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On the Convergence Properties of a Majorized Alternating Direction Method of Multipliers for Linearly Constrained Convex Optimization Problems with Coupled Objective Functions

Author

Listed:
  • Ying Cui

    (National University of Singapore)

  • Xudong Li

    (National University of Singapore)

  • Defeng Sun

    (National University of Singapore)

  • Kim-Chuan Toh

    (National University of Singapore)

Abstract

In this paper, we establish the convergence properties for a majorized alternating direction method of multipliers for linearly constrained convex optimization problems, whose objectives contain coupled functions. Our convergence analysis relies on the generalized Mean-Value Theorem, which plays an important role to properly control the cross terms due to the presence of coupled objective functions. Our results, in particular, show that directly applying two-block alternating direction method of multipliers with a large step length of the golden ratio to the linearly constrained convex optimization problem with a quadratically coupled objective function is convergent under mild conditions. We also provide several iteration complexity results for the algorithm.

Suggested Citation

  • Ying Cui & Xudong Li & Defeng Sun & Kim-Chuan Toh, 2016. "On the Convergence Properties of a Majorized Alternating Direction Method of Multipliers for Linearly Constrained Convex Optimization Problems with Coupled Objective Functions," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1013-1041, June.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:3:d:10.1007_s10957-016-0877-2
    DOI: 10.1007/s10957-016-0877-2
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    Cited by:

    1. Radu Ioan Bot & Dang-Khoa Nguyen, 2020. "The Proximal Alternating Direction Method of Multipliers in the Nonconvex Setting: Convergence Analysis and Rates," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 682-712, May.
    2. V. A. Adona & M. L. N. Gonçalves & J. G. Melo, 2019. "Iteration-complexity analysis of a generalized alternating direction method of multipliers," Journal of Global Optimization, Springer, vol. 73(2), pages 331-348, February.
    3. Yu-Hong Dai & Fangfang Xu & Liwei Zhang, 2023. "Alternating direction method of multipliers for linear hyperspectral unmixing," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 97(3), pages 289-310, June.
    4. Max L. N. Gonçalves & Maicon Marques Alves & Jefferson G. Melo, 2018. "Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 177(2), pages 448-478, May.
    5. Deren Han & Defeng Sun & Liwei Zhang, 2018. "Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Programming," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 622-637, May.

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