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Understanding the Douglas–Rachford splitting method through the lenses of Moreau-type envelopes

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  • Felipe Atenas

    (The University of Melbourne)

Abstract

We analyze the Douglas–Rachford splitting method for weakly convex optimization problems, by the token of the Douglas–Rachford envelope, a merit function akin to the Moreau envelope. First, we use epi-convergence techniques to show that this artifact approximates the original objective function via epigraphs. Secondly, we present how global convergence and local linear convergence rates for Douglas–Rachford splitting can be obtained using such envelope, under mild regularity assumptions. The keystone of the convergence analysis is the fact that the Douglas–Rachford envelope satisfies a sufficient descent inequality alongside the generated sequence, a feature that allows us to use arguments usually employed for descent methods. We report numerical experiments that use weakly convex penalty functions, which are comparable with the known behavior of the method in the convex case.

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  • Felipe Atenas, 2025. "Understanding the Douglas–Rachford splitting method through the lenses of Moreau-type envelopes," Computational Optimization and Applications, Springer, vol. 90(3), pages 881-910, April.
    • Handle: RePEc:spr:coopap:v:90:y:2025:i:3:d:10.1007_s10589-024-00646-9
    DOI: 10.1007/s10589-024-00646-9
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    References listed on IDEAS

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    1. Pierre Frankel & Guillaume Garrigos & Juan Peypouquet, 2015. "Splitting Methods with Variable Metric for Kurdyka–Łojasiewicz Functions and General Convergence Rates," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 874-900, June.
    2. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    3. Jonathan Eckstein, 2017. "A Simplified Form of Block-Iterative Operator Splitting and an Asynchronous Algorithm Resembling the Multi-Block Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 155-182, April.
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