IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i5p811-d1602587.html
   My bibliography  Save this article

An Improvement of the Alternating Direction Method of Multipliers to Solve the Convex Optimization Problem

Author

Listed:
  • Jingjing Peng

    (College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China)

  • Zhijie Wang

    (College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China)

  • Siting Yu

    (College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China)

  • Zengao Tang

    (College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China)

Abstract

The alternating direction method is one of the attractive approaches for solving convex optimization problems with linear constraints and separable objective functions. Experience with applications has shown that the number of iterations depends significantly on the penalty parameter for the linear constraint. The penalty parameters in the classical alternating direction method are a constant. In this paper, an improved alternating direction method is proposed, which not only adaptively adjusts the penalty parameters per iteration based on the iteration message but also adds relaxation factors to the Lagrange multiplier update steps. Preliminary numerical experiments show that the technique of adaptive adjusting of the penalty parameters per iteration and attaching relaxation factors in the Lagrange multiplier updating steps are effective in practical applications.

Suggested Citation

  • Jingjing Peng & Zhijie Wang & Siting Yu & Zengao Tang, 2025. "An Improvement of the Alternating Direction Method of Multipliers to Solve the Convex Optimization Problem," Mathematics, MDPI, vol. 13(5), pages 1-19, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:811-:d:1602587
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/5/811/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/5/811/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. B. S. He & L. Z. Liao, 2002. "Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 111-128, January.
    3. 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.
    4. B. S. He & H. Yang & S. L. Wang, 2000. "Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 337-356, August.
    5. Bin Gao & Feng Ma, 2018. "Symmetric Alternating Direction Method with Indefinite Proximal Regularization for Linearly Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 178-204, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Dolgopolik, Maksim V., 2021. "The alternating direction method of multipliers for finding the distance between ellipsoids," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    3. Zhao, Baining & Qian, Tong & Li, Weiwei & Xin, Yanli & Zhao, Wei & Lin, Zekang & Tang, Wenhu & Jin, Xin & Cao, Wangzhang & Pan, Tingzhe, 2024. "Fast distributed co-optimization of electricity and natural gas systems hedging against wind fluctuation and uncertainty," Energy, Elsevier, vol. 298(C).
    4. Maryam Yashtini, 2022. "Convergence and rate analysis of a proximal linearized ADMM for nonconvex nonsmooth optimization," Journal of Global Optimization, Springer, vol. 84(4), pages 913-939, December.
    5. Zamani, Moslem & Abbaszadehpeivasti, Hadi & de Klerk, Etienne, 2024. "The exact worst-case convergence rate of the alternating direction method of multipliers," Other publications TiSEM f30ae9e6-ed19-423f-bd1e-0, Tilburg University, School of Economics and Management.
    6. M.A. Noor, 2002. "Proximal Methods for Mixed Quasivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 453-459, November.
    7. Bingsheng He & Li-Zhi Liao & Xiang Wang, 2012. "Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods," Computational Optimization and Applications, Springer, vol. 51(2), pages 649-679, March.
    8. William W. Hager & Hongchao Zhang, 2020. "Convergence rates for an inexact ADMM applied to separable convex optimization," Computational Optimization and Applications, Springer, vol. 77(3), pages 729-754, December.
    9. Zhong-bao Wang & Xue Chen & Jiang Yi & Zhang-you Chen, 2022. "Inertial projection and contraction algorithms with larger step sizes for solving quasimonotone variational inequalities," Journal of Global Optimization, Springer, vol. 82(3), pages 499-522, March.
    10. 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.
    11. Kun Jin & Yevgeniy Vorobeychik & Mingyan Liu, 2021. "Multi-Scale Games: Representing and Solving Games on Networks with Group Structure," Papers 2101.08314, arXiv.org.
    12. Myungjin Kim & Li Wang & Yuyu Zhou, 2021. "Spatially Varying Coefficient Models with Sign Preservation of the Coefficient Functions," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 26(3), pages 367-386, September.
    13. Yao, Yu & Zhu, Xiaoning & Dong, Hongyu & Wu, Shengnan & Wu, Hailong & Carol Tong, Lu & Zhou, Xuesong, 2019. "ADMM-based problem decomposition scheme for vehicle routing problem with time windows," Transportation Research Part B: Methodological, Elsevier, vol. 129(C), pages 156-174.
    14. Bingsheng He & Feng Ma & Xiaoming Yuan, 2020. "Optimally linearizing the alternating direction method of multipliers for convex programming," Computational Optimization and Applications, Springer, vol. 75(2), pages 361-388, March.
    15. Min Zhang & Deren Han & Gang Qian & Xihong Yan, 2012. "A New Decomposition Method for Variational Inequalities with Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 675-695, March.
    16. Lijun Xu & Bo Yu & Yin Zhang, 2017. "An alternating direction and projection algorithm for structure-enforced matrix factorization," Computational Optimization and Applications, Springer, vol. 68(2), pages 333-362, November.
    17. Bingsheng He & Li-Zhi Liao & Xiang Wang, 2012. "Proximal-like contraction methods for monotone variational inequalities in a unified framework II: general methods and numerical experiments," Computational Optimization and Applications, Springer, vol. 51(2), pages 681-708, March.
    18. Xingju Cai & Guoyong Gu & Bingsheng He, 2014. "On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators," Computational Optimization and Applications, Springer, vol. 57(2), pages 339-363, March.
    19. Zheng Peng & Wenxing Zhu, 2013. "An Alternating Direction Method for Nash Equilibrium of Two-Person Games with Alternating Offers," Journal of Optimization Theory and Applications, Springer, vol. 157(2), pages 533-551, May.
    20. L. Z. Liao, 2005. "A Continuous Method for Convex Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 207-226, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:811-:d:1602587. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.