Strongly Nonexpansive Mappings Revisited: Uniform Monotonicity and Operator Splitting
本文系统研究了一致单调算子及其反射预解式，揭示其与强非扩张算子的联系，并应用于Douglas-Rachford和Peaceman-Rachford分裂方法，提供收敛性分析。
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., the 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 strongly nonexpansive operators introduced by Bruck and Reich. Connections to duality via inverse operators are systematically studied. We provide applications to the Douglas–Rachford and Peaceman–Rachford methods. Examples that illustrate and tighten our results are presented.