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Global optimization based on bisection of rectangles, function values at diagonals, and a set of Lipschitz constants

Author

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  • Remigijus Paulavičius

    (Vilnius University)

  • Lakhdar Chiter

    (University of Sétif 1)

  • Julius Žilinskas

    (Vilnius University)

Abstract

We consider a global optimization problem for Lipschitz-continuous functions with an unknown Lipschitz constant. Our approach is based on the well-known DIRECT (DIviding RECTangles) algorithm and motivated by the diagonal partitioning strategy. One of the main advantages of the diagonal partitioning scheme is that the objective function is evaluated at two points at each hyper-rectangle and, therefore, more comprehensive information about the objective function is considered than using the central sampling strategy used in most DIRECT-type algorithms. In this paper, we introduce a new DIRECT-type algorithm, which we call BIRECT (BIsecting RECTangles). In this algorithm, a bisection is used instead of a trisection which is typical for diagonal-based and DIRECT-type algorithms. The bisection is preferable to the trisection because of the shapes of hyper-rectangles, but usual evaluation of the objective function at the center or at the endpoints of the diagonal are not favorable for bisection. In the proposed algorithm the objective function is evaluated at two points on the diagonal equidistant between themselves and the endpoints of a diagonal. This sampling strategy enables reuse of the sampling points in descendant hyper-rectangles. The developed algorithm gives very competitive numerical results compared to the DIRECT algorithm and its well know modifications.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:71:y:2018:i:1:d:10.1007_s10898-016-0485-6
    DOI: 10.1007/s10898-016-0485-6
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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.
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    8. Qunfeng Liu & Wanyou Cheng, 2014. "A modified DIRECT algorithm with bilevel partition," Journal of Global Optimization, Springer, vol. 60(3), pages 483-499, November.
    9. 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.
    10. 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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    Cited by:

    1. Yue Chen & Jian Shi & Xiao-Jian Yi, 2021. "Design Improvement for Complex Systems with Uncertainty," Mathematics, MDPI, vol. 9(11), pages 1-20, May.
    2. 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.
    3. Khalid Abdulaziz Alnowibet & Salem Mahdi & Ahmad M. Alshamrani & Karam M. Sallam & Ali Wagdy Mohamed, 2022. "A Family of Hybrid Stochastic Conjugate Gradient Algorithms for Local and Global Minimization Problems," Mathematics, MDPI, vol. 10(19), pages 1-37, October.
    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. 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.
    6. Linas Stripinis & Remigijus Paulavičius, 2023. "Novel Algorithm for Linearly Constrained Derivative Free Global Optimization of Lipschitz Functions," Mathematics, MDPI, vol. 11(13), pages 1-19, June.
    7. 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.

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