A direct proof of convergence of Davis-Yin splitting algorithm allowing larger stepsizes

Abstract

This note is devoted to the splitting algorithm proposed by Davis and Yin in 2017 for computing a zero of the sum of three maximally monotone operators, with one of them being cocoercive. We provide a direct proof that guarantees its convergence when the stepsizes are smaller than four times the cocoercivity constant, thus doubling the size of the interval established by Davis and Yin. As a by-product, the same conclusion applies to the forward-backward splitting algorithm. Further, we use the notion of "strengthening" of a set-valued operator to derive a new splitting algorithm for computing the resolvent of the sum. Last but not least, we provide some numerical experiments illustrating the importance of appropriately choosing the stepsize and relaxation parameters of the algorithms.

Distributed Forward-Backward Methods without Central Coordination

“…, f m : H → (−∞, +∞) are convex and differentiable with L-Lipschitz continuous gradients. Through its first order optimality condition, (2) can be posed as (1) with…”

“…, Φ m : H 1 × H 2 → (−∞, +∞] are differentiable convex-concave functions with Lipschitz continuous gradient. Assuming a saddle-point exists, (3) can be posed as (1) in the space H := H 1 × H 2 with…”

“…When n = 2, there are also many methods. For instance, if F is cocoercive, Davis-Yin splitting [12,11,1] which takes the form…”

“…Other methods for (1) with n > 2 include the generalised forward-backward method [18] and those from the projective splitting family [13,14]. Indisputably, product space reformulations such as (6) provide a convenient tool that makes the derivation of algorithms for n > 2 operators an almost mechanical procedure.…”

“…Our contribution. We propose and analyse algorithms of forward-backward-type for solving (1) which exploit problem structure. Note that by using the zero operator in (1) if necessary, we can always assume that m = n − 1.…”