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

Author

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  • Dirk A. Lorenz

    (TU Braunschweig)

  • Quoc Tran-Dinh

    (University of North Carolina at Chapel Hill (UNC-Chapel Hill))

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.

Suggested Citation

  • Dirk A. Lorenz & Quoc Tran-Dinh, 2019. "Non-stationary Douglas–Rachford and alternating direction method of multipliers: adaptive step-sizes and convergence," Computational Optimization and Applications, Springer, vol. 74(1), pages 67-92, September.
    • Handle: RePEc:spr:coopap:v:74:y:2019:i:1:d:10.1007_s10589-019-00106-9
    DOI: 10.1007/s10589-019-00106-9
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    References listed on IDEAS

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    1. B. S. He & H. Yang & S. L. Wang, 2000. "Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 337-356, August.
    2. Jingwei Liang & Jalal Fadili & Gabriel Peyré, 2017. "Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 874-913, March.
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    Cited by:

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    2. Martina Cerulli & Marianna Santis & Elisabeth Gaar & Angelika Wiegele, 2021. "Improving ADMMs for solving doubly nonnegative programs through dual factorization," 4OR, Springer, vol. 19(3), pages 415-448, September.

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