Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  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)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  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)

  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)

  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)

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

    MATH  Google Scholar 

  12. Ye, Y.: Interior Point Algorithms, Theory and Analysis. Wiley, Chichester (1997)

    Book  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

  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)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  17. Cho, G.M.: An interior point algorithm for linear optimization based on a new barrier function. Appl. Math. Comput. 218, 386–395 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mousaab Bouafia.

Additional information

Communicated by Suliman Saleh Al-Homidan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bouafia, M., Benterki, D. & Yassine, A. An Efficient Primal–Dual Interior Point Method for Linear Programming Problems Based on a New Kernel Function with a Trigonometric Barrier Term. J Optim Theory Appl 170, 528–545 (2016). https://doi.org/10.1007/s10957-016-0895-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-0895-0

Keywords

  • Linear optimization
  • Kernel function
  • Interior point methods
  • Complexity bound

Mathematics Subject Classification

  • 90C05
  • 90C31
  • 90C51