Finding the Forward-Douglas–Rachford-Forward Method

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.