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A Convergent Variant of the Nelder–Mead Algorithm

Author

Listed:
  • C.J. Price

    (University of Canterbury)

  • I.D. Coope

    (University of Canterbury)

  • D. Byatt

    (University of Canterbury)

Abstract

The Nelder–Mead algorithm (1965) for unconstrained optimization has been used extensively to solve parameter estimation and other problems. Despite its age, it is still the method of choice for many practitioners in the fields of statistics, engineering, and the physical and medical sciences because it is easy to code and very easy to use. It belongs to a class of methods which do not require derivatives and which are often claimed to be robust for problems with discontinuities or where the function values are noisy. Recently (1998), it has been shown that the method can fail to converge or converge to nonsolutions on certain classes of problems. Only very limited convergence results exist for a restricted class of problems in one or two dimensions. In this paper, a provably convergent variant of the Nelder–Mead simplex method is presented and analyzed. Numerical results are included to show that the modified algorithm is effective in practice.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:113:y:2002:i:1:d:10.1023_a:1014849028575
    DOI: 10.1023/A:1014849028575
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    References listed on IDEAS

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    1. 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:

    1. Ian Coope & Rachael Tappenden, 2019. "Efficient calculation of regular simplex gradients," Computational Optimization and Applications, Springer, vol. 72(3), pages 561-588, April.
    2. C.J. Price & I.D. Coope, 2003. "Frame-Based Ray Search Algorithms in Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 116(2), pages 359-377, February.
    3. Hsun-Heng Tsai & Chyun-Chau Fuh & Jeng-Rong Ho & Chih-Kuang Lin, 2021. "Design of Optimal Controllers for Unknown Dynamic Systems through the Nelder–Mead Simplex Method," Mathematics, MDPI, vol. 9(16), pages 1-14, August.
    4. 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.
    5. Henri Bonnel & C. Yalçın Kaya, 2010. "Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 93-112, October.
    6. Žiga Rojec & Tadej Tuma & Jernej Olenšek & Árpád Bűrmen & Janez Puhan, 2022. "Meta-Optimization of Dimension Adaptive Parameter Schema for Nelder–Mead Algorithm in High-Dimensional Problems," Mathematics, MDPI, vol. 10(13), pages 1-16, June.
    7. Min Xi & Wenyu Sun & Yannan Chen & Hailin Sun, 2020. "A derivative-free algorithm for spherically constrained optimization," Journal of Global Optimization, Springer, vol. 76(4), pages 841-861, April.

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