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An efficient partial parallel method with scaling step size strategy for three-block convex optimization problems

Author

Listed:
  • Yaning Jiang

    (Nanjing University)

  • Deren Han

    (Beihang University)

  • Xingju Cai

    (Nanjing Normal University)

Abstract

A popular optimization model arising from image processing is the separable optimization problem whose objective function is the sum of three independent functions, and the variables are coupled by a linear equality. In this paper, we propose to solve such problem by a new partially parallel splitting method, whose step sizes for the primal and the dual variables in correction step are not necessarily identical. We establish the global convergence, and study the convergence rate on this varying ADMM-based prediction-correction method named as VAPCM. We derive the worst-case O(1/t) convergence rate in both the ergodic and non-ergodic senses. We also show that the convergence rate can be improved to o(1/t). Moreover, under the error bound assumptions, we establish the global linear convergence of VAPCM. We apply the new method to solve problems in robust principal component analysis and image decomposition. Numerical results indicate that the new method is efficient.

Suggested Citation

  • Yaning Jiang & Deren Han & Xingju Cai, 2022. "An efficient partial parallel method with scaling step size strategy for three-block convex optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(3), pages 383-419, December.
    • Handle: RePEc:spr:mathme:v:96:y:2022:i:3:d:10.1007_s00186-022-00796-8
    DOI: 10.1007/s00186-022-00796-8
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Liusheng Hou & Hongjin He & Junfeng Yang, 2016. "A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA," Computational Optimization and Applications, Springer, vol. 63(1), pages 273-303, January.
    4. Caihua Chen & Yuan Shen & Yanfei You, 2013. "On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, October.
    5. 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.
    6. K. Wang & D. R. Han & L. L. Xu, 2013. "A Parallel Splitting Method for Separable Convex Programs," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 138-158, October.
    7. Deren Han & Xiaoming Yuan, 2012. "A Note on the Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 227-238, October.
    8. 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.
    9. Xingju Cai & Deren Han & Xiaoming Yuan, 2017. "On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function," Computational Optimization and Applications, Springer, vol. 66(1), pages 39-73, January.
    10. Bing-Sheng He, 2009. "Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities," Computational Optimization and Applications, Springer, vol. 42(2), pages 195-212, March.
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