a recently introduced extension of Lipschitz differentiability. By way of example, we apply our theory to locally interpret joint alternating Bregman minimization with proximal regularization as a Bregman proximal gradient algorithm, applied to a smooth adaptable function.

中文翻译：

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相对近端正则性下的 Bregman 近端映射和 Bregman-Moreau 包络

我们系统地研究了非凸函数的 Bregman 近端映射的局部单值性和非凸函数的 Bregman-Moreau 包络的局部平滑性，该函数最初是由 Poliquin 和 Rockafellar 引入的。由于 Bregman 距离通常是不对称的，根据 Bauschke 等人的说法，很自然地考虑 Bregman 近端映射的两种变体，根据参数的顺序，它们被称为左 Bregman 近端映射和右 Bregman 近端映射。我们首先考虑左 Bregman 近端映射。然后，通过翻译结果，我们获得了右侧 Bregman 近端映射的模拟（部分清晰）结果。相对接近正则函数的类别显着扩展了最近考虑的相对低凸函数的类别。特别是，相对近似正则性允许具有可能非凸域的函数。此外，作为示例的主要来源并类似于经典设置，我们引入了相对适用的函数，即凸复合函数，对于这些函数，内部非线性映射是组件方向平滑自适应的，这是最近引入的 Lipschitz 可微性的扩展。举例来说，我们将我们的理论应用到局部将具有近端正则化的联合交替 Bregman 最小化解释为 Bregman 近端梯度算法，应用于平滑的自适应函数。对于其中内部非线性映射是组件方式平滑自适应的，这是最近引入的 Lipschitz 可微性的扩展。举例来说，我们将我们的理论应用到局部将具有近端正则化的联合交替 Bregman 最小化解释为 Bregman 近端梯度算法，应用于平滑的自适应函数。对于其中内部非线性映射是组件方式平滑自适应的，这是最近引入的 Lipschitz 可微性的扩展。举例来说，我们将我们的理论应用到局部将具有近端正则化的联合交替 Bregman 最小化解释为 Bregman 近端梯度算法，应用于平滑的自适应函数。