Iteration complexity of an inexact Douglas-Rachford method and of a Douglas-Rachford-Tseng’s F-B four-operator splitting method for solving monotone inclusions
Iteration complexity of an inexact Douglas-Rachford method and of a Douglas-Rachford-Tseng’s F-B four-operator splitting method for solving monotone inclusions
复制标题
求解单调包含的不精确 Douglas-Rachford 方法和 Douglas-Rachford-Tseng 的 F-B 四算子分裂方法的迭代复杂度
DOI：
10.1007/s11075-018-0604-1
复制
发表时间：
2017
影响因子：
2.1
通讯作者：
Marina Geremia
中科院分区：
文献类型：
--
作者：
M. Alves;Marina Geremia
本文提出并研究了求解双算子和四算子单调包含的不精确Douglas-Rachford分裂(DRS)方法和Douglas-Rachford-Tseng向前-向后(F-B)分裂方法。前一种方法(尽管基于略有不同的迭代机制)是由J.Eackstein和W.姚最近的工作所启发的，其中不精确DRS方法是从Solodov和Svaiter的混合近端外梯度(HPE)方法的一个特殊情况导出的，而后者结合了所提出的不精确DRS方法(用作外迭代)和Tseng的F-B分裂型方法(用作内迭代)来求解相应的子问题。我们证明了这两种算法在逐点(非遍历)和遍历意义下的迭代复杂性界，证明了它们允许两种不同的迭代：一种可以嵌入到HPE方法中，其迭代复杂度从Monteiro和Svaiter的工作中就已知，另一种需要单独分析。最后，我们进行了简单的数值实验，与其他已有算法进行了比较，验证了所提方法的性能。
In this paper, we propose and study the iteration complexity of an inexact Douglas-Rachford splitting (DRS) method and a Douglas-Rachford-Tseng’s forward-backward (F-B) splitting method for solving two-operator and four-operator monotone inclusions, respectively. The former method (although based on a slightly different mechanism of iteration) is motivated by the recent work of J. Eckstein and W. Yao, in which an inexact DRS method is derived from a special instance of the hybrid proximal extragradient (HPE) method of Solodov and Svaiter, while the latter one combines the proposed inexact DRS method (used as an outer iteration) with a Tseng’s F-B splitting-type method (used as an inner iteration) for solving the corresponding subproblems. We prove iteration complexity bounds for both algorithms in the pointwise (non-ergodic) as well as in the ergodic sense by showing that they admit two different iterations: one that can be embedded into the HPE method, for which the iteration complexity is known since the work of Monteiro and Svaiter, and another one which demands a separate analysis. Finally, we perform simple numerical experiments to show the performance of the proposed methods when compared with other existing algorithms.
影响因子：
2.7
作者：
Bot, Radu Ioan;Csetnek, Erno Robert;Hendrich, Christopher
通讯作者：
Hendrich, Christopher