Journal
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
Volume 184, Issue 3, Pages 858-876Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10957-019-01601-z
Keywords
Douglas-Rachford; Forward-backward-forward; Forward-reflected-backward; Monotone inclusion
Funding
- AFOSR MURI [FA9550-18-1-0502]
- NSF [DMS-1720237]
- ONR [N000141712162]
- Vietnam National Foundation for Science and Technology Development (NAFOSTED) [102.01-2017.05]
- National Research Foundation of Korea [2020R1F1A1A01072877] Funding Source: Korea Institute of Science & Technology Information (KISTI), National Science & Technology Information Service (NTIS)
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We consider the monotone inclusion problem with a sum of 3 operators, in which 2 are monotone and 1 is monotone-Lipschitz. The classical Douglas-Rachford and forward-backward-forward methods, respectively, solve the monotone inclusion problem with a sum of 2 monotone operators and a sum of 1 monotone and 1 monotone-Lipschitz operators. We first present a method that naturally combines Douglas-Rachford and forward-backward-forward and show that it solves the 3-operator problem under further assumptions, but fails in general. We then present a method that naturally combines Douglas-Rachford and forward-reflected-backward, a recently proposed alternative to forward-backward-forward by Malitsky and Tam (A forward-backward splitting method for monotone inclusions without cocoercivity, 2018. ). We show that this second method solves the 3-operator problem generally, without further assumptions.
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