On a New Analysis Framework for the Linear Convergence of Relaxed Operator Splitting Methods
Received:March 20, 2021  Revised:October 16, 2021
Key Word: averaged operator   negatively averaged operator   relaxed forward-backward splitting method   proximal point algorithm  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11801455; 11871059; 11971238), China Postdoctoral Science Foundation (Grant Nos.2019M663459; 2020T130081), the Applied Basic Project of Sichuan Province (Grant No.2020YJ0111) and the Open Project of Key Laboratory of School of Mathematical Sciences, Chongqing Normal University (Grant No.CSSXKFKTM202004).
Author NameAffiliation
Qingjing LI School of Mathematics and Information, China West Normal University, Sichuan 637002, P. R. China 
Ke GUO School of Mathematics and Information, China West Normal University, Sichuan 637002, P. R. China 
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Abstract:
      In this paper, we propose a new analysis framework to study the linear convergence of relaxed operator splitting methods, which can be viewed as an extension of the classic Krasnosel'ski$\breve{\mbox{i}}$-Mann iteration and Banach-Picard contraction. As applications, we derive the linear convergence of the generalized proximal point algorithm and the relaxed forward-backward splitting method in a simple and elegant way.
Citation:
DOI:10.3770/j.issn:2095-2651.2022.02.010
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