IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v208y2026i1d10.1007_s10957-025-02832-z.html
   My bibliography  Save this article

Derivative-Free Optimization on Riemannian Manifolds Using Simplex Gradient Approximations

Author

Listed:
  • Shahabeddin Najafi

    (Shahid Beheshti University)

  • Masoud Hajarian

    (Shahid Beheshti University)

Abstract

In optimization problems with complex or unknown gradients, using a derivative-free algorithm is an efficient approach. In this paper, we present a new derivative-free optimization method designed for problems in which the search space is a Riemannian manifold. The method utilizes a simplex gradient approximation and incorporates a line search strategy. We state the conditions under which the proposed algorithm is well-defined and establish its convergence to critical points on Riemannian manifolds from any starting point. Lastly, we demonstrate the practical implementation of this technique on two commonly used manifolds and compare its performance to some existing Riemannian derivative-free methods.

Suggested Citation

  • Shahabeddin Najafi & Masoud Hajarian, 2026. "Derivative-Free Optimization on Riemannian Manifolds Using Simplex Gradient Approximations," Journal of Optimization Theory and Applications, Springer, vol. 208(1), pages 1-22, January.
    • Handle: RePEc:spr:joptap:v:208:y:2026:i:1:d:10.1007_s10957-025-02832-z
    DOI: 10.1007/s10957-025-02832-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-025-02832-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-025-02832-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. W. L. Hare & Y. Lucet, 2014. "Derivative-Free Optimization Via Proximal Point Methods," Journal of Optimization Theory and Applications, Springer, vol. 160(1), pages 204-220, January.
    2. Jiaxiang Li & Krishnakumar Balasubramanian & Shiqian Ma, 2023. "Stochastic Zeroth-Order Riemannian Derivative Estimation and Optimization," Mathematics of Operations Research, INFORMS, vol. 48(2), pages 1183-1211, May.
    3. G. Cocchi & G. Liuzzi & A. Papini & M. Sciandrone, 2018. "An implicit filtering algorithm for derivative-free multiobjective optimization with box constraints," Computational Optimization and Applications, Springer, vol. 69(2), pages 267-296, March.
    4. G. Di Pillo & G. Liuzzi & S. Lucidi & V. Piccialli & F. Rinaldi, 2016. "A DIRECT-type approach for derivative-free constrained global optimization," Computational Optimization and Applications, Springer, vol. 65(2), pages 361-397, November.
    5. A. M. Bagirov & B. Karasözen & M. Sezer, 2008. "Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 317-334, May.
    6. 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.
    7. Shahabeddin Najafi & Masoud Hajarian, 2024. "Multiobjective BFGS method for optimization on Riemannian manifolds," Computational Optimization and Applications, Springer, vol. 87(2), pages 337-354, March.
    8. 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.
    9. Shahabeddin Najafi & Masoud Hajarian, 2025. "A Globally Convergent Riemannian Memory Gradient Method," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-17, August.
    10. Giampaolo Liuzzi & Stefano Lucidi & Francesco Rinaldi, 2015. "Derivative-Free Methods for Mixed-Integer Constrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 933-965, March.
    11. Tatsiana Levina & Yuri Levin & Jeff McGill & Mikhail Nediak, 2009. "Dynamic Pricing with Online Learning and Strategic Consumers: An Application of the Aggregating Algorithm," Operations Research, INFORMS, vol. 57(2), pages 327-341, April.
    12. Gianni Pillo & Stefano Lucidi & Francesco Rinaldi, 2015. "A Derivative-Free Algorithm for Constrained Global Optimization Based on Exact Penalty Functions," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 862-882, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. M. Fernanda P. Costa & Ana Maria A. C. Rocha & Edite M. G. P. Fernandes, 2018. "Filter-based DIRECT method for constrained global optimization," Journal of Global Optimization, Springer, vol. 71(3), pages 517-536, July.
    2. Donald R. Jones & Joaquim R. R. A. Martins, 2021. "The DIRECT algorithm: 25 years Later," Journal of Global Optimization, Springer, vol. 79(3), pages 521-566, March.
    3. Candelieri Antonio, 2021. "Sequential model based optimization of partially defined functions under unknown constraints," Journal of Global Optimization, Springer, vol. 79(2), pages 281-303, February.
    4. Ubaldo M. García Palomares, 2023. "Convergence of derivative-free nonmonotone Direct Search Methods for unconstrained and box-constrained mixed-integer optimization," Computational Optimization and Applications, Springer, vol. 85(3), pages 821-856, July.
    5. Najmeh Hoseini Monjezi, 2025. "Bundle trust region algorithm based on linear subproblem," Journal of Global Optimization, Springer, vol. 92(1), pages 87-109, May.
    6. Arthur Charpentier & Romuald Élie & Carl Remlinger, 2023. "Reinforcement Learning in Economics and Finance," Computational Economics, Springer;Society for Computational Economics, vol. 62(1), pages 425-462, June.
    7. Adil M. Bagirov & Julien Ugon & Hijran G. Mirzayeva, 2015. "Nonsmooth Optimization Algorithm for Solving Clusterwise Linear Regression Problems," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 755-780, March.
    8. Miranda-Pinto, Jorge & Murphy, Daniel & Walsh, Kieran James & Young, Eric R., 2025. "A model of expenditure shocks," Journal of Monetary Economics, Elsevier, vol. 154(C).
    9. Hélène Le Cadre & Bernardo Pagnoncelli & Tito Homem-De-Mello & Olivier Beaude, 2018. "Designing Coalition-Based Fair and Stable Pricing Mechanisms Under Private Information on Consumers' Reservation Prices," Working Papers hal-01353763, HAL.
    10. Jean Bigeon & Sébastien Le Digabel & Ludovic Salomon, 2021. "DMulti-MADS: mesh adaptive direct multisearch for bound-constrained blackbox multiobjective optimization," Computational Optimization and Applications, Springer, vol. 79(2), pages 301-338, June.
    11. Omar Besbes & Denis Sauré, 2014. "Dynamic Pricing Strategies in the Presence of Demand Shifts," Manufacturing & Service Operations Management, INFORMS, vol. 16(4), pages 513-528, October.
    12. G. Liuzzi & S. Lucidi & V. Piccialli, 2016. "Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization," Computational Optimization and Applications, Springer, vol. 65(2), pages 449-475, November.
    13. J. Michael Harrison & N. Bora Keskin & Assaf Zeevi, 2012. "Bayesian Dynamic Pricing Policies: Learning and Earning Under a Binary Prior Distribution," Management Science, INFORMS, vol. 58(3), pages 570-586, March.
    14. Jiang, Yuanchun & Liu, Yezheng & Shang, Jennifer & Yildirim, Pinar & Zhang, Qingfu, 2018. "Optimizing online recurring promotions for dual-channel retailers: Segmented markets with multiple objectives," European Journal of Operational Research, Elsevier, vol. 267(2), pages 612-627.
    15. Juliane Müller & Joshua D. Woodbury, 2017. "GOSAC: global optimization with surrogate approximation of constraints," Journal of Global Optimization, Springer, vol. 69(1), pages 117-136, September.
    16. E. F. Campana & M. Diez & G. Liuzzi & S. Lucidi & R. Pellegrini & V. Piccialli & F. Rinaldi & A. Serani, 2018. "A multi-objective DIRECT algorithm for ship hull optimization," Computational Optimization and Applications, Springer, vol. 71(1), pages 53-72, September.
    17. Zhao, Li & Tian, Peng & Xiangyong Li, 2012. "Dynamic pricing in the presence of consumer inertia," Omega, Elsevier, vol. 40(2), pages 137-148, April.
    18. Le Cadre, Hélène & Pagnoncelli, Bernardo & Homem-de-Mello, Tito & Beaude, Olivier, 2019. "Designing coalition-based fair and stable pricing mechanisms under private information on consumers’ reservation prices," European Journal of Operational Research, Elsevier, vol. 272(1), pages 270-291.
    19. K. C. Kiwiel, 2010. "Improved Convergence Result for the Discrete Gradient and Secant Methods for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 69-75, January.
    20. Linas Stripinis & Remigijus Paulavičius, 2024. "Lipschitz-inspired HALRECT algorithm for derivative-free global optimization," Journal of Global Optimization, Springer, vol. 88(1), pages 139-169, January.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:208:y:2026:i:1:d:10.1007_s10957-025-02832-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.