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

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Ernest Ryu was partially supported by AFOSR MURI FA9550-18-1-0502, NSF Grant DMS-1720237, and ONR Grant N000141712162. Bằng Công Vũ’s research work was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 102.01-2017.05.

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Correspondence to Ernest K. Ryu.

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Communicated by Jalal Fadili.

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Ryu, E.K., Vũ, B.C. Finding the Forward-Douglas–Rachford-Forward Method. J Optim Theory Appl 184, 858–876 (2020).

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