Skip to main content
SpringerLink
Log in

A wide neighborhood interior-point algorithm for linear optimization based on a specific kernel function

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

This paper presents an interior point algorithm for solving linear optimization problems in a wide neighborhood of the central path introduced by Ai and Zhang (SIAM J Optim 16:400–417, 2005). In each iteration, the algorithm computes the new search directions by using a specific kernel function. The convergence of the algorithm is shown and it is proved that the algorithm has the same iteration bound as the best short-step algorithms. We demonstrate the computational efficiency of the proposed algorithm by testing some Netlib problems in standard form. To best our knowledge, this is the first wide neighborhood path-following interior-point method with the same complexity as the best small neighborhood path-following interior-point methods that uses the kernel function.

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. W. Ai, S. Zhang, An \(O(\sqrt{n}L)\) iteration primal-dual path-following method, based on wide neighborhoods and large updates, for monotone LCP. SIAM J. Optim. 16, 400–417 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Y.Q. Bai, M. El Ghami, C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization. SIAM J. Optim. 15(1), 101–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Z. Darvay, P.R. Takács, Large-step interior-point algorithm for linear optimization based on a new wide neighborhood. Cent. Eur. J. Oper. Res. 26(3), 551–563 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Z. Feng, L. Fang, A new \(O(\sqrt{n}L)\)-iteration predictor–corrector algorithm with wide neighborhood for semidefinite programming. J. Comput. Appl. Math. 256, 65–76 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. N.K. Karmarkar, A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Kheirfam, A predictor–corrector infeasible-interior-point algorithm for semidefinite optimization in a wide neighborhood. Fundam. Inform. 152(1), 33–50 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. B. Kheirfam, M. Chitsaz, Polynomial convergence of two higher order interior-point methods for \(P_*(\kappa )\)-LCP in a wide neighborhood of the central path. Period. Math. Hung. 76(2), 243–264 (2018)

    Article  MATH  Google Scholar 

  8. B. Kheirfam, M. Chitsaz, A new second-order corrector interior-point algorithm for \(P_*(\kappa )\)-LCP. Filomat 31(20), 6379–6391 (2017)

    Article  MathSciNet  Google Scholar 

  9. B. Kheirfam, M. Mohammadi-Sanghachin, A wide neighborhood second-order predictor–corrector interior-point algorithm for semidefinite optimization with modified corrector directions. Fundam. Inform. 153(4), 327–346 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Y. Li, T. Terlaky, A new class of large neighborhood path-following interior point algorithms for semidefinite optimization with \(O(\sqrt{n}\log (\frac{{\rm tr}(X^0S^0)}{\epsilon }))\) iteration complexity. SIAM J. Optim. 20(6), 2853–2875 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Liu, H.W. Liu, W. Cong, An \(O(\sqrt{n}L)\) iteration primal-dual second-order corrector algorithm for linear programming. Optim. Lett. 5, 729–743 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Megiddo, Pathways to the optimal set in linear programming, in Progress in Mathematical Programming, Interior-Point and Related Methods, ed. by N. Megiddo (Springer, New York, 1989), pp. 131–158

    Chapter  Google Scholar 

  13. J. Peng, C. Roos, T. Terlaky, Self-Regular Functions: A New Paradigm for Primal-Dual Interior-Point Methods (Princeton University Press, Princeton, 2002)

    MATH  Google Scholar 

  14. F.A. Potra, Interior point methods for sufficient horizontal LCP in a wide neighborhood of the central path with best known iteration complexity. SIAM J. Optim. 24(1), 1–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Renegar, A polynomial-time algorithm, based on Newton’s method, for linear programming. Math. Program. 40(1–3), 59–93 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gy. Sonnevend, An ”analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming, in System Modelling and Optimization: Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985, vol. 84, Lecture Notes in Control and Information Sciences, ed. by A. Prékopa, J. Szelezsán, B. Strazicky (Springer, Berlin, 1986), pp. 866–876

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

    Book  MATH  Google Scholar 

  18. X. Yang, Y. Zhang, H. Liu, A wide neighborhood infeasible-interior-point method with arc-search for linear programming. J. Appl. Math. Comput. 51(1), 209–225 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Y. Ye, M. Todd, S. Mizuno, An \({\cal{O}}(\sqrt{n}L)\)-iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19, 53–67 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors 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 & Masoumeh Haghighi

Authors
  1. Behrouz Kheirfam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masoumeh Haghighi
    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., Haghighi, M. A wide neighborhood interior-point algorithm for linear optimization based on a specific kernel function. Period Math Hung 79, 94–105 (2019). https://doi.org/10.1007/s10998-018-00271-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-018-00271-0

Keywords

Search

Navigation