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Out of the Niche: Using Direct Search Methods to Find Multiple Global Optima

Author

Listed:
  • Javier Cano

    (Department of Computer Science and Statistics, Rey Juan Carlos University, 28933 Madrid, Spain
    Department of Statistics, University of Auckland, Auckland 1010, New Zealand)

  • Cesar Alfaro

    (Department of Computer Science and Statistics, Rey Juan Carlos University, 28933 Madrid, Spain)

  • Javier Gomez

    (Department of Computer Science and Statistics, Rey Juan Carlos University, 28933 Madrid, Spain)

  • Abraham Duarte

    (Department of Computer Science and Statistics, Rey Juan Carlos University, 28933 Madrid, Spain)

Abstract

Multimodal optimization deals with problems where multiple feasible global solutions coexist. Despite sharing a common objective function value, some global optima may be preferred to others for various reasons. In such cases, it is paramount to devise methods that are able to find as many global optima as possible within an affordable computational budget. Niching strategies have received an overwhelming attention in recent years as the most suitable technique to tackle these kinds of problems. In this paper we explore a different approach, based on a systematic yet versatile use of traditional direct search methods. When tested over reference benchmark functions, our proposal, despite its apparent simplicity, noticeably resists the comparison with state-of-the-art niching methods in most cases, both in the number of global optima found and in the number of function evaluations required. However, rather than trying to outperform niching methods—far more elaborated—our aim is to enrich them with the knowledge gained from exploiting the distinctive features of direct search methods. To that end, we propose two new performance measures that can be used to evaluate, compare and monitor the progress of optimization algorithms of (possibly) very different nature in their effort to find as many global optima of a given multimodal objective function as possible. We believe that adopting these metrics as reference criteria could lead to more sophisticated and computationally-efficient algorithms, which could benefit from the brute force of derivative-free local search methods.

Suggested Citation

  • Javier Cano & Cesar Alfaro & Javier Gomez & Abraham Duarte, 2022. "Out of the Niche: Using Direct Search Methods to Find Multiple Global Optima," Mathematics, MDPI, vol. 10(9), pages 1-20, April.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1494-:d:806484
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    References listed on IDEAS

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    1. Zhao, Zhiwei & Yang, Jingming & Hu, Ziyu & Che, Haijun, 2016. "A differential evolution algorithm with self-adaptive strategy and control parameters based on symmetric Latin hypercube design for unconstrained optimization problems," European Journal of Operational Research, Elsevier, vol. 250(1), pages 30-45.
    2. Pierre Hansen & Nenad Mladenović & José Moreno Pérez, 2010. "Variable neighbourhood search: methods and applications," Annals of Operations Research, Springer, vol. 175(1), pages 367-407, March.
    3. Luis Rios & Nikolaos Sahinidis, 2013. "Derivative-free optimization: a review of algorithms and comparison of software implementations," Journal of Global Optimization, Springer, vol. 56(3), pages 1247-1293, July.
    4. 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.
    5. Chang, Kuo-Hao, 2012. "Stochastic Nelder–Mead simplex method – A new globally convergent direct search method for simulation optimization," European Journal of Operational Research, Elsevier, vol. 220(3), pages 684-694.
    6. Abraham Duarte & Rafael Martí & Fred Glover & Francisco Gortazar, 2011. "Hybrid scatter tabu search for unconstrained global optimization," Annals of Operations Research, Springer, vol. 183(1), pages 95-123, March.
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    Cited by:

    1. Árpád Bűrmen & Tadej Tuma, 2022. "Preface to the Special Issue on “Optimization Theory and Applications”," Mathematics, MDPI, vol. 10(24), pages 1-3, December.

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