A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity

Abstract

In this work, we propose a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators. Our method converges under the same assumptions as Tseng's forward-backward-forward method, namely, it does not require cocoercivity of the single-valued operator. Moreover, each iteration only uses one forward evaluation rather than two as is the case for Tseng's method. Variants of the method incorporating a linesearch, relaxation and inertia, or a structured three operator inclusion are also discussed.

Scaled Relative Graph: Nonexpansive operators via 2D Euclidean Geometry

“…In [6,7,68] the maximal monotone extension theorem was translated to extension theorems of other operator classes. Corollary C.2 shows that this approach will not work for M ∩ L L , a class of operators considered by the extragradient method [41], forward-backward-forward splitting [72], and other related methods [14,49]. In fact, [68] shows that certain simple interpolation condition for M fails for M ∩ L L .…”

Convergence of sequences: a survey

“…Application of Lemma 3.3 and Lemma 3.18. Lemma 3.3 is used in [51,35] to prove convergence in the inclusion problem:…”

Single-Forward-Step Projective Splitting: Exploiting Cocoercivity

“…• The recently proposed forward-reflected-backward method [29], applied to this same primal-dual inclusion 0 ∈ Ãp + Bp specified by (57)-(58). We call this method frb-pd.…”

“…If we strengthen the assumption to L −1cocoercivity, we can use FB, which only needs one forward step per iteration and allows stepsizes bounded away from 2L −1 . One departure from this pattern is the recently developed method of [29], which only requires Lipschitz continuity but uses just one forward step per iteration. While this property is remarkable, it should be noted that its stepsizes must be bounded by (1/2)L −1 , which is half the allowable stepsize for EG or FBF.…”