SIAM Journal on Optimization, Volume 31, Issue 1, Page 404-424, January 2021.
Recently, a new kind of distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this kind of distance specializes under modest assumptions to the classical Bregman distance. We name this new kind of distance the generalized Bregman distance, and we shed light on it with examples that utilize the other two most natural representative functions: the Fitzpatrick function and its conjugate. We provide sufficient conditions for convexity, coercivity, and supercoercivity: properties which are essential for implementation in proximal point type algorithms. We establish these results for both the left and right variants of this new kind of distance. We construct examples closely related to the Kullback--Leibler divergence, which was previously considered in the context of Bregman distances and whose importance in information theory is well known. In so doing, we demonstrate how to compute a difficult Fitzpatrick conjugate function, and we discover natural occurrences of the Lambert ${\mathcal W}$ function, whose importance in optimization is of growing interest.

中文翻译：

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广义布雷格曼距离

SIAM优化杂志，第31卷，第1期，第404-424页，2021年1月。
最近，为两个点对集算子的图形引入了一种新型距离，其中一个最大为单调。当两个算子都是适当的下半连续凸函数的次微分时，这种距离在适度的假设下专门针对经典的Bregman距离。我们将这种新型距离命名为广义Bregman距离，并通过利用其他两个最自然的代表性函数：菲茨帕特里克函数及其共轭的例子来阐明这一点。我们为凸度，矫顽力和超矫顽力提供了充分的条件：这些属性对于在近端点类型算法中实现至关重要。我们为这种新型距离的左右变体建立了这些结果。我们构建了与Kullback-Leibler散度密切相关的示例，Kullback-Leibler散度以前在Bregman距离的背景下被考虑过，并且其在信息论中的重要性众所周知。这样，我们演示了如何计算困难的Fitzpatrick共轭函数，并发现了Lambert $ {\ mathcal W} $函数的自然出现，该函数在优化中的重要性日益受到关注。