Journal of Optimization Theory and Applications

, Volume 166, Issue 2, pp 572–587 | Cite as

A New Strategy in the Complexity Analysis of an Infeasible-Interior-Point Method for Symmetric Cone Programming

Article
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Abstract

In this paper, we give a new strategy in the complexity analysis of an infeasible-interior-point method for symmetric cone programming. Using the strategy, we improve the theoretical complexity bound of an infeasible-interior-point method. Convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the \(xs\) and \(sx\) directions.

Keywords

Jordan algebra Symmetric cone programming Infeasible-interior-point method Polynomial complexity 

Mathematics Subject Classification

90C05 90C51 
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Notes

Acknowledgments

Authors would like to thank the anonymous referees and editor for their useful comments and suggestions. Authors would also like to the supports of National Natural Science Foundation of China (NNSFC) under Grant No. 61179040 and No. 61303030 and the Scientific Research of the Higher Education Institutions of Guangxi under Grant No. ZD2014050.

References

  1. 1.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica. 4, 302–311 (1984)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Peng, J., Roos, C., Terlaky, T.: Self-Regularity. A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton (2002)Google Scholar
  3. 3.
    Roos, C., Terlaky, T., Vial, J.P.: Theory and Algorithms for Linear Optimization: An Interior Point Approach. Wiley, Chichester (1997)MATHGoogle Scholar
  4. 4.
    Vanderbei, R.J.: Linear Programming: Foundations and Extensions. Kluwer Academic Publishers, Boston (1996)Google Scholar
  5. 5.
    Nesterov, Y., Todd, M.: Primal-dual interior-point methods for self-scaled cones. SIAM J. Optim. 8, 324–364 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Nesterov, Y., Todd, M.: Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22, 1–42 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Güler, O.: Barrier functions in interior-point methods. Math. Oper. Res. 21, 860–885 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Faybusovich, L.: Linear systems in Jordan algebras and primal-dual interior-point algorithms. J. Comput. Appl. Math. 86, 149–175 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior point algorithm to symmetric cones. Math. Program. 96, 409–438 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Vieira, M.V.C.: Jordan algebras approach to symmetric optimization. Ph.D. thesis, Delft University of Technology (2007)Google Scholar
  11. 11.
    Vieira, M.V.C.: Interior-point methods based on kernel functions for symmetric optimization. Optim. Methods Softw. 27, 513–537 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wang, G., Bai, Y.: A new full Nesterov-Todd step primal-dual path-following interior-point algorithm for symmetric optimization. J. Optim. Theory Appl. 154, 966–985 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lustig, I.J.: Feasible issues in a primal-dual interior-point method. Math. Program. 67, 145–162 (1990)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Tanabe, K.: Centered Newton method for linear programming: interior and ‘exterior’ point method (in Janpanese). In: Tone, K. (ed.) New Methods for Linear Programming 3, pp. 98–100. Wiley, New York (1990)Google Scholar
  16. 16.
    Mizuno, S.: Polynomiality of infeasible-interior-point algorithms for linear programming. Math. Program. 67, 109–119 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Potra, F.A., Sheng, R.: Superlinear convergence of interior-point algorithms for semidefinite programming. J. Optim. Theory Appl. 99, 103–119 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Potra, F.A., Sheng, R.: A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming. SIAM J. Optim. 8, 1007–1028 (1998)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhang, Y.: On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming. SIAM J. Optim. 8, 365–386 (1998)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Rangarajan, B.K.: Polynomial convergence of infeasible-interior-point methods over symmetric cones. SIAM J. Optim. 16, 1211–1229 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhang, J., Zhang, K.: Polynomial complexity of an interior point algorithm with a second order corrector step for symmetric cone programming. Math. Meth. Oper. Res. 73, 75–90 (2011)CrossRefGoogle Scholar
  22. 22.
    Potra, F.A.: An infeasible interior point method for linear complementarity problems over symmetric cones. Numer. Anal. Appl. Math. 1168, 1403–1406 (2009)CrossRefGoogle Scholar
  23. 23.
    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)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Faraut, J., Korányi, A.: Analysis on Symmetric Cone. Oxford University Press, New York (1994)Google Scholar
  25. 25.
    Luo, Z., Xiu, N.: Path-following interior point algorithms for the Cartesian \(P_*(\kappa )\)-LCP over symmetric cones. Sci. China. Ser. A 52, 1769–1784 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Gu, G., Zangiabadi, M., Roos, C.: Full Nesterov-Todd step infeasible interior-point method for symmetric optimization. Eur. J. Oper. Res. 214, 473–484 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Liu, C.: Study on complexity of some interior-point algorithms in conic programming (in chinese). Ph.D. thesis, Xidian University (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China
  2. 2.School of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeoples Republic of China

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