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An efficient augmented Lagrangian method with applications to total variation minimization

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  • Chengbo Li
  • Wotao Yin
  • Hong Jiang
  • Yin Zhang

Abstract

Based on the classic augmented Lagrangian multiplier method, we propose, analyze and test an algorithm for solving a class of equality-constrained non-smooth optimization problems (chiefly but not necessarily convex programs) with a particular structure. The algorithm effectively combines an alternating direction technique with a nonmonotone line search to minimize the augmented Lagrangian function at each iteration. We establish convergence for this algorithm, and apply it to solving problems in image reconstruction with total variation regularization. We present numerical results showing that the resulting solver, called TVAL3, is competitive with, and often outperforms, other state-of-the-art solvers in the field. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Chengbo Li & Wotao Yin & Hong Jiang & Yin Zhang, 2013. "An efficient augmented Lagrangian method with applications to total variation minimization," Computational Optimization and Applications, Springer, vol. 56(3), pages 507-530, December.
    • Handle: RePEc:spr:coopap:v:56:y:2013:i:3:p:507-530
    DOI: 10.1007/s10589-013-9576-1
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

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    2. Paula A. Kikuchi & Aurelio R. L. Oliveira, 2020. "New Preconditioners Applied to Linear Programming and the Compressive Sensing Problems," SN Operations Research Forum, Springer, vol. 1(4), pages 1-20, December.
    3. Ya-Feng Liu & Xin Liu & Shiqian Ma, 2019. "On the Nonergodic Convergence Rate of an Inexact Augmented Lagrangian Framework for Composite Convex Programming," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 632-650, May.
    4. Zhen Wei & Qiurong Yan & Xiaoqiang Lu & Yongjian Zheng & Shida Sun & Jian Lin, 2023. "Compression Reconstruction Network with Coordinated Self-Attention and Adaptive Gaussian Filtering Module," Mathematics, MDPI, vol. 11(4), pages 1-17, February.
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    6. Kruse, René-Marcel & Silbersdorff, Alexander & Säfken, Benjamin, 2022. "Model averaging for linear mixed models via augmented Lagrangian," Computational Statistics & Data Analysis, Elsevier, vol. 167(C).
    7. Keshvari, Abolfazl, 2017. "A penalized method for multivariate concave least squares with application to productivity analysis," European Journal of Operational Research, Elsevier, vol. 257(3), pages 1016-1029.

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