Abstract
This note is devoted to the splitting algorithm proposed by Davis and Yin (Set-valued Var. Anal. 25(4), 829–858, 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.
Highlights
We provide a direct proof that guarantees its convergence when the stepsizes are smaller than four times the cocoercivity constant, doubling the size of the interval established by Davis and Yin
We provide some numerical experiments illustrating the importance of appropriately choosing the stepsize and relaxation parameters of the algorithms
We have presented an alternative proof of convergence for the Davis–Yin splitting algorithm without requiring the Davis–Yin operator (2) to be averaged
Summary
When a problem has certain structure, it is normally useful to take advantage of it. Following the divide-and-conquer paradigm, splitting algorithms iteratively solve simpler problems which are defined by separately using some parts of the original problem. And (PA(xk))k∈N converges to the minimum norm point in A ∩ B (the normal cone sum rule holds) If γ ∈ ]0, 2β[, overrelaxed steps (i.e., λk > 1) are allowed in (3), while only underrelaxed steps can be taken when γ ≥ 2β The fact that both the stepsize and the relaxation parameters are important is especially apparent when one considers the particular case of A = B = 0 and T = ∇f for a differentiable function f whose gradient is Lipschitz continuous with constant L = β1.
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