Skip to main content

Advertisement

Springer Nature Link
Log in

A Large-Update Interior-Point Method for Cartesian P ∗(κ)-LCP Over Symmetric Cones

  • Published:
Journal of Mathematical Modelling and Algorithms in Operations Research

Abstract

In this paper, we propose a new large-update interior point algorithm for the Cartesian P ∗(κ) linear complementarity problem over symmetric cones (SCLCP) based on a parametric kernel function, which determines both search directions and the proximity measure between the iterate and the μ-center. Using Euclidean Jordan algebras, we derive the iteration bound that match the currently best known iteration bound for large update methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Faybusovich, L.: Euclidean Jordan algebras and interior-point algorithms. Positivity 1(4), 331–357 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Zangiabadi, M., Gu, G., Roos, C.: A full nesterovtodd step infeasible interior-point method for second-order cone optimization. J. Optim. Theory Appl. 158, 816–858 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gu, G., Zangiabadi, M., Roos, C.: Full Nesterov-Todd step interior-point methods for symmetric optimization. Eur. J. Oper. Res. 214(3), 473–484 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Muramatsu, M.: On a commutative class of search directions for linear programming over symmetric cones. J. Optim. Theory Appl. 112, 595–625 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior-point algorithms to symmetric cones. Math. Program. 96(3), 409–438 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Luo, Z.Y., Xiu, N.H.: Solution Existence and Boundedness of Symmetric Cone Linear Complementarity Problems with the Cartesian P ∗(κ)-Property. Preprint, Department of Applied Mathematics, Beijing Jiaotong University (2007)

    Google Scholar 

  7. Mansouri, H., Zangiabadi, M., Pirhaji, M.: A full-Newton step O (n) infeasible interior-point algorithm for linear complementarity problems. Nonlinear Anal. Real. World. Appl. 12, 545–561 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zangiabadi, M., Mansouri, H.: Improved infeasible-interior-point algorithm for linear complementarity problems. Bulletin Iranian Math. Soc. 38(3), 787–803 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A unified approach to interior point algorithms for linear complementarity problems. In: Lecture Notes in Computer Science, vol 538. Springer, New York (1991)

    Book  Google Scholar 

  10. Potra, F.A., Sheng, R.Q.: Predictor-corrector algorithms for solving P ∗(κ)-matrix LCP from arbitrary positive starting points. Math. Program 76(1), 223–244 (1996)

    Article  MathSciNet  Google Scholar 

  11. Pan, S.H., Chen, J.S.: A regularization method for the second-order cone complementarity problem with the Cartesian P0-property. Nonlinear Anal. 70(4), 1475–1491 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, X., Qi, H.D.: Cartesian P-property and its applications to the semidefinite linear complementarity problem. Math. Program. 106(1), 177–201 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Choi, B.K., Lee, G.M.: On complexity analysis of the primal-dual interior-point methods for semidefinite optimization problem based on a new proximity function. Nonlinear Anal. Theory Meth. Appl. 71, 2628–2640 (2009)

    Article  MathSciNet  Google Scholar 

  14. Choi, B.K., Lee, G.M.: On complexity analysis of the primal-dual interior-point method for second-order cone optimization problem. J. Korean Soc. Ind. Appl. Math. 14, 93–111 (2010)

    MathSciNet  Google Scholar 

  15. Bai, Y.Q., El Ghami, M., Roos, C.: A comparative study of kernel functions for primaldual interior-point algorithms in linear optimization. SIAM J. Optim. 15, 101–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lesaja, G., Wang, G.Q., Zhu, D.T.: Interior-point methods for Cartesian P ∗(κ)-linear complementarity problems over symmetric cones based on the eligible kernel functions. Optim. Meth. Soft. 27(4–5), 827–843 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, G.Q.: A new polynomial interior-point algorithm for the monotone linear complementarity problem over symmetric cones with full NT-steps. Asia-Pac. J. Oper. Res. 29(2), 1250015 (2012)

    Article  MathSciNet  Google Scholar 

  18. Wang, G.Q., Bai, Y.Q.: A class of polynomial interior-point algorithms for the Cartesian P-Matrix linear complementarity problem over symmetric cones. J. Optim. Theory Appl. 152(3), 739–772 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, G.Q., Lesaja, G.: Full Nesterov-Todd step feasible interior-point method for the Cartesian P ∗(κ)-SCLCP. Optim. Meth. Soft. 28(3), 600–618 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Choi, B.K., Lee, G.M.: New complexity analysis for primal-dual interior-point methods for self-scaled optimization problems. Fixed Point Theory Appl. 1186(10), 1687–1812 (2012)

    Google Scholar 

  21. Faraut, J., Koranyi. A.: Analysis on Symmetric Cones. Oxford University Press, New York (1994)

    MATH  Google Scholar 

  22. Faybusovich, L.: A Jordan-algebraic approach to potential-reduction algorithms. Math. Z. 1, 117–129 (2002)

    Article  MathSciNet  Google Scholar 

  23. Luo, Z.Y., Xiu, N.H.: Path-following interior point algorithms for the Cartesian P ∗(κ)-LCP over symmetric cones. Sci. China Ser. A 52(8), 1769–1784 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Vieira, M.V.: Interior-point methods based on kernel functions for symmetric optimization. Optim. Methods Soft. 27, 513–537 (2011)

    Article  MathSciNet  Google Scholar 

  25. Peng, J., Roos, C., Terlaky, T.: Self-regular functions and new search directions for linear and semidefinite optimization. Math. Program. 93(1), 129–171 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran

    M. Zangiabadi, M. Sayadi Shahraki & H. Mansouri

Authors
  1. M. Zangiabadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Sayadi Shahraki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Mansouri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Mansouri.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zangiabadi, M., Sayadi Shahraki, M. & Mansouri, H. A Large-Update Interior-Point Method for Cartesian P ∗(κ)-LCP Over Symmetric Cones. J Math Model Algor 13, 537–556 (2014). https://doi.org/10.1007/s10852-013-9246-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10852-013-9246-4

Keywords