IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v73y2011i1p75-90.html
   My bibliography  Save this article

Polynomial complexity of an interior point algorithm with a second order corrector step for symmetric cone programming

Author

Listed:
  • Jian Zhang
  • Kecun Zhang

Abstract

In this paper, we propose a second order interior point algorithm for symmetric cone programming using a wide neighborhood of the central path. The convergence is shown for commutative class of search directions. The complexity bound is $${O(r^{3/2}\,\log\epsilon^{-1})}$$ for the NT methods, and $${O(r^{2}\,\log\epsilon^{-1})}$$ for the XS and SX methods, where r is the rank of the associated Euclidean Jordan algebra and $${\epsilon\,{ > }\,0}$$ is a given tolerance. If the staring point is strictly feasible, then the corresponding bounds can be reduced by a factor of r 3/4 . The theory of Euclidean Jordan algebras is a basic tool in our analysis. Copyright Springer-Verlag 2011

Suggested Citation

  • Jian Zhang & Kecun Zhang, 2011. "Polynomial complexity of an interior point algorithm with a second order corrector step for symmetric cone programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(1), pages 75-90, February.
    • Handle: RePEc:spr:mathme:v:73:y:2011:i:1:p:75-90
    DOI: 10.1007/s00186-010-0334-1
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00186-010-0334-1
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00186-010-0334-1?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. S. H. Schmieta & F. Alizadeh, 2001. "Associative and Jordan Algebras, and Polynomial Time Interior-Point Algorithms for Symmetric Cones," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 543-564, August.
    2. 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.
    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. 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.
    2. M. Sayadi Shahraki & H. Mansouri & M. Zangiabadi, 2017. "Two wide neighborhood interior-point methods for symmetric cone optimization," Computational Optimization and Applications, Springer, vol. 68(1), pages 29-55, September.
    3. 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.
    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. Changhe Liu & Hongwei Liu & Xinze Liu, 2012. "Polynomial Convergence of Second-Order Mehrotra-Type Predictor-Corrector Algorithms over Symmetric Cones," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 949-965, September.

    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. Behrouz Kheirfam, 2015. "A Corrector–Predictor Path-Following Method for Convex Quadratic Symmetric Cone Optimization," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 246-260, January.
    2. G. Q. Wang & Y. Q. Bai, 2012. "A Class of Polynomial Interior Point Algorithms for the Cartesian P-Matrix Linear Complementarity Problem over Symmetric Cones," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 739-772, March.
    3. Gu, G. & Zangiabadi, M. & Roos, C., 2011. "Full Nesterov-Todd step infeasible interior-point method for symmetric optimization," European Journal of Operational Research, Elsevier, vol. 214(3), pages 473-484, November.
    4. 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.
    5. 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.
    6. 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.
    7. Cardoso, Domingos Moreira & Vieira, Luis Almeida, 2006. "On the optimal parameter of a self-concordant barrier over a symmetric cone," European Journal of Operational Research, Elsevier, vol. 169(3), pages 1148-1157, March.
    8. Terlaky, Tamas, 2001. "An easy way to teach interior-point methods," European Journal of Operational Research, Elsevier, vol. 130(1), pages 1-19, April.
    9. Michael Orlitzky, 2021. "Gaddum’s test for symmetric cones," Journal of Global Optimization, Springer, vol. 79(4), pages 927-940, April.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. G. Q. Wang & Y. Q. Bai & X. Y. Gao & D. Z. Wang, 2015. "Improved Complexity Analysis of Full Nesterov–Todd Step Interior-Point Methods for Semidefinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 242-262, April.
    19. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    20. Mehdi Karimi & Levent Tunçel, 2020. "Primal–Dual Interior-Point Methods for Domain-Driven Formulations," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 591-621, 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:mathme:v:73:y:2011:i:1:p:75-90. 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.