IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v148y2011i1d10.1007_s10957-010-9746-6.html
   My bibliography  Save this article

Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems

Author

Listed:
  • Chee-Khian Sim

    (The Hong Kong Polytechnic University)

Abstract

An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), these off-central paths are shown to be well-defined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007; J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of off-central paths corresponding to the HKM direction is studied. In particular, in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), the authors study the asymptotic behavior of these paths for a simple example, while, in Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study off-central paths corresponding to another well-known direction, the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-central paths to be analytic w.r.t. $\sqrt{\mu}$ and then w.r.t. μ, at solutions of a general SDLCP. Also, as in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), we present off-central path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:148:y:2011:i:1:d:10.1007_s10957-010-9746-6
    DOI: 10.1007/s10957-010-9746-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-010-9746-6
    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/s10957-010-9746-6?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. F. A. Potra & R. Sheng, 1998. "Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 103-119, October.
    2. 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.
    3. Yu. E. Nesterov & M. J. Todd, 1997. "Self-Scaled Barriers and Interior-Point Methods for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 1-42, February.
    4. Sanjay Mehrotra, 1993. "Quadratic Convergence in a Primal-Dual Method," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 741-751, August.
    5. 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.
    6. 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.
    7. F. A. Potra & R. Sheng, 1998. "Superlinearly Convergent Infeasible-Interior-Point Algorithm for Degenerate LCP," Journal of Optimization Theory and Applications, Springer, vol. 97(2), pages 249-269, May.
    8. Josef Stoer & Martin Wechs & Shinji Mizuno, 1998. "High Order Infeasible-Interior-Point Methods for Solving Sufficient Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 832-862, November.
    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, 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, 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.
    2. Ximei Yang & Hongwei Liu & Yinkui Zhang, 2015. "A New Strategy in the Complexity Analysis of an Infeasible-Interior-Point Method for Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 572-587, August.
    3. 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.
    4. Hongwei Liu & Ximei Yang & Changhe Liu, 2013. "A New Wide Neighborhood Primal–Dual Infeasible-Interior-Point Method for Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 796-815, September.
    5. Sungwoo Park & Dianne P. O’Leary, 2015. "A Polynomial Time Constraint-Reduced Algorithm for Semidefinite Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 558-571, August.
    6. 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.
    7. 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.
    8. Baha Alzalg & Khaled Badarneh & Ayat Ababneh, 2019. "An Infeasible Interior-Point Algorithm for Stochastic Second-Order Cone Optimization," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 324-346, April.
    9. Sturm, J.F., 2001. "Avoiding Numerical Cancellation in the Interior Point Method for Solving Semidefinite Programs," Other publications TiSEM 949fb20a-a2c6-4d87-85ea-8, Tilburg University, School of Economics and Management.
    10. Robert Chares & François Glineur, 2008. "An interior-point method for the single-facility location problem with mixed norms using a conic formulation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(3), pages 383-405, December.
    11. Illes, Tibor & Nagy, Marianna, 2007. "A Mizuno-Todd-Ye type predictor-corrector algorithm for sufficient linear complementarity problems," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1097-1111, September.
    12. Terlaky, Tamas, 2001. "An easy way to teach interior-point methods," European Journal of Operational Research, Elsevier, vol. 130(1), pages 1-19, April.
    13. Michael Orlitzky, 2021. "Gaddum’s test for symmetric cones," Journal of Global Optimization, Springer, vol. 79(4), pages 927-940, April.
    14. B.V. Halldórsson & R.H. Tütüncü, 2003. "An Interior-Point Method for a Class of Saddle-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 116(3), pages 559-590, March.
    15. G. Q. Wang & Y. Q. Bai, 2012. "A New Full Nesterov–Todd Step Primal–Dual Path-Following Interior-Point Algorithm for Symmetric Optimization," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 966-985, September.
    16. G. Q. Wang & L. C. Kong & J. Y. Tao & G. Lesaja, 2015. "Improved Complexity Analysis of Full Nesterov–Todd Step Feasible Interior-Point Method for Symmetric Optimization," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 588-604, August.
    17. E. A. Yıldırım, 2003. "An Interior-Point Perspective on Sensitivity Analysis in Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 649-676, November.
    18. Xiao-Kang Wang & Wen-Hui Hou & Chao Song & Min-Hui Deng & Yong-Yi Li & Jian-Qiang Wang, 2021. "BW-MaxEnt: A Novel MCDM Method for Limited Knowledge," Mathematics, MDPI, vol. 9(14), pages 1-17, July.
    19. Héctor Ramírez & David Sossa, 2017. "On the Central Paths in Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 649-668, February.
    20. Vasile L. Basescu & John E. Mitchell, 2008. "An Analytic Center Cutting Plane Approach for Conic Programming," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 529-551, August.

    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:148:y:2011:i:1:d:10.1007_s10957-010-9746-6. 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.