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An Optimal ADMM for Unilateral Obstacle Problems

Author

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  • Shougui Zhang

    (School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China)

  • Xiyong Cui

    (CISDI Information Technology Co., Ltd., Chongqing 401120, China)

  • Guihua Xiong

    (School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China)

  • Ruisheng Ran

    (School of Computer and Information Science, Chongqing Normal University, Chongqing 401331, China)

Abstract

We propose a new alternating direction method of multipliers (ADMM) with an optimal parameter for the unilateral obstacle problem. We first use the five-point difference scheme to discretize the problem. Then, we present an augmented Lagrangian by introducing an auxiliary unknown, and an ADMM is applied to the corresponding saddle-point problem. Through eliminating the primal and auxiliary unknowns, a pure dual algorithm is then used. The convergence of the proposed method is analyzed, and a simple strategy is presented for selecting the optimal parameter, with the largest and smallest eigenvalues of the iterative matrix. Several numerical experiments confirm the theoretical findings of this study.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1901-:d:1418091
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    References listed on IDEAS

    as
    1. Dolgopolik, Maksim V., 2021. "The alternating direction method of multipliers for finding the distance between ellipsoids," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    2. Wang, Fei & Eichholz, Joseph & Han, Weimin, 2018. "A two level algorithm for an obstacle problem," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 65-76.
    3. Zhang, Shougui, 2018. "Two projection methods for the solution of Signorini problems," Applied Mathematics and Computation, Elsevier, vol. 326(C), pages 75-86.
    4. David G. Luenberger & Yinyu Ye, 2016. "Linear and Nonlinear Programming," International Series in Operations Research and Management Science, Springer, edition 4, number 978-3-319-18842-3, December.
    5. Xu, Chao & Shi, Dongyang, 2019. "Superconvergence analysis of low order nonconforming finite element methods for variational inequality problem with displacement obstacle," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 1-11.
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