Finding the Forward-Douglas–Rachford-Forward Method

Ernest K. Ryu () and Bằng Công Vũ ()
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Ernest K. Ryu: University of California
Bằng Công Vũ: Vietnam National University

Journal of Optimization Theory and Applications, 2020, vol. 184, issue 3, No 8, 858-876

Abstract: Abstract 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. arXiv:1808.04162). We show that this second method solves the 3-operator problem generally, without further assumptions.

Keywords: Douglas–Rachford; Forward–backward–forward; Forward–reflected–backward; Monotone inclusion; 47H05; 47H09; 90C25 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-019-01601-z

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