IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v141y2009i1d10.1007_s10957-008-9480-5.html
   My bibliography  Save this article

On the Analyticity of Underlying HKM Paths for Monotone Semidefinite Linear Complementarity Problems

Author

Listed:
  • C. K. Sim

    (The Hong Kong Polytechnic University)

Abstract

An interior point method (IPM) defines a search direction at an interior point of the feasible region. These search directions form a direction field, which in turn defines a system of ordinary differential equations (ODEs). The solutions of the system of ODEs are called off-central paths, underlying paths lying in the interior of the feasible region. It is known that not all off-central paths are analytic, whether w.r.t. μ or $\sqrt{\mu}$ , where μ represents the duality gap, at a solution of a given semidefinite linear complementarity problem, SDLCP (Sim and Zhao, Math. Program. 110:475–499, 2007). In Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008), we give a necessary and sufficient condition for when an off-central path is analytic as a function of $\sqrt{\mu}$ at a solution of a general SDLCP. It is then natural to ask about the analyticity of a SDLCP off-central path at a solution, as a function of μ. We investigate this in the current paper. Again, we work under the assumption that the given SDLCP satisfies strict complementarity condition.

Suggested Citation

  • C. K. Sim, 2009. "On the Analyticity of Underlying HKM Paths for Monotone Semidefinite Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(1), pages 193-215, April.
    • Handle: RePEc:spr:joptap:v:141:y:2009:i:1:d:10.1007_s10957-008-9480-5
    DOI: 10.1007/s10957-008-9480-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-008-9480-5
    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/s10957-008-9480-5?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. Jos F. Sturm, 1999. "Superlinear Convergence of an Algorithm for Monotone Linear Complementarity Problems, When No Strictly Complementary Solution Exists," Mathematics of Operations Research, INFORMS, vol. 24(1), pages 72-94, February.
    2. Sanjay Mehrotra, 1993. "Quadratic Convergence in a Primal-Dual Method," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 741-751, August.
    3. C. K. Sim & G. Zhao, 2008. "Asymptotic Behavior of Helmberg-Kojima-Monteiro (HKM) Paths in Interior-Point Methods for Monotone Semidefinite Linear Complementarity Problems: General Theory," Journal of Optimization Theory and Applications, Springer, vol. 137(1), pages 11-25, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Chee-Khian Sim, 2011. "Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 79-106, January.
    2. Chee-Khian Sim, 2019. "Interior point method on semi-definite linear complementarity problems using the Nesterov–Todd (NT) search direction: polynomial complexity and local convergence," Computational Optimization and Applications, Springer, vol. 74(2), pages 583-621, November.

    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. Chee-Khian Sim, 2011. "Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 79-106, January.
    2. C. K. Sim & G. Zhao, 2008. "Asymptotic Behavior of Helmberg-Kojima-Monteiro (HKM) Paths in Interior-Point Methods for Monotone Semidefinite Linear Complementarity Problems: General Theory," Journal of Optimization Theory and Applications, Springer, vol. 137(1), pages 11-25, April.
    3. Chee-Khian Sim, 2019. "Interior point method on semi-definite linear complementarity problems using the Nesterov–Todd (NT) search direction: polynomial complexity and local convergence," Computational Optimization and Applications, Springer, vol. 74(2), pages 583-621, November.
    4. Filiz Gurtuna & Cosmin Petra & Florian Potra & Olena Shevchenko & Adrian Vancea, 2011. "Corrector-predictor methods for sufficient linear complementarity problems," Computational Optimization and Applications, Springer, vol. 48(3), pages 453-485, April.
    5. Halicka, Margareta, 2002. "Analyticity of the central path at the boundary point in semidefinite programming," European Journal of Operational Research, Elsevier, vol. 143(2), pages 311-324, December.

    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:141:y:2009:i:1:d:10.1007_s10957-008-9480-5. 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.