IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v201y2024i1d10.1007_s10957-024-02403-8.html
   My bibliography  Save this article

Riemannian Interior Point Methods for Constrained Optimization on Manifolds

Author

Listed:
  • Zhijian Lai

    (University of Tsukuba)

  • Akiko Yoshise

    (University of Tsukuba)

Abstract

We extend the classical primal-dual interior point method from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point method, is for solving Riemannian constrained optimization problems. We establish its local superlinear and quadratic convergence under the standard assumptions. Moreover, we show its global convergence when it is combined with a classical line search. Our method is a generalization of the classical framework of primal-dual interior point methods for nonlinear nonconvex programming. Numerical experiments show the stability and efficiency of our method.

Suggested Citation

  • Zhijian Lai & Akiko Yoshise, 2024. "Riemannian Interior Point Methods for Constrained Optimization on Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 201(1), pages 433-469, April.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:1:d:10.1007_s10957-024-02403-8
    DOI: 10.1007/s10957-024-02403-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02403-8
    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/s10957-024-02403-8?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Marcio Antônio de A. Bortoloti & Teles A. Fernandes & Orizon P. Ferreira & Jinyun Yuan, 2020. "Damped Newton’s method on Riemannian manifolds," Journal of Global Optimization, Springer, vol. 77(3), pages 643-660, July.
    2. C. Durazzi & V. Ruggiero, 2004. "Global Convergence of the Newton Interior-Point Method for Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 199-208, January.
    3. Xiaojing Zhu & Hiroyuki Sato, 2020. "Riemannian conjugate gradient methods with inverse retraction," Computational Optimization and Applications, Springer, vol. 77(3), pages 779-810, December.
    4. Teles A. Fernandes & Orizon P. Ferreira & Jinyun Yuan, 2017. "On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 828-843, June.
    5. Yuya Yamakawa & Hiroyuki Sato, 2022. "Sequential optimality conditions for nonlinear optimization on Riemannian manifolds and a globally convergent augmented Lagrangian method," Computational Optimization and Applications, Springer, vol. 81(2), pages 397-421, March.
    6. C. Durazzi, 2000. "On the Newton Interior-Point Method for Nonlinear Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 104(1), pages 73-90, January.
    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. Yasushi Narushima & Shummin Nakayama & Masashi Takemura & Hiroshi Yabe, 2023. "Memoryless Quasi-Newton Methods Based on the Spectral-Scaling Broyden Family for Riemannian Optimization," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 639-664, May.
    2. Brennan McCann & Morad Nazari & Christopher Petersen, 2024. "Numerical Approaches for Constrained and Unconstrained, Static Optimization on the Special Euclidean Group SE(3)," Journal of Optimization Theory and Applications, Springer, vol. 201(3), pages 1116-1150, June.
    3. Fabiana R. Oliveira & Fabrícia R. Oliveira, 2021. "A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 259-273, July.
    4. Caroline Geiersbach & Tim Suchan & Kathrin Welker, 2024. "Stochastic Augmented Lagrangian Method in Riemannian Shape Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 165-195, October.
    5. Marcio Antônio de A. Bortoloti & Teles A. Fernandes & Orizon P. Ferreira & Jinyun Yuan, 2020. "Damped Newton’s method on Riemannian manifolds," Journal of Global Optimization, Springer, vol. 77(3), pages 643-660, July.
    6. Benedetta Morini & Valeria Simoncini, 2017. "Stability and Accuracy of Inexact Interior Point Methods for Convex Quadratic Programming," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 450-477, November.
    7. C. Durazzi & V. Ruggiero & G. Zanghirati, 2001. "Parallel Interior-Point Method for Linear and Quadratic Programs with Special Structure," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 289-313, August.
    8. Hiroyuki Sakai & Hideaki Iiduka, 2024. "Modified Memoryless Spectral-Scaling Broyden Family on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 202(2), pages 834-853, August.
    9. Yuya Yamakawa & Hiroyuki Sato, 2022. "Sequential optimality conditions for nonlinear optimization on Riemannian manifolds and a globally convergent augmented Lagrangian method," Computational Optimization and Applications, Springer, vol. 81(2), pages 397-421, March.
    10. Zhou Sheng & Gonglin Yuan, 2024. "Convergence and worst-case complexity of adaptive Riemannian trust-region methods for optimization on manifolds," Journal of Global Optimization, Springer, vol. 89(4), pages 949-974, August.
    11. Hiroyuki Sato, 2023. "Riemannian optimization on unit sphere with p-norm and its applications," Computational Optimization and Applications, Springer, vol. 85(3), pages 897-935, July.
    12. Fabiana R. de Oliveira & Orizon P. Ferreira, 2020. "Newton Method for Finding a Singularity of a Special Class of Locally Lipschitz Continuous Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 522-539, May.

    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:201:y:2024:i:1:d:10.1007_s10957-024-02403-8. 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.