General Resolvents for Monotone Operators: Characterization and Extension
General Resolvents for Monotone Operators: Characterization and Extension
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发表时间：
2008-10
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通讯作者：
Heinz H. Bauschke;Xianfu Wang;Liangjin Yao
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作者：
Heinz H. Bauschke;Xianfu Wang;Liangjin Yao
单调算子，特别是次微分算子，在最优化中有着重要的意义。众所周知，自Minty，Rockafellar和Bertsekas-Eckstein以来，在希尔伯特空间中，单调算子可以从强非扩张映射的替代观点来理解和分析，这些映射被发现恰好是单调算子的预解式。例如，Moreau意义下的邻近映射恰好是次微分算子的预解式。更一般的“预解式”，“邻近映射”和“坚定非扩张”的概念进行了研究。一个重要的类，主要是由阿尔伯，由神村和高桥，以及由Kohnikh和高桥推广，是基于正规化对偶映射。此外，Censor和Lent开创性地在Bregman距离框架中使用表现良好的凸函数的梯度。众所周知，预解式是坚定的非扩张，但匡威一直是后者的框架开放的问题。在本说明中，我们建立在最近的最大单调性的特征，由于Mart 'onez-Legaz提供了一个框架，研究坚定的非扩张映射总是预解式。这个框架包括经典的预解式，基于正规化对偶映射的预解式，基于Bregman距离的预解式，甚至基于（非对称）旋转子的预解式。作为最近工作的一个副产品的邻近平均，我们得到了一个建设性的Kirszbraun-Valentine推广结果的广义坚定非扩张映射。几个例子说明了我们的结果。
Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators. For example, the proximal mappings in the sense of Moreau are precisely the resolvents of subdifferential operators. More general notions of “resolvent”, “proximal mapping” and “firmly nonexpansive” have been studied. One important class, popularized chiefly by Alber, by Kamimura and Takahashi, and by Kohsaka and Takahashi, is based on the normalized duality mapping. Furthermore, Censor and Lent pioneered the use of the gradient of a well behaved convex functions in a Bregman-distance based framework. It is known that resolvents are firmly nonexpansive, but the converse has been an open problem for the latter framework. In this note, we build on the very recent characterization of maximal monotonicity due to Mart´onez-Legaz to provide a framework for studying resolvents in which firmly nonexpansive mappings are always resolvents. This framework includes classical resolvents, resolvents based on the normalized duality mapping, resolvents based on Bregman distances, and even resolvents based on (nonsymmetric) rotators. As a by-product of recent work on the proximal average, we obtain a constructive Kirszbraun-Valentine extension result for generalized firmly nonexpansive mappings. Several examples illustrate our results.