Recently, several convergence rate results for Douglas-Rachford splitting and the alternating direction method of multipliers (ADMM) have been presented in the literature. In this paper, we show global linear convergence rate bounds for Douglas-Rachford splitting and ADMM under strong convexity and smoothness assumptions. We further show that the rate bounds are tight for the class of problems under consideration for all feasible algorithm parameters. For problems that satisfy the assumptions, we show how to select step-size and metric for the algorithm that optimize the derived convergence rate bounds. For problems with a similar structure that do not satisfy the assumptions, we present heuristic step-size and metric selection methods.
In [Theorem 6] Davis_Yin_2014 , dual Douglas-Rachford splitting with @math is shown to converge linearly at least with rate @math . For @math they recover the bound in @cite_0 . In Figure , the former (better) of these rate bounds is plotted. We see that it is more conservative than the one in Corollary .
@cite_0
2019569173
Splitting algorithms for the sum of two monotone operators.We study two splitting algorithms for (stationary and evolution) problems involving the sum of two monotone operators. These algorithms ar...