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MrDIRECT: a multilevel robust DIRECT algorithm for global optimization problems

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  • Qunfeng Liu
  • Jinping Zeng
  • Gang Yang

Abstract

Although DIRECT global optimization algorithm quickly gets close to the basin of the optimum, it often takes much longer to refine the solution to a high degree of accuracy. This behavior of DIRECT is similar to the “smooth mode phenomenon” encountered when solving linear systems discretized from partial differential equation (PDE). In the case of PDE, this smooth mode phenomenon can be eliminated efficiently by the multigrid algorithm in which the PDE solver is applied at different levels of discretization. In this paper we adapt the multigrid approach to a robust version of DIRECT algorithm, obtaining a “multilevel” robust DIRECT (MrDIRECT) algorithm. Although additional parameters are needed, our numerical results show that MrDIRECT is insensitive to the parameters, and the parameters setting proposed in this paper performs very well on the tested sets of benchmark problems, in terms of the speed with which the global optimum is found to a high degree of accuracy. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Qunfeng Liu & Jinping Zeng & Gang Yang, 2015. "MrDIRECT: a multilevel robust DIRECT algorithm for global optimization problems," Journal of Global Optimization, Springer, vol. 62(2), pages 205-227, June.
    • Handle: RePEc:spr:jglopt:v:62:y:2015:i:2:p:205-227
    DOI: 10.1007/s10898-014-0241-8
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    References listed on IDEAS

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    1. Giampaolo Liuzzi & Stefano Lucidi & Veronica Piccialli, 2010. "A partition-based global optimization algorithm," Journal of Global Optimization, Springer, vol. 48(1), pages 113-128, September.
    2. Qunfeng Liu, 2013. "Linear scaling and the DIRECT algorithm," Journal of Global Optimization, Springer, vol. 56(3), pages 1233-1245, July.
    3. Qunfeng Liu & Wanyou Cheng, 2014. "A modified DIRECT algorithm with bilevel partition," Journal of Global Optimization, Springer, vol. 60(3), pages 483-499, November.
    4. Remigijus Paulavičius & Yaroslav Sergeyev & Dmitri Kvasov & Julius Žilinskas, 2014. "Globally-biased Disimpl algorithm for expensive global optimization," Journal of Global Optimization, Springer, vol. 59(2), pages 545-567, July.
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    Citations

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    Cited by:

    1. Nazih-Eddine Belkacem & Lakhdar Chiter & Mohammed Louaked, 2024. "A Novel Approach to Enhance DIRECT -Type Algorithms for Hyper-Rectangle Identification," Mathematics, MDPI, vol. 12(2), pages 1-24, January.
    2. Albertas Gimbutas & Antanas Žilinskas, 2018. "An algorithm of simplicial Lipschitz optimization with the bi-criteria selection of simplices for the bi-section," Journal of Global Optimization, Springer, vol. 71(1), pages 115-127, May.
    3. Qunfeng Liu & Guang Yang & Zhongzhi Zhang & Jinping Zeng, 2017. "Improving the convergence rate of the DIRECT global optimization algorithm," Journal of Global Optimization, Springer, vol. 67(4), pages 851-872, April.
    4. Stripinis, Linas & Žilinskas, Julius & Casado, Leocadio G. & Paulavičius, Remigijus, 2021. "On MATLAB experience in accelerating DIRECT-GLce algorithm for constrained global optimization through dynamic data structures and parallelization," Applied Mathematics and Computation, Elsevier, vol. 390(C).
    5. Remigijus Paulavičius & Lakhdar Chiter & Julius Žilinskas, 2018. "Global optimization based on bisection of rectangles, function values at diagonals, and a set of Lipschitz constants," Journal of Global Optimization, Springer, vol. 71(1), pages 5-20, May.
    6. Donald R. Jones & Joaquim R. R. A. Martins, 2021. "The DIRECT algorithm: 25 years Later," Journal of Global Optimization, Springer, vol. 79(3), pages 521-566, March.
    7. M. Fernanda P. Costa & Ana Maria A. C. Rocha & Edite M. G. P. Fernandes, 2018. "Filter-based DIRECT method for constrained global optimization," Journal of Global Optimization, Springer, vol. 71(3), pages 517-536, July.
    8. Kaiwen Ma & Luis Miguel Rios & Atharv Bhosekar & Nikolaos V. Sahinidis & Sreekanth Rajagopalan, 2023. "Branch-and-Model: a derivative-free global optimization algorithm," Computational Optimization and Applications, Springer, vol. 85(2), pages 337-367, June.
    9. Linas Stripinis & Remigijus Paulavičius, 2022. "Experimental Study of Excessive Local Refinement Reduction Techniques for Global Optimization DIRECT-Type Algorithms," Mathematics, MDPI, vol. 10(20), pages 1-18, October.

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