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Convergence study on strictly contractive Peaceman–Rachford splitting method for nonseparable convex minimization models with quadratic coupling terms

Author

Listed:
  • Peixuan Li

    (City University of Hong Kong)

  • Yuan Shen

    (Nanjing University of Finance and Economics)

  • Suhong Jiang

    (Nanjing University of Finance and Economics)

  • Zehua Liu

    (Nanjing University)

  • Caihua Chen

    (Nanjing University)

Abstract

The alternating direction method of multipliers (ADMM) and Peaceman Rachford splitting method (PRSM) are two popular splitting algorithms for solving large-scale separable convex optimization problems. Though problems with nonseparable structure appear frequently in practice, researches on splitting methods for these problems remain to be scarce. Very recently, Chen et al. (Math Program 173(1–2):37–77, 2019) extended the 2-block ADMM to linearly constrained nonseparable models with quadratic coupling terms and established its convergence. However, theoretical researches about nonseparable PRSM or its variants are still lacking. To fill the gap, in this paper we focus on the strictly contractive PRSM (SC-PRSM) applied to 2-block linearly constrained convex minimization problems with quadratic coupling objective functions. Under mild conditions, we prove the convergence of our proposed SC-PRSM and establish its o(1/k) convergence rate. Moreover, we implement the SC-PRSM to solve a problem of calculating the Euclidian distance between two ellipsoids, and compare its performance with three ADMM type algorithms. The results show the nonseparable SC-PRSM outperforms the other three algorithms in terms of both the iteration numbers and CPU time.

Suggested Citation

  • Peixuan Li & Yuan Shen & Suhong Jiang & Zehua Liu & Caihua Chen, 2021. "Convergence study on strictly contractive Peaceman–Rachford splitting method for nonseparable convex minimization models with quadratic coupling terms," Computational Optimization and Applications, Springer, vol. 78(1), pages 87-124, January.
    • Handle: RePEc:spr:coopap:v:78:y:2021:i:1:d:10.1007_s10589-020-00229-4
    DOI: 10.1007/s10589-020-00229-4
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    References listed on IDEAS

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    1. Min Li & Defeng Sun & Kim-Chuan Toh, 2015. "A Convergent 3-Block Semi-Proximal ADMM for Convex Minimization Problems with One Strongly Convex Block," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 32(04), pages 1-19.
    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.
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    5. Zhongming Wu & Min Li & David Z. W. Wang & Deren Han, 2017. "A Symmetric Alternating Direction Method of Multipliers for Separable Nonconvex Minimization Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 34(06), pages 1-27, December.
    6. Guoyong Gu & Bingsheng He & Junfeng Yang, 2014. "Inexact Alternating-Direction-Based Contraction Methods for Separable Linearly Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 163(1), pages 105-129, October.
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