Convergence of a randomized Douglas-Rachford method for linear system

Hu, Leyu; Cai, Xingju

doi:10.3934/naco.2020045

Cited by 3 publications

(1 citation statement)

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“…Recently, Hu and Cai [34] studied the classical randomized Douglas-Rachford (RDR) method in a simple case where r = 2. They proved that E[x k ] → x * as k → ∞, however, without convergence rates analysis.…”

Section: Introductionmentioning

confidence: 99%

Randomized Douglas-Rachford method for linear systems: Improved accuracy and efficiency

Han¹,

Su²,

Xie³

2022

Preprint

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The Douglas-Rachford (DR) method is a widely used method for finding a point in the intersection of two closed convex sets (feasibility problem). However, the method converges weakly and the associated rate of convergence is hard to analyze in general. In addition, the direct extension of the DR method for solving more-than-two-sets feasibility problems, called the r-sets-DR method, is not necessarily convergent. Recently, the introduction of randomization and momentum techniques in optimization algorithms has attracted increasing attention as it makes the algorithms more efficient. In this paper, we show that randomization is powerful for the DR method and can make the divergent r-sets-DR method converge. Specifically, we propose the randomized r-sets-DR (RrDR) method for solving the linear system Ax = b and prove its linear convergence in expectation, and the convergence rate does not depend on the dimension of the matrix A. We also study RrDR with heavy ball momentum and show that it converges at an accelerated linear rate. Numerical experiments are provided to confirm our results and demonstrate the significant improvement in accuracy and efficiency of the DR method that randomization and momentum bring.C i where C i := {x : a i , x = b i }.

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Section: Introductionmentioning

confidence: 99%