Douglas-Rachford分裂算法的Mann迭代形式的收敛性及其应用  

Convergence and Applications of the Mann Iiterative Methods of Douglas-Rachford Algorithm

作　　者：赵旭 ZHAO Xu(School of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637009)

机构地区：[1]西华师范大学数学与信息学院,四川南充637009

出　　处：《绵阳师范学院学报》2021年第11期20-25,共6页Journal of Mianyang Teachers＇ College

基　　金：国家自然科学基金(11871059,11371015);四川省高校科研创新团队(16TD0019);西华师范大学英才科研基金(17YC379).

摘　　要：在寻找两个次微分算子和为零时,以及更一般的情况,两个算子是两个极大单调算子时,Douglas-Rachford分裂算法是一种流行的分割技术.最近基于不同的假设条件下,关于Douglas-Rachford算法的收敛性已有很多成果.在此基础上,本文考虑Douglas-Rachford算法收敛的情况下其凸组合形式以及Mann迭代形式的收敛性.本文证明Douglas-Rachford算法的凸组合形式收敛于实的Hilbert空间中一点,Douglas-Rachford算法的Mann迭代形式弱收敛于Douglas-Rachford算法的不动点集中一点.此外,将结果应用于变分不等式问题,可得到Douglas-Rachford算法的Mann迭代形式的弱收敛性.When finding the sum of two subdifferential operators is 0,and more generally,the sum of two maximal monotone operators,Douglas-Rachford algorithm is a popular splitting technique.Recently,there are many results on this topic.This paper investigates the convergence of Douglas-Rachford algorithm in the case of its convex combination form and Mann iteration form.It is proved that the convex combination form of the Douglas-Rachford algorithm converges to a point in the real Hilbert space,and the mann iteration form of the Douglas-Rachford algorithm converges weakly to a point in the fixed point set of the Douglas-Rachford operator.Finally,we apply these results to the variational inequality involving the sum of a monotone operator and a strongly monotone operator,and obtain the weak convergence of the iterative form of the Douglas-Rachford algorithm.

关 键 词：Douglas-Rachford算法 MANN迭代 LIPSCHITZ连续 强单调 弱收敛 

分 类 号：O221[理学—运筹学与控制论]