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

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  • 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
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    References listed on IDEAS

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    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. 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.
    5. 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.
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    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.

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