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A Unified Alternating Direction Method of Multipliers by Majorization Minimization.

Canyi Lu, Jiashi Feng, Shuicheng Yan

    IEEE Transactions on Pattern Analysis and Machine Intelligence
    |April 4, 2017
    PubMed
    Summary
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    New Alternating Direction Method of Multipliers (ADMM) frameworks, including Gauss-Seidel and Jacobian ADMMs, offer unified solutions for convex problems. These methods improve convergence speed and handle non-separable objectives, demonstrating superior performance in compressed sensing applications.

    Area of Science:

    • Optimization Algorithms
    • Signal Processing
    • Applied Mathematics

    Background:

    • Compressed sensing relies heavily on optimization solvers.
    • Alternating Direction Method of Multipliers (ADMM) is a popular choice for linearly constrained convex problems.
    • Existing ADMMs often have case-by-case convergence proofs and limitations with non-separable objectives.

    Purpose of the Study:

    • To develop unified frameworks for ADMM that generalize existing methods.
    • To improve the convergence speed and applicability of ADMMs.
    • To address challenges in solving convex problems with non-separable objectives.

    Main Methods:

    • Inspired by majorization minimization, proposed unified Gauss-Seidel ADMMs and Jacobian ADMMs.
    • Introduced Mixed Gauss-Seidel and Jacobian ADMM (M-ADMM) to combine advantages and accelerate convergence.

    Related Experiment Videos

  • Developed Proximal Gauss-Seidel ADMM for multi-block problems, applicable to non-strongly convex objectives.

    • Main Results:

    • Demonstrated that tighter majorant functions lead to faster ADMM convergence.
    • Showcased the ability of the new frameworks to handle non-separable objectives.
    • Empirically validated the superiority of the proposed ADMM variants on synthesized and real-world data.

    Conclusions:

    • The proposed unified ADMM frameworks offer significant improvements over existing methods.
    • M-ADMM effectively alleviates slow convergence issues, with further enhancements possible through backtracking and variable partitioning.
    • The developed ADMMs and accompanying toolbox provide efficient solutions for compressed sensing and related problems.