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An ADM-based splitting method for separable convex programming

Author

Listed:
  • Deren Han
  • Xiaoming Yuan
  • Wenxing Zhang
  • Xingju Cai

Abstract

We consider the convex minimization problem with linear constraints and a block-separable objective function which is represented as the sum of three functions without coupled variables. To solve this model, it is empirically effective to extend straightforwardly the alternating direction method of multipliers (ADM for short). But, the convergence of this straightforward extension of ADM is still not proved theoretically. Based on ADM’s straightforward extension, this paper presents a new splitting method for the model under consideration, which is empirically competitive to the straightforward extension of ADM and meanwhile the global convergence can be proved under standard assumptions. At each iteration, the new method corrects the output of the straightforward extension of ADM by some slight correction computation to generate a new iterate. Thus, the implementation of the new method is almost as easy as that of ADM’s straightforward extension. We show the numerical efficiency of the new method by some applications in the areas of image processing and statistics. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Deren Han & Xiaoming Yuan & Wenxing Zhang & Xingju Cai, 2013. "An ADM-based splitting method for separable convex programming," Computational Optimization and Applications, Springer, vol. 54(2), pages 343-369, March.
    • Handle: RePEc:spr:coopap:v:54:y:2013:i:2:p:343-369
    DOI: 10.1007/s10589-012-9510-y
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Sun, Jie & Zhang, Su, 2010. "A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1210-1220, December.
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    Cited by:

    1. Zhangquan Wang & Shanshan Huo & Xinlong Xiong & Ke Wang & Banteng Liu, 2023. "A Maximally Split and Adaptive Relaxed Alternating Direction Method of Multipliers for Regularized Extreme Learning Machines," Mathematics, MDPI, vol. 11(14), pages 1-16, July.
    2. X. Wang & S. Li & X. Kou & Q. Zhang, 2015. "A new alternating direction method for linearly constrained nonconvex optimization problems," Journal of Global Optimization, Springer, vol. 62(4), pages 695-709, August.
    3. 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.
    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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