论文标题
重新审视的无用映射：均匀的单调性和操作员分裂
Strongly nonexpansive mappings revisited: uniform monotonicity and operator splitting
论文作者
论文摘要
通过后者的反射分辨率，非专业映射类别与最大单调算子类别之间的对应关系在分裂方法的收敛分析中发挥了工具作用。确实，其中一些方法的性能，例如，道格拉斯·拉赫福德（Douglas-Rachford）和和平人 - 拉赫福德（Peaceman-Rachford）方法取决于迭代与单个运营商相关的所谓分裂操作员。该分裂操作员是基础运算符反射分辨率的组成的函数。在本文中，我们提供了一项统一单调操作员及其相应反射回散的全面研究。我们表明，后者与布鲁克和赖希（Bruck and Reich）介绍的强烈非专业运营商的类别密切相关。通过逆操作员进行与二元性的连接。我们向道格拉斯·拉赫福德（Douglas-Rachford）和和平人 - 拉赫福德（Peaceman-Rachford）方法提供申请。提出了说明和收紧我们的结果的示例。
The correspondence between the class of nonexpansive mappings and the class of maximally monotone operators via the reflected resolvents of the latter has played an instrumental role in the convergence analysis of the splitting methods. Indeed, the performance of some of these methods, e.g., Douglas-Rachford and Peaceman-Rachford methods hinges on iterating the so-called splitting operator associated with the individual operators. This splitting operator is a function of the composition of the reflected resolvents of the underlying operators. In this paper, we provide a comprehensive study of the class of uniformly monotone operators and their corresponding reflected resolvents. We show that the latter is closely related to the class of the strongly nonexpansive operators introduced by Bruck and Reich. Connections to duality via inverse operators are systematically studied. We provide applications to Douglas-Rachford and Peaceman-Rachford methods. Examples that illustrate and tighten our results are presented.
