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Linear scaling and the DIRECT algorithm

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  • Qunfeng Liu

Abstract

In this paper, we discuss the performance of the DIRECT global optimization algorithm on problems with linear scaling. We show with computations that the performance of DIRECT can be affected by linear scaling of the objective function. We also provide a theoretical result which shows that DIRECT does not perform well when the absolute value of the objective function is large enough. Then we present DIRECT-a, a modification of DIRECT, to eliminate the sensitivity to linear scaling of the objective function. We prove theoretically that linear scaling of the objective function does not affect the performance of DIRECT-a. Similarly, we prove that some modifications of DIRECT are also unaffected by linear scaling of the objective function, while the original DIRECT algorithm is sensitive to linear scaling. Numerical results in this paper show that DIRECT-a is more robust than the original DIRECT algorithm, which support the theoretical results. Numerical results also show that careful choices of the parameter ε can help DIRECT perform well when the objective function is poorly linearly scaled. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Qunfeng Liu, 2013. "Linear scaling and the DIRECT algorithm," Journal of Global Optimization, Springer, vol. 56(3), pages 1233-1245, July.
    • Handle: RePEc:spr:jglopt:v:56:y:2013:i:3:p:1233-1245
    DOI: 10.1007/s10898-012-9952-x
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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. Locatelli, Marco & Schoen, Fabio, 2012. "Local search based heuristics for global optimization: Atomic clusters and beyond," European Journal of Operational Research, Elsevier, vol. 222(1), pages 1-9.
    3. Eligius M.T. Hendrix & Boglárka G.-Tóth, 2010. "Introduction to Nonlinear and Global Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-88670-1, September.
    4. Weitao Sun & Yuan Dong, 2011. "Study of multiscale global optimization based on parameter space partition," Journal of Global Optimization, Springer, vol. 49(1), pages 149-172, January.
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    Cited by:

    1. 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.
    2. Jonas Mockus & Remigijus Paulavičius & Dainius Rusakevičius & Dmitrij Šešok & Julius Žilinskas, 2017. "Application of Reduced-set Pareto-Lipschitzian Optimization to truss optimization," Journal of Global Optimization, Springer, vol. 67(1), pages 425-450, January.
    3. 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.
    4. 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.
    5. 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.
    6. Qunfeng Liu & Wanyou Cheng, 2014. "A modified DIRECT algorithm with bilevel partition," Journal of Global Optimization, Springer, vol. 60(3), pages 483-499, November.
    7. 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.
    8. Qunfeng Liu & Jinping Zeng, 2015. "Global optimization by multilevel partition," Journal of Global Optimization, Springer, vol. 61(1), pages 47-69, January.

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