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A Novel Approach to Enhance DIRECT -Type Algorithms for Hyper-Rectangle Identification

Author

Listed:
  • Nazih-Eddine Belkacem

    (Department of Mathematics, Faculty of Sciences, Ferhat-Abbas University of Sétif 1, Sétif 19000, Algeria)

  • Lakhdar Chiter

    (Department of Mathematics, Faculty of Sciences, Ferhat-Abbas University of Sétif 1, Sétif 19000, Algeria
    Fundamental and Numerical Mathematics Laboratory (LMFN), Ferhat-Abbas University, Sétif 19000, Algeria)

  • Mohammed Louaked

    (Laboratoire de Mathématiques Nicolas Oresme, Université de Caen, Campus II, Boulevard Maréchal Juin, B.P. 5186, 14032 Caen, France)

Abstract

This paper introduces novel enhancements to the most recent versions of DIRECT-type algorithms, especially when dealing with solutions located at the hyper-rectangle vertices. The BIRECT algorithm encounters difficulties in efficiently sampling points at the boundaries of the feasible region, leading to potential slowdowns in convergence. This issue is particularly pronounced when the optimal solution resides near the boundary. Our research explores diverse approaches, with a primary focus on incorporating a grouping strategy for hyper-rectangles of similar sizes. This categorization into different classes, constrained by a predefined threshold, aims to enhance computational efficiency, particularly involving a substantial number of hyper-rectangles of varying sizes. To further improve the new algorithm’s efficiency, we implemented a mechanism to prevent oversampling and mitigate redundancy in sampling at shared vertices within descendant sub-regions. Comparisons with several DIRECT-type algorithms highlight the promising nature of the proposed algorithms as a global optimization solution. Statistical analyses, including Friedman and Wilcoxon tests, demonstrated the effectiveness of the improvements introduced in this new algorithm.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:283-:d:1319616
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. G. Liuzzi & S. Lucidi & V. Piccialli, 2016. "Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization," Computational Optimization and Applications, Springer, vol. 65(2), pages 449-475, November.
    4. Remigijus Paulavičius & Julius Žilinskas, 2014. "Simplicial Lipschitz optimization without the Lipschitz constant," Journal of Global Optimization, Springer, vol. 59(1), pages 23-40, May.
    5. 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.
    6. 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.
    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.
    8. Anatoly Zhigljavsky & Antanas Žilinskas, 2008. "Stochastic Global Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-74740-8, June.
    9. 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.
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