IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v68y2017i1d10.1007_s10589-017-9905-x.html
   My bibliography  Save this article

Two wide neighborhood interior-point methods for symmetric cone optimization

Author

Listed:
  • M. Sayadi Shahraki

    (Institute for Research in Fundamental Sciences (IPM))

  • H. Mansouri

    (Shahrekord University)

  • M. Zangiabadi

    (Shahrekord University)

Abstract

In this paper, we present two primal–dual interior-point algorithms for symmetric cone optimization problems. The algorithms produce a sequence of iterates in the wide neighborhood $$\mathcal {N}(\tau ,\,\beta )$$ N ( τ , β ) of the central path. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. We derive that these two path-following algorithms have $$\begin{aligned} \text{ O }\left( \sqrt{r\text{ cond }(G)}\log \varepsilon ^{-1}\right) , \text{ O }\left( \sqrt{r}\left( \text{ cond }(G)\right) ^{1/4}\log \varepsilon ^{-1}\right) \end{aligned}$$ O r cond ( G ) log ε - 1 , O r cond ( G ) 1 / 4 log ε - 1 iteration complexity bounds, respectively. The obtained complexity bounds are the best result in regard to the iteration complexity bound in the context of the path-following methods for symmetric cone optimization. Numerical results show that the algorithms are efficient for this kind of problems.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:coopap:v:68:y:2017:i:1:d:10.1007_s10589-017-9905-x
    DOI: 10.1007/s10589-017-9905-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-017-9905-x
    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/s10589-017-9905-x?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. Shinji Mizuno & Michael J. Todd & Yinyu Ye, 1993. "On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 964-981, November.
    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.
    3. 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.
    4. Satoshi Kakihara & Atsumi Ohara & Takashi Tsuchiya, 2014. "Curvature integrals and iteration complexities in SDP and symmetric cone programs," Computational Optimization and Applications, Springer, vol. 57(3), pages 623-665, April.
    5. 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.
    6. M. Zangiabadi & G. Gu & C. Roos, 2013. "A Full Nesterov–Todd Step Infeasible Interior-Point Method for Second-Order Cone Optimization," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 816-858, September.
    7. 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.
    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. 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. Soodabeh Asadi & Hossein Mansouri & Zsolt Darvay & Maryam Zangiabadi & Nezam Mahdavi-Amiri, 2019. "Large-Neighborhood Infeasible Predictor–Corrector Algorithm for Horizontal Linear Complementarity Problems over Cartesian Product of Symmetric Cones," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 811-829, March.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. Ali Mohammad-Nezhad & Tamás Terlaky, 2017. "A polynomial primal-dual affine scaling algorithm for symmetric conic optimization," Computational Optimization and Applications, Springer, vol. 66(3), pages 577-600, April.
    14. Petra Renáta Rigó & Zsolt Darvay, 2018. "Infeasible interior-point method for symmetric optimization using a positive-asymptotic barrier," Computational Optimization and Applications, Springer, vol. 71(2), pages 483-508, November.
    15. Behrouz Kheirfam, 2013. "A new infeasible interior-point method based on Darvay’s technique for symmetric optimization," Annals of Operations Research, Springer, vol. 211(1), pages 209-224, December.
    16. 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.
    17. 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.
    18. 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.
    19. Terlaky, Tamas, 2001. "An easy way to teach interior-point methods," European Journal of Operational Research, Elsevier, vol. 130(1), pages 1-19, April.
    20. Michael Orlitzky, 2021. "Gaddum’s test for symmetric cones," Journal of Global Optimization, Springer, vol. 79(4), pages 927-940, April.

    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:coopap:v:68:y:2017:i:1:d:10.1007_s10589-017-9905-x. 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.