IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v176y2018i3d10.1007_s10957-018-1225-5.html
   My bibliography  Save this article

Direct Search Methods on Reductive Homogeneous Spaces

Author

Listed:
  • David W. Dreisigmeyer

    (United States Census Bureau
    Colorado State University)

Abstract

Direct search methods are mainly designed for use in problems with no equality constraints. However, there are many instances where the feasible set is of measure zero in the ambient space and no mesh point lies within it. There are methods for working with feasible sets that are (Riemannian) manifolds, but not all manifolds are created equal. In particular, reductive homogeneous spaces seem to be the most general space that can be conveniently optimized over. The reason is that a ‘law of motion’ over the feasible region is also given. Examples include $$\mathbb {R}^{n}$$ R n and its linear subspaces, Lie groups, and coset manifolds such as Grassmannians and Stiefel manifolds. These are important arenas for optimization, for example, in the areas of image processing and data mining. We demonstrate optimization procedures over general reductive homogeneous spaces utilizing maps from the tangent space to the manifold. A concrete implementation of the probabilistic descent direct search method is shown. This is then extended to a procedure that works solely with the manifold elements, eliminating the need for the use of the tangent space.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:joptap:v:176:y:2018:i:3:d:10.1007_s10957-018-1225-5
    DOI: 10.1007/s10957-018-1225-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-018-1225-5
    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/s10957-018-1225-5?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. 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.
    2. 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.
    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. Ubaldo M. García-Palomares, 2020. "Non-monotone derivative-free algorithm for solving optimization models with linear constraints: extensions for solving nonlinearly constrained models via exact penalty methods," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 599-625, October.
    2. Benjamin Dyke & Thomas J. Asaki, 2013. "Using QR Decomposition to Obtain a New Instance of Mesh Adaptive Direct Search with Uniformly Distributed Polling Directions," Journal of Optimization Theory and Applications, Springer, vol. 159(3), pages 805-821, December.
    3. A. Sanchez & Diego Martinez, 2011. "Optimization in Non-Standard Problems. An Application to the Provision of Public Inputs," Computational Economics, Springer;Society for Computational Economics, vol. 37(1), pages 13-38, January.
    4. 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.
    5. 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.
    6. Y. Diouane & S. Gratton & L. Vicente, 2015. "Globally convergent evolution strategies for constrained optimization," Computational Optimization and Applications, Springer, vol. 62(2), pages 323-346, November.
    7. Benjamin Van Dyke, 2014. "Equal Angle Distribution of Polling Directions in Direct-Search Methods," Journal of Optimization, Hindawi, vol. 2014, pages 1-15, July.
    8. Javaid Ali & Muhammad Saeed & Muhammad Farhan Tabassam & Shaukat Iqbal, 2019. "Controlled showering optimization algorithm: an intelligent tool for decision making in global optimization," Computational and Mathematical Organization Theory, Springer, vol. 25(2), pages 132-164, June.
    9. Árpád Bűrmen & Jernej Olenšek & Tadej Tuma, 2015. "Mesh adaptive direct search with second directional derivative-based Hessian update," Computational Optimization and Applications, Springer, vol. 62(3), pages 693-715, December.
    10. Charles Audet & Andrew R. Conn & Sébastien Le Digabel & Mathilde Peyrega, 2018. "A progressive barrier derivative-free trust-region algorithm for constrained optimization," Computational Optimization and Applications, Springer, vol. 71(2), pages 307-329, November.
    11. S. Gratton & C. W. Royer & L. N. Vicente & Z. Zhang, 2019. "Direct search based on probabilistic feasible descent for bound and linearly constrained problems," Computational Optimization and Applications, Springer, vol. 72(3), pages 525-559, April.
    12. Árpád Bűrmen & Iztok Fajfar, 2019. "Mesh adaptive direct search with simplicial Hessian update," Computational Optimization and Applications, Springer, vol. 74(3), pages 645-667, December.
    13. 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.

    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:176:y:2018:i:3:d:10.1007_s10957-018-1225-5. 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.