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A parallel splitting ALM-based algorithm for separable convex programming

Author

Listed:
  • Shengjie Xu

    (Harbin Institute of Technology
    Southern University of Science and Technology)

  • Bingsheng He

    (Nanjing University)

Abstract

The augmented Lagrangian method (ALM) provides a benchmark for solving the canonical convex optimization problem with linear constraints. The direct extension of ALM for solving the multiple-block separable convex minimization problem, however, is proved to be not necessarily convergent in the literature. It has thus inspired a number of ALM-variant algorithms with provable convergence. This paper presents a novel parallel splitting method for the multiple-block separable convex optimization problem with linear equality constraints, which enjoys a larger step size compared with the existing parallel splitting methods. We first show that a fully Jacobian decomposition of the regularized ALM can contribute a descent direction yielding the contraction of proximity to the solution set; then, the new iterate is generated via a simple correction step with an ignorable computational cost. We establish the convergence analysis for the proposed method, and then demonstrate its numerical efficiency by solving an application problem arising in statistical learning.

Suggested Citation

  • Shengjie Xu & Bingsheng He, 2021. "A parallel splitting ALM-based algorithm for separable convex programming," Computational Optimization and Applications, Springer, vol. 80(3), pages 831-851, December.
    • Handle: RePEc:spr:coopap:v:80:y:2021:i:3:d:10.1007_s10589-021-00321-3
    DOI: 10.1007/s10589-021-00321-3
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    References listed on IDEAS

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    1. Bingsheng He & Feng Ma & Xiaoming Yuan, 2020. "Optimally linearizing the alternating direction method of multipliers for convex programming," Computational Optimization and Applications, Springer, vol. 75(2), pages 361-388, March.
    2. Bingsheng He & Min Tao & Xiaoming Yuan, 2017. "Convergence Rate Analysis for the Alternating Direction Method of Multipliers with a Substitution Procedure for Separable Convex Programming," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 662-691, August.
    3. Jianchao Bai & Jicheng Li & Fengmin Xu & Hongchao Zhang, 2018. "Generalized symmetric ADMM for separable convex optimization," Computational Optimization and Applications, Springer, vol. 70(1), pages 129-170, May.
    4. Bingsheng He & Xiaoming Yuan, 2018. "A class of ADMM-based algorithms for three-block separable convex programming," Computational Optimization and Applications, Springer, vol. 70(3), pages 791-826, July.
    5. William W. Hager & Hongchao Zhang, 2020. "Convergence rates for an inexact ADMM applied to separable convex optimization," Computational Optimization and Applications, Springer, vol. 77(3), pages 729-754, December.
    6. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
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