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Accelerated primal–dual proximal block coordinate updating methods for constrained convex optimization

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  • Yangyang Xu

    (Rensselaer Polytechnic Institute)

  • Shuzhong Zhang

    (University of Minnesota)

Abstract

Block coordinate update (BCU) methods enjoy low per-update computational complexity because every time only one or a few block variables would need to be updated among possibly a large number of blocks. They are also easily parallelized and thus have been particularly popular for solving problems involving large-scale dataset and/or variables. In this paper, we propose a primal–dual BCU method for solving linearly constrained convex program with multi-block variables. The method is an accelerated version of a primal–dual algorithm proposed by the authors, which applies randomization in selecting block variables to update and establishes an O(1 / t) convergence rate under convexity assumption. We show that the rate can be accelerated to $$O(1/t^2)$$ O ( 1 / t 2 ) if the objective is strongly convex. In addition, if one block variable is independent of the others in the objective, we then show that the algorithm can be modified to achieve a linear rate of convergence. The numerical experiments show that the accelerated method performs stably with a single set of parameters while the original method needs to tune the parameters for different datasets in order to achieve a comparable level of performance.

Suggested Citation

  • Yangyang Xu & Shuzhong Zhang, 2018. "Accelerated primal–dual proximal block coordinate updating methods for constrained convex optimization," Computational Optimization and Applications, Springer, vol. 70(1), pages 91-128, May.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:1:d:10.1007_s10589-017-9972-z
    DOI: 10.1007/s10589-017-9972-z
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    References listed on IDEAS

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    1. Min Li & Defeng Sun & Kim-Chuan Toh, 2015. "A Convergent 3-Block Semi-Proximal ADMM for Convex Minimization Problems with One Strongly Convex Block," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 32(04), pages 1-19.
    2. NESTEROV, Yurii, 2012. "Efficiency of coordinate descent methods on huge-scale optimization problems," LIDAM Reprints CORE 2511, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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