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Regularized Jacobi-type ADMM-methods for a class of separable convex optimization problems in Hilbert spaces

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  • Eike Börgens

    (University of Würzburg)

  • Christian Kanzow

    (University of Würzburg)

Abstract

We consider a regularized version of a Jacobi-type alternating direction method of multipliers (ADMM) for the solution of a class of separable convex optimization problems in a Hilbert space. The analysis shows that this method is equivalent to the standard proximal-point method applied in a Hilbert space with a transformed scalar product. The method therefore inherits the known convergence results from the proximal-point method and allows suitable modifications to get a strongly convergent variant. Some additional properties are also shown by exploiting the particular structure of the ADMM-type solution method. Applications and numerical results are provided for the domain decomposition method and potential (generalized) Nash equilibrium problems in a Hilbert space setting.

Suggested Citation

  • Eike Börgens & Christian Kanzow, 2019. "Regularized Jacobi-type ADMM-methods for a class of separable convex optimization problems in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 73(3), pages 755-790, July.
    • Handle: RePEc:spr:coopap:v:73:y:2019:i:3:d:10.1007_s10589-019-00087-9
    DOI: 10.1007/s10589-019-00087-9
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    References listed on IDEAS

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    1. Yunda Dong, 2015. "Comments on “The Proximal Point Algorithm Revisited”," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 343-349, July.
    2. Yunda Dong, 2014. "The Proximal Point Algorithm Revisited," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 478-489, May.
    3. Guoyong Gu & Bingsheng He & Xiaoming Yuan, 2014. "Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach," Computational Optimization and Applications, Springer, vol. 59(1), pages 135-161, October.
    4. Min Tao & Xiaoming Yuan, 2012. "An inexact parallel splitting augmented Lagrangian method for monotone variational inequalities with separable structures," Computational Optimization and Applications, Springer, vol. 52(2), pages 439-461, June.
    5. Bingsheng He & Min Tao & Xiaoming Yuan, 2017. "Convergence Rate Analysis for the Alternating Direction Method of Multipliers with a Substitution Procedure for Separable Convex Programming," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 662-691, August.
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    Cited by:

    1. Jinbao Jian & Chen Zhang & Jianghua Yin & Linfeng Yang & Guodong Ma, 2020. "Monotone Splitting Sequential Quadratic Optimization Algorithm with Applications in Electric Power Systems," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 226-247, July.

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