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Retraction-Based Direct Search Methods for Derivative Free Riemannian Optimization

Author

Listed:
  • Vyacheslav Kungurtsev

    (Czech Technical University)

  • Francesco Rinaldi

    (Università di Padova)

  • Damiano Zeffiro

    (Università di Padova)

Abstract

Direct search methods represent a robust and reliable class of algorithms for solving black-box optimization problems. In this paper, the application of those strategies is exported to Riemannian optimization, wherein minimization is to be performed with respect to variables restricted to lie on a manifold. More specifically, classic and linesearch extrapolated variants of direct search are considered, and tailored strategies are devised for the minimization of both smooth and nonsmooth functions, by making use of retractions. A class of direct search algorithms for minimizing nonsmooth objectives on a Riemannian manifold without having access to (sub)derivatives is analyzed for the first time in the literature. Along with convergence guarantees, a set of numerical performance illustrations on a standard set of problems is provided.

Suggested Citation

  • Vyacheslav Kungurtsev & Francesco Rinaldi & Damiano Zeffiro, 2024. "Retraction-Based Direct Search Methods for Derivative Free Riemannian Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1710-1735, November.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:2:d:10.1007_s10957-023-02268-3
    DOI: 10.1007/s10957-023-02268-3
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    References listed on IDEAS

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    1. David W. Dreisigmeyer, 2018. "Direct Search Methods on Reductive Homogeneous Spaces," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 585-604, March.
    2. S. Hosseini & M. R. Pouryayevali, 2013. "Nonsmooth Optimization Techniques on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 328-342, August.
    3. Charles Audet & Sébastien Le Digabel & Mathilde Peyrega, 2015. "Linear equalities in blackbox optimization," Computational Optimization and Applications, Springer, vol. 61(1), pages 1-23, May.
    4. Yurii NESTEROV & Vladimir SPOKOINY, 2017. "Random gradient-free minimization of convex functions," LIDAM Reprints CORE 2851, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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