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Accelerated sampling Kaczmarz Motzkin algorithm for the linear feasibility problem

Author

Listed:
  • Md Sarowar Morshed

    (Northeastern University)

  • Md Saiful Islam

    (Northeastern University)

  • Md. Noor-E-Alam

    (Northeastern University)

Abstract

The Sampling Kaczmarz Motzkin (SKM) algorithm is a generalized method for solving large-scale linear systems of inequalities. Having its root in the relaxation method of Agmon, Schoenberg, and Motzkin and the randomized Kaczmarz method, SKM outperforms the state-of-the-art methods in solving large-scale Linear Feasibility (LF) problems. Motivated by SKM’s success, in this work, we propose an Accelerated Sampling Kaczmarz Motzkin (ASKM) algorithm which achieves better convergence compared to the standard SKM algorithm on ill-conditioned problems. We provide a thorough convergence analysis for the proposed accelerated algorithm and validate the results with various numerical experiments. We compare the performance and effectiveness of ASKM algorithm with SKM, Interior Point Method (IPM) and Active Set Method (ASM) on randomly generated instances as well as Netlib LPs. In most of the test instances, the proposed ASKM algorithm outperforms the other state-of-the-art methods.

Suggested Citation

  • Md Sarowar Morshed & Md Saiful Islam & Md. Noor-E-Alam, 2020. "Accelerated sampling Kaczmarz Motzkin algorithm for the linear feasibility problem," Journal of Global Optimization, Springer, vol. 77(2), pages 361-382, June.
    • Handle: RePEc:spr:jglopt:v:77:y:2020:i:2:d:10.1007_s10898-019-00850-6
    DOI: 10.1007/s10898-019-00850-6
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. S. Bellavia, 1998. "Inexact Interior-Point Method," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 109-121, January.
    3. Chengjing Wang & Aimin Xu, 2013. "An Inexact Accelerated Proximal Gradient Method and a Dual Newton-CG Method for the Maximal Entropy Problem," Journal of Optimization Theory and Applications, Springer, vol. 157(2), pages 436-450, May.
    4. 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).
    5. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Zhang, Yanjun & Li, Hanyu, 2023. "Splitting-based randomized iterative methods for solving indefinite least squares problem," Applied Mathematics and Computation, Elsevier, vol. 446(C).

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