Skip to main content
SpringerLink
Log in

A Modified and Simplified Full Nesterov–Todd Step \(\mathcal {O}(N)\) Infeasible Interior-Point Method for Second-Order Cone Optimization

  • Published:
Journal of the Operations Research Society of China Aims and scope Submit manuscript

Abstract

We present a modified and simplified version of an infeasible interior-point method for second-order cone optimization published in 2013 (Zangiabadi et al. in J Optim Theory Appl, 2013). In the earlier version, each iteration consisted of one so-called feasibility step and a few centering steps. Here, each iteration consists of only a feasibility step. Thus, the new algorithm improves the number of iterations and the improvement is due to a lemma which gives an upper bound for the proximity after the feasibility step. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods.

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

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. 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 

  4. Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princton (2002)

    MATH  Google Scholar 

  5. 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(10), 3584–3602 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Kojima, M., Megiddo, N., Mizuno, S.: A primal-dual infeasible-interior-point algorithm for linear programming. Math. Program. 61(3), 263–280 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lustig, I.J.: Feasible issues in a primal-dual interior-point method. Math. Program. 67, 145–162 (1990)

    Article  MATH  Google Scholar 

  10. Mizuno, S.: Polynomiality of infeasible-interior-point algorithms for linear programming. Math. Program. 67, 109–119 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Potra, F.A.: An \(O(nL)\) infeasible-interior-point algorithm for LCP with quadratic convergence. Ann. Oper. Res. 62, 81–102 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Potra, F.A., Sheng, R.Q.: A large-step infeasible-interior-point method for the \(P_*\)-matrix LCP. SIAM J. Optim. 7(2), 318–335 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    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. Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997)

    Book  MATH  Google Scholar 

  16. Zhang, Y.: On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem. SIAM J. Optim. 4(1), 208–227 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, C.H., Wu, D., Shang, Y.L.: A new infeasible-interior-point algorithm based on wide neighborhoods for symmetric cone programming. J. Oper. Res. Soc. China 4(2), 147–165 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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 

  19. Zangiabadi, M., Gu, G., Roos, C.: A full Nesterov–Todd step infeasible interior-point method for second-order cone optimization. J. Optim. Theory Appl. 158(3), 816–858 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kheirfam, B., Mahdavi-Amiri, N.: A full Nesterov–Todd step infeasible interior-point algorithm for symmetric cone linear complementarity problem. Bull. Irani. Math. Soc. 40(3), 541–564 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Roos, C.: An improved and simplified full-Newton step \(O(n)\) infeasible interior-point method for linear optimization. SIAM J. Optim. 25(1), 102–114 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lim, Y.: Geometric means on symmetric cones. Arch. Math. (Basel) 75(1), 39–45 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Roos, C., Terlaky, T., Vial, J.-P.: Theory and Algorithms for Linear Optimization. An Interior-Point Approach. Wiley, Chichester (1997)

    MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the editors and the anonymous referees for their useful comments and suggestions, which helped to improve the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

    Behrouz Kheirfam

Authors
  1. Behrouz Kheirfam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Behrouz Kheirfam.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kheirfam, B. A Modified and Simplified Full Nesterov–Todd Step \(\mathcal {O}(N)\) Infeasible Interior-Point Method for Second-Order Cone Optimization. J. Oper. Res. Soc. China 6, 301–315 (2018). https://doi.org/10.1007/s40305-017-0168-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40305-017-0168-0

Keywords

Mathematics Subject Classification

Search

Navigation