Non-stationary Douglas–Rachford and alternating direction method of multipliers: adaptive step-sizes and convergence

Dirk A. Lorenz () and Quoc Tran-Dinh ()
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Dirk A. Lorenz: TU Braunschweig
Quoc Tran-Dinh: University of North Carolina at Chapel Hill (UNC-Chapel Hill)

Computational Optimization and Applications, 2019, vol. 74, issue 1, No 3, 67-92

Abstract: Abstract We revisit the classical Douglas–Rachford (DR) method for finding a zero of the sum of two maximal monotone operators. Since the practical performance of the DR method crucially depends on the step-sizes, we aim at developing an adaptive step-size rule. To that end, we take a closer look at a linear case of the problem and use our findings to develop a step-size strategy that eliminates the need for step-size tuning. We analyze a general non-stationary DR scheme and prove its convergence for a convergent sequence of step-sizes with summable increments in the case of maximally monotone operators. This, in turn, proves the convergence of the method with the new adaptive step-size rule. We also derive the related non-stationary alternating direction method of multipliers. We illustrate the efficiency of the proposed methods on several numerical examples.

Keywords: Douglas–Rachford method; Alternating direction methods of multipliers; Maximal monotone inclusions; Adaptive step-size; Non-stationary iteration; 90C25; 65K05; 65J15; 47H05 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10589-019-00106-9

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