Numerical Algorithms

, Volume 78, Issue 2, pp 535–552 | Cite as

A wide neighborhood primal-dual predictor-corrector interior-point method for symmetric cone optimization

  • M. Sayadi Shahraki
  • H. Mansouri
  • M. Zangiabadi
  • N. Mahdavi-Amiri
Original Paper
  • 31 Downloads

Abstract

We present a primal-dual predictor-corrector interior-point method for symmetric cone optimization. The proposed algorithm is based on the Nesterov-Todd search directions and a wide neighborhood, which is an even wider neighborhood than a given negative infinity neighborhood. At each iteration, the method computes two corrector directions in addition to the Ai and Zhang directions (SIAM J. Optim. 16, 400–417, 2005), in order to improve performance. Moreover, we derive the complexity bound of the wide neighborhood predictor-corrector interior-point method for symmetric cone optimization that coincides with the currently best known theoretical complexity bounds for the short step algorithm. Finally, some numerical experiments are provided to reveal the effectiveness of the proposed method.

Keywords

Symmetric cone Euclidean Jordan algebra Predictor-corrector interior-point method Central path 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The research of the first author was in part supported by a grant from IPM (No. 96900076). The second and third authors thank Shahrekord University and the fourth author thanks Research Council of Sharif University of Technology for financial support. The second and third authors were also partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Shahrekord, Iran.

References

  1. 1.
    Karmarkar, N.K.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford University Press, New York (1994)MATHGoogle Scholar
  3. 3.
    Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior-point algorithms to symmetric cones. Math. Program. 96, 409–438 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Gu, G., Zangiabadi, M., Roos, C.: Full Nesterov-Todd step interior-point methods for symmetric optimization. Eur. J. Oper. Res. 214, 473–484 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Zhang, J., Zhang, K.: Polynomiality complexity of an interior point algorithm with a second order corrector step for symmetric cone programming. Math. Meth. Oper. Res. 73, 75–90 (2011)CrossRefMATHGoogle Scholar
  6. 6.
    Liu, H., Yang, X., Liu, C.: A new wide neighborhood primal-dual infeasible-interior-point method for symmetric cone programming. J. Optim. Theory Appl. 158, 796–815 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ai, W.: Neighborhood-following algorithms for linear programming. Sci. China Ser. A 47, 812–820 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ai, W., Zhang, S.: An \(O(\sqrt {n}L)\) iteration primal-dual path-following method, based on wide neighborhoods and large updates, for monotone LCP. SIAM J. Optim. 16, 400–417 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Li, Y., Terlaky, T.: A new class of large neighborhood path-following interior point algorithms for semidefinite optimization with \(O(\sqrt {n}\log (Tr(X^{0} S^{0})/\varepsilon ))\) iteration complexity. SIAM J. Optim. 20, 2853–2875 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Feng, Z.Z.: A new \(O(\sqrt {n}L)\) iteration large update primal-dual interior-point method for second-order cone programming. Numer. Funct. Anal. Optim. 33, 397–414 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Feng, Z.Z., Fang, L.: A new \(O(\sqrt {n}L)\)-iteration predictor-corrector algorithm with wide neighborhood for semidefinite programming. J. Comp. App. Math. 256, 65–76 (2014)CrossRefMATHGoogle Scholar
  12. 12.
    Sayadi Shahraki, M., Mansouri, H., Zangiabadi, M.: A new primal-dual predictor-corrector interior-point method for linear programming based on a wide neighborhood. J. Optim. Theory Appl.  https://doi.org/10.1007/s10957-016-0927-9 (2016)
  13. 13.
    Liu, C., Liu, H., Cong, W.: An \(O(\sqrt {n}L)\) iteration primal-dual second-order corrector algorithm for linear programming. Optim. Lett. 5, 729–743 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Sayadi Shahraki, M., Mansouri, H., Zangiabadi, M.: A new wide neighborhood primal-dual predictor-corrector interior-point method for linear programming. To appear in Numer. Funct. Anal. Optim.  https://doi.org/10.1080/01630563.2016.1138128 (2015)
  15. 15.
    Nesterov, Y.E., Todd, M.J.: Primal-dual interior-point methods for self-scaled cones. SIAM J. Optim. 8, 324–364 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Faybusovich, L.: A Jordan-algebraic approach to potential-reduction algorithms. Math. Z. 239, 117–129 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nesterov, Y.E., Nemirovskii, A.S.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)CrossRefMATHGoogle Scholar
  18. 18.
    Liu, C.: Study on Complexity of Some Interior-Point Algorithms in Conic Programming. Ph.D. thesis, Xidian University, in Chinese (2012)Google Scholar
  19. 19.
    Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997)CrossRefMATHGoogle Scholar
  20. 20.
    Todd, M.J., Toh, K. C., Tautauncau, R.H.: On the Nesterov-Todd direction in semidefinite programming. Optimization 8, 769–796 (1998)MathSciNetMATHGoogle Scholar
  21. 21.
    Liu, C., Liu, H., Liu, X.: Polynomial convergence of Second-Order Mehrotra-Type Predictor-Corrector algorithms over symmetric cones. J. Optim. Theory Appl. 154, 949–965 (2012)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zhang, Y., Zhang, D.: On polynomial of the Mehrotra-type predictor-corrector interior-point algorithms. Math. Program. 68, 303–318 (1995)CrossRefMATHGoogle Scholar
  23. 23.
    Bai, Y.Q., Wang, G.Q., Roos, C.: Primal-dual interior-point algorithms for second-order cone optimization based on kernel functions. Nonlinear Anal. 70, 3584–3602 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • M. Sayadi Shahraki
    • 1
  • H. Mansouri
    • 2
  • M. Zangiabadi
    • 2
  • N. Mahdavi-Amiri
    • 3
  1. 1.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran
  2. 2.Department of Applied Mathematics, Faculty of Mathematical SciencesShahrekord UniversityShahrekordIran
  3. 3.Department of Mathematical SciencesSharif University of TechnologyTehranIran

Personalised recommendations