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Mesh-based Nelder–Mead algorithm for inequality constrained optimization

Author

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  • Charles Audet

    (École Polytechnique de Montréal)

  • Christophe Tribes

    (École Polytechnique de Montréal)

Abstract

Despite the lack of theoretical and practical convergence support, the Nelder–Mead (NM) algorithm is widely used to solve unconstrained optimization problems. It is a derivative-free algorithm, that attempts iteratively to replace the worst point of a simplex by a better one. The present paper proposes a way to extend the NM algorithm to inequality constrained optimization. This is done through a search step of the mesh adaptive direct search (Mads) algorithm, inspired by the NM algorithm. The proposed algorithm does not suffer from the NM lack of convergence, but instead inherits from the totality of the Mads convergence analysis. Numerical experiments show an important improvement in the quality of the solutions produced using this search step.

Suggested Citation

  • Charles Audet & Christophe Tribes, 2018. "Mesh-based Nelder–Mead algorithm for inequality constrained optimization," Computational Optimization and Applications, Springer, vol. 71(2), pages 331-352, November.
    • Handle: RePEc:spr:coopap:v:71:y:2018:i:2:d:10.1007_s10589-018-0016-0
    DOI: 10.1007/s10589-018-0016-0
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    References listed on IDEAS

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    1. C.J. Price & I.D. Coope & D. Byatt, 2002. "A Convergent Variant of the Nelder–Mead Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 5-19, April.
    2. Fuchang Gao & Lixing Han, 2012. "Implementing the Nelder-Mead simplex algorithm with adaptive parameters," Computational Optimization and Applications, Springer, vol. 51(1), pages 259-277, January.
    3. 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.
    4. Charles Audet & Michael Kokkolaras & Sébastien Le Digabel & Bastien Talgorn, 2018. "Order-based error for managing ensembles of surrogates in mesh adaptive direct search," Journal of Global Optimization, Springer, vol. 70(3), pages 645-675, March.
    5. I. D. Coope & C. J. Price, 2000. "Frame Based Methods for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 107(2), pages 261-274, November.
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    Cited by:

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    3. Charles Audet & Edward Hallé-Hannan & Sébastien Le Digabel, 2023. "A General Mathematical Framework for Constrained Mixed-variable Blackbox Optimization Problems with Meta and Categorical Variables," SN Operations Research Forum, Springer, vol. 4(1), pages 1-37, March.
    4. Vladimir Dmitrievskii & Vladimir Prakht & Vadim Kazakbaev, 2019. "Design Optimization of a Permanent-Magnet Flux-Switching Generator for Direct-Drive Wind Turbines," Energies, MDPI, vol. 12(19), pages 1-15, September.
    5. Vladimir Prakht & Vladimir Dmitrievskii & Vadim Kazakbaev & Ekaterina Andriushchenko, 2021. "Comparison of Flux-Switching and Interior Permanent Magnet Synchronous Generators for Direct-Driven Wind Applications Based on Nelder–Mead Optimal Designing," Mathematics, MDPI, vol. 9(7), pages 1-16, March.
    6. Charles Audet & Jean Bigeon & Romain Couderc, 2021. "Combining Cross-Entropy and MADS Methods for Inequality Constrained Global Optimization," SN Operations Research Forum, Springer, vol. 2(3), pages 1-26, September.

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