Skip to main content
SpringerLink
Log in

A New Full Nesterov–Todd Step Primal–Dual Path-Following Interior-Point Algorithm for Symmetric Optimization

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we generalize a primal–dual path-following interior-point algorithm for linear optimization to symmetric optimization by using Euclidean Jordan algebras. The proposed algorithm is based on a new technique for finding the search directions and the strategy of the central path. At each iteration, we use only full Nesterov–Todd steps. Moreover, we derive the currently best known iteration bound for the small-update method. This unifies the analysis for linear, second-order cone, and semidefinite optimizations.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Nesterov, Y.E., Todd, M.J.: Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22(1), 1–42 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Nesterov, Y.E., Todd, M.J.: Primal–dual interior-point methods for self-scaled cones. SIAM J. Optim. 8(2), 324–364 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Potra, F.A., Sheng, R.: A superlinearly convergent primal–dual infeasible-interior-point algorithm for semidefinite programming. SIAM J. Optim. 8(4), 1007–1028 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kojima, M., Shida, M., Shindoh, S.: Local convergence of predictor–corrector infeasible-interior-point algorithms for SDPs and SDLCPs. Math. Program. 96(80), 129–161 (1998)

    MathSciNet  Google Scholar 

  5. Ji, J., Potra, F.A., Sheng, R.: On the local convergence of a predictor–corrector method for semidefinite programming. SIAM J. Optim. 10(1), 195–210 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Monteiro, R.D.C.: Primal–dual path-following algorithms for semidefinite programming. SIAM J. Optim. 7(3), 663–678 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Potra, F.A., Sheng, R.: On homogeneous interior-point algorithms for semidefinite programming. Optim. Methods Softw. 9(1–3), 161–184 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Luo, Z.Q., Sturm, J.F., Zhang, S.: Conic convex programming and self-dual embedding. Optim. Methods Softw. 14(3), 169–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semi-definite Programming, Theory, Algorithms, and Applications. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  10. De Klerk, E.: Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  11. Güler, O.: Barrier functions in interior point methods. Math. Oper. Res. 21(4), 860–885 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Faybusovich, L.: Linear system in Jordan algebras and primal-dual interior-point algorithms. J. Comput. Appl. Math. 86(1), 149–175 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Muramatsu, M.: On a commutative class of search directions for linear programming over symmetric cones. J. Optim. Theory Appl. 112(3), 595–625 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rangarajan, B.K.: Polynomial convergence of infeasible-interior-point methods over symmetric cones. SIAM J. Optim. 16(4), 1211–1229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Vieira, M.V.C.: Jordan algebraic approach to symmetric optimization. Ph.D. thesis, Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, The Netherlands (2007)

  17. Darvay, Z.: New interior-point algorithms in linear programming. Adv. Model. Optim. 5(1), 51–92 (2003)

    MathSciNet  MATH  Google Scholar 

  18. Achache, M.: A new primal–dual path-following method for convex quadratic programming. Comput. Appl. Math. 25(1), 97–110 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, G.Q., Bai, Y.Q.: A primal–dual path-following interior-point algorithm for second-order cone optimization with full Nesterov–Todd step. Appl. Math. Comput. 215(3), 1047–1061 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, G.Q., Bai, Y.Q.: A new primal–dual path-following interior-point algorithm for semidefinite optimization. J. Math. Anal. Appl. 353(1), 339–349 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Roos, C.: A full-Newton step O(n) infeasible interior-point algorithm for linear optimization. SIAM J. Optim. 16(4), 1110–1136 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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 

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

    MATH  Google Scholar 

  24. Sturm, J.F.: Similarity and other spectral relations for symmetric cones. Linear Algebra Appl. 312(1–3), 135–154 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Faybusovich, L.: A Jordan-algebraic approach to potential-reduction algorithms. Math. Z. 239(1), 117–129 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Professor Florian A. Potra and the anonymous referees for their useful suggestions which helped improve the presentation of this paper. This work was supported by National Natural Science Foundation of China (Nos. 11001169, 11071158), China Postdoctoral Science Foundation (No. 20100480604) and the Key Disciplines of Shanghai Municipality (No. S30104).

Author information

Authors and Affiliations

  1. College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, China

    G. Q. Wang

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Y. Q. Bai

Authors
  1. G. Q. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. Q. Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Q. Wang.

Additional information

Communicated by Florian A. Potra.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, G.Q., Bai, Y.Q. A New Full Nesterov–Todd Step Primal–Dual Path-Following Interior-Point Algorithm for Symmetric Optimization. J Optim Theory Appl 154, 966–985 (2012). https://doi.org/10.1007/s10957-012-0013-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0013-x

Keywords

Search

Navigation