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Inexact Alternating-Direction-Based Contraction Methods for Separable Linearly Constrained Convex Optimization

Author

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  • Guoyong Gu

    (Nanjing University)

  • Bingsheng He

    (Nanjing University)

  • Junfeng Yang

    (Nanjing University)

Abstract

Alternating direction method of multipliers has been well studied in the context of linearly constrained convex optimization. In the last few years, we have witnessed a number of novel applications arising from image processing, compressive sensing and statistics, etc., where the approach is surprisingly efficient. In the early applications, the objective function of the linearly constrained convex optimization problem is separable into two parts. Recently, the alternating direction method of multipliers has been extended to the case where the number of the separable parts in the objective function is finite. However, in each iteration, the subproblems are required to be solved exactly. In this paper, by introducing some reasonable inexactness criteria, we propose two inexact alternating-direction-based contraction methods, which substantially broaden the applicable scope of the approach. The convergence and complexity results for both methods are derived in the framework of variational inequalities.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:163:y:2014:i:1:d:10.1007_s10957-013-0489-z
    DOI: 10.1007/s10957-013-0489-z
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    Cited by:

    1. Jiao-fen Li & Wen Li & Ru Huang, 2016. "An efficient method for solving a matrix least squares problem over a matrix inequality constraint," Computational Optimization and Applications, Springer, vol. 63(2), pages 393-423, March.
    2. Jianchao Bai & William W. Hager & Hongchao Zhang, 2022. "An inexact accelerated stochastic ADMM for separable convex optimization," Computational Optimization and Applications, Springer, vol. 81(2), pages 479-518, March.
    3. Jueyou Li & Zhiyou Wu & Changzhi Wu & Qiang Long & Xiangyu Wang, 2016. "An Inexact Dual Fast Gradient-Projection Method for Separable Convex Optimization with Linear Coupled Constraints," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 153-171, January.
    4. 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.

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