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Worst Case Complexity Bounds for Linesearch-Type Derivative-Free Algorithms

Author

Listed:
  • Andrea Brilli

    (Sapienza University of Rome)

  • Morteza Kimiaei

    (Universität Wien)

  • Giampaolo Liuzzi

    (Sapienza University of Rome)

  • Stefano Lucidi

    (Sapienza University of Rome)

Abstract

This paper is devoted to the analysis of worst case complexity bounds for linesearch-type derivative-free algorithms for the minimization of general non-convex smooth functions. We consider a derivative-free algorithm based on a linesearch extrapolation technique. First we prove that it enjoys the same complexity properties which have been proved for pattern and direct search algorithms, namely that the number of iterations and of function evaluations required to drive the norm of the gradient of the objective function below a given threshold $$\epsilon $$ ϵ for the first time is $${{\mathcal {O}}}(\epsilon ^{-2})$$ O ( ϵ - 2 ) in the worst case. This is the first contribution proving worst-case complexity properties for derivative-free linesearch-type algorithms. Then we show that the lineasearch approach used by the described algorithm allows us to guarantee the further property that the number of iterations such that the norm of the gradient is bigger than $$\epsilon $$ ϵ is $$\mathcal{O}(\epsilon ^{-2})$$ O ( ϵ - 2 ) in the worst case.

Suggested Citation

  • Andrea Brilli & Morteza Kimiaei & Giampaolo Liuzzi & Stefano Lucidi, 2024. "Worst Case Complexity Bounds for Linesearch-Type Derivative-Free Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 419-454, October.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:1:d:10.1007_s10957-024-02519-x
    DOI: 10.1007/s10957-024-02519-x
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    References listed on IDEAS

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    1. 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.
    2. Charles Audet & Kwassi Joseph Dzahini & Michael Kokkolaras & Sébastien Le Digabel, 2021. "Stochastic mesh adaptive direct search for blackbox optimization using probabilistic estimates," Computational Optimization and Applications, Springer, vol. 79(1), pages 1-34, May.
    3. V. S. Amaral & R. Andreani & E. G. Birgin & D. S. Marcondes & J. M. Martínez, 2022. "On complexity and convergence of high-order coordinate descent algorithms for smooth nonconvex box-constrained minimization," Journal of Global Optimization, Springer, vol. 84(3), pages 527-561, November.
    4. Jeffrey Larson & Stephen C. Billups, 2016. "Stochastic derivative-free optimization using a trust region framework," Computational Optimization and Applications, Springer, vol. 64(3), pages 619-645, July.
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