IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v409y2021ics0096300321004768.html
   My bibliography  Save this article

The alternating direction method of multipliers for finding the distance between ellipsoids

Author

Listed:
  • Dolgopolik, Maksim V.

Abstract

We study several versions of the alternating direction method of multipliers (ADMM) for solving the convex problem of finding the distance between two ellipsoids and the nonconvex problem of finding the distance between the boundaries of two ellipsoids. In the convex case we present the ADMM with and without automatic penalty updates and demonstrate via numerical experiments on problems of various dimensions that our methods significantly outperform all other existing methods for finding the distance between ellipsoids. In the nonconvex case we propose a heuristic rule for updating the penalty parameter and a heuristic restarting procedure (a heuristic choice of a new starting for point for the second run of the algorithm). The restarting procedure was verified numerically with the use of a global method based on KKT optimality conditions. The results of numerical experiments on various test problems showed that this procedure always allows one to find a globally optimal solution in the nonconvex case. Furthermore, the numerical experiments also demonstrated that our version of the ADMM significantly outperforms existing methods for finding the distance between the boundaries of ellipsoids on problems of moderate and high dimensions.

Suggested Citation

  • Dolgopolik, Maksim V., 2021. "The alternating direction method of multipliers for finding the distance between ellipsoids," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    • Handle: RePEc:eee:apmaco:v:409:y:2021:i:c:s0096300321004768
    DOI: 10.1016/j.amc.2021.126387
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300321004768
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2021.126387?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    4. Davood Hajinezhad & Qingjiang Shi, 2018. "Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications," Journal of Global Optimization, Springer, vol. 70(1), pages 261-288, January.
    5. Zheng Peng & Jianli Chen & Wenxing Zhu, 2015. "A proximal alternating direction method of multipliers for a minimization problem with nonconvex constraints," Journal of Global Optimization, Springer, vol. 62(4), pages 711-728, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Shougui Zhang & Xiyong Cui & Guihua Xiong & Ruisheng Ran, 2024. "An Optimal ADMM for Unilateral Obstacle Problems," Mathematics, MDPI, vol. 12(12), pages 1-16, June.

    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. Le Thi Khanh Hien & Duy Nhat Phan & Nicolas Gillis, 2022. "Inertial alternating direction method of multipliers for non-convex non-smooth optimization," Computational Optimization and Applications, Springer, vol. 83(1), pages 247-285, September.
    2. Kun Jin & Yevgeniy Vorobeychik & Mingyan Liu, 2021. "Multi-Scale Games: Representing and Solving Games on Networks with Group Structure," Papers 2101.08314, arXiv.org.
    3. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "A Block Inertial Bregman Proximal Algorithm for Nonsmooth Nonconvex Problems with Application to Symmetric Nonnegative Matrix Tri-Factorization," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 234-258, July.
    4. 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.
    5. Zhiqing Meng & Min Jiang & Rui Shen & Leiyan Xu & Chuangyin Dang, 2021. "An objective penalty function method for biconvex programming," Journal of Global Optimization, Springer, vol. 81(3), pages 599-620, November.
    6. 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.
    7. Eyal Cohen & Nadav Hallak & Marc Teboulle, 2022. "A Dynamic Alternating Direction of Multipliers for Nonconvex Minimization with Nonlinear Functional Equality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 324-353, June.
    8. Hou, Yanqiu & Bao, Minglei & Sang, Maosheng & Ding, Yi, 2024. "A market framework to exploit the multi-energy operating reserve of smart energy hubs in the integrated electricity-gas systems," Applied Energy, Elsevier, vol. 357(C).
    9. Z. K. Jiang & X. M. Yuan, 2010. "New Parallel Descent-like Method for Solving a Class of Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 311-323, May.
    10. Dirk A. Lorenz & Quoc Tran-Dinh, 2019. "Non-stationary Douglas–Rachford and alternating direction method of multipliers: adaptive step-sizes and convergence," Computational Optimization and Applications, Springer, vol. 74(1), pages 67-92, September.
    11. Yunhai Xiao & Hong Zhu & Soon-Yi Wu, 2013. "Primal and dual alternating direction algorithms for ℓ 1 -ℓ 1 -norm minimization problems in compressive sensing," Computational Optimization and Applications, Springer, vol. 54(2), pages 441-459, March.
    12. Zhongming Wu & Min Li, 2019. "General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 129-158, May.
    13. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 79(3), pages 681-715, July.
    14. Yong-Jin Liu & Jing Yu, 2022. "A Semismooth Newton-based Augmented Lagrangian Algorithm for Density Matrix Least Squares Problems," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 749-779, December.
    15. Wu, Yiqian & Zhang, Xuan & Sun, Hongbin, 2021. "A multi-time-scale autonomous energy trading framework within distribution networks based on blockchain," Applied Energy, Elsevier, vol. 287(C).
    16. Xianfu Wang & Ziyuan Wang, 2022. "Malitsky-Tam forward-reflected-backward splitting method for nonconvex minimization problems," Computational Optimization and Applications, Springer, vol. 82(2), pages 441-463, June.
    17. 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.
    18. Abbaszadehpeivasti, Hadi, 2024. "Performance analysis of optimization methods for machine learning," Other publications TiSEM 3050a62d-1a1f-494e-99ef-7, Tilburg University, School of Economics and Management.
    19. Utsav Sadana & Erick Delage, 2023. "The Value of Randomized Strategies in Distributionally Robust Risk-Averse Network Interdiction Problems," INFORMS Journal on Computing, INFORMS, vol. 35(1), pages 216-232, January.
    20. N. Aybat & G. Iyengar, 2015. "An alternating direction method with increasing penalty for stable principal component pursuit," Computational Optimization and Applications, Springer, vol. 61(3), pages 635-668, July.

    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:eee:apmaco:v:409:y:2021:i:c:s0096300321004768. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.