Journal of Optimization Theory and Applications

, Volume 170, Issue 2, pp 528–545 | Cite as

An Efficient Primal–Dual Interior Point Method for Linear Programming Problems Based on a New Kernel Function with a Trigonometric Barrier Term

Article

Abstract

In this paper, we present a primal–dual interior point method for linear optimization problems based on a new efficient kernel function with a trigonometric barrier term. We derive the complexity bounds for large and small-update methods, respectively. We obtain the best known complexity bound for large update, which improves significantly the so far obtained complexity results based on a trigonometric kernel function given by Peyghami et al. The results obtained in this paper are the first to reach this goal.

Keywords

Linear optimization Kernel function Interior point methods Complexity bound 

Mathematics Subject Classification

90C05 90C31 90C51 

Notes

Acknowledgments

The authors are very grateful and would like to thank the anonymous referees for their suggestions and helpful comments, which significantly improved the presentation of this paper.

References

  1. 1.
    El Ghami, M., Ivanov, I.D., Roos, C., Steihaug, T.: A polynomial-time algorithm for LO based on generalized logarithmic barrier functions. Int. J. Appl. Math. 21, 99–115 (2008)MathSciNetMATHGoogle Scholar
  2. 2.
    Roos, C., Terlaky, T., Vial, J.Ph: Theory and Algorithms for Linear Optimization, An Interior Point Approach. Wiley, Chichester (1997)MATHGoogle Scholar
  3. 3.
    Peng, J., Roos, C., Terlaky, T.: A new and efficient large-update interior point method for linear optimization. J. Comput. Technol. 6, 61–80 (2001)MathSciNetMATHGoogle Scholar
  4. 4.
    Bai, Y.Q., El Ghami, M., Roos, C.: A comparative study of kernel functions for primal-dual interior point algorithms in linear optimization. SIAM. J. Optim. 15(1), 101–128 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    El Ghami, M., Guennoun, Z.A., Bouali, S., Steihaug, T.: Interior point methods for linear optimization based on a kernel function with a trigonometric barrier term. J. Comput. Appl. Math. 236, 3613–3623 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Peyghami, M.R., Hafshejani, S.F., Shirvani, L.: Complexity of interior point methods for linear optimization based on a new trigonometric kernel function. J. Comput. Appl. Math. 255, 74–85 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Peyghami, M.R., Hafshejani, S.F.: Complexity analysis of an interior point algorithm for linear optimization based on a new proximity function. Numer. Algorithms 67, 33–48 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cai, X.Z., Wang, G.Q., El Ghami, M., Yue, Y.J.: Complexity analysis of primal-dual interior-point methods for linear optimization based on a new parametric kernel function with a trigonometric barrier term. Abstr. Appl. Anal., Art. ID 710158, 11 (2014)Google Scholar
  9. 9.
    Karmarkar, N.K.: A new polynomial-time algorithm for linear programming. In: Proceedings of the 16th Annual ACM Symposium on Theory of Computing, vol. 4, pp. 373–395 (1984)Google Scholar
  10. 10.
    Bai, Y.Q., Roos, C.: A primal-dual interior point method based on a new kernel function with linear growth rate. In: Proceedings of the 9th Australian Optimization Day, Perth, Australia (2002)Google Scholar
  11. 11.
    Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior Point Algorithms. Princeton University Press, Princeton (2002)MATHGoogle Scholar
  12. 12.
    Ye, Y.: Interior Point Algorithms, Theory and Analysis. Wiley, Chichester (1997)CrossRefMATHGoogle Scholar
  13. 13.
    Megiddo, N.: Pathways to the optimal set in linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 131–158. Springer, New York (1989)CrossRefGoogle Scholar
  14. 14.
    Sonnevend, G.: An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In: Prekopa, A., Szelezsan, J., Strazicky, B. (eds.) System Modelling and Optimization: Proceedings of the 12th IFIP-Conference, Budapest, Hungary, 1985. Lecture Notes in Control and Information Science, vol. 84, pp. 866–876. Springer, Berlin (1986)Google Scholar
  15. 15.
    Kheirfam, B., Moslem, M.: A polynomial-time algorithm for linear optimization based on a new kernel function with trigonometric barrier term. YUJOR 25(2), 233–250 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Li, X., Zhang, M.: Interior-point algorithm for linear optimization based on a new trigonometric kernel function. Oper. Res. Lett 43(5), 471–475 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cho, G.M.: An interior point algorithm for linear optimization based on a new barrier function. Appl. Math. Comput. 218, 386–395 (2011)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Mousaab Bouafia
    • 1
    • 2
  • Djamel Benterki
    • 3
  • Adnan Yassine
    • 2
  1. 1.LabCAV, Laboratory of Advanced ControlUniversity of 8 May 1945 GuelmaGuelmaAlgeria
  2. 2.LMAH - ULH, FR CNRS 3335Normandie UniversityLe HavreFrance
  3. 3.LMFN, Laboratory of Fundamental and Numerical MathematicsUniversity Setif 1SétifAlgeria

Personalised recommendations