Skip to main content
SpringerLink
Log in

A wide neighborhood infeasible-interior-point method with arc-search for linear programming

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we propose an arc-search infeasible-interior-point method for linear programming. The proposed algorithm is based on a wide neighborhood of the central path and searches the optimizers along the ellipses that approximate the entire the central path. We also establish polynomial complexity of the proposed algorithm. Finally, numerical experiments show that the proposed algorithm is efficient and reliable.

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. Ai, W.: Neighborhood-following algorithms for linear programming. Sci. China Ser. A 47, 812–820 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ai, W., Zhang, S.: 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 

  3. Anstreicher, K., Bosch, R.: A new infinity-norm path following algorithm for linear programming. SIAM J. Optim. 5, 236–246 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Browne, S., Dongarra, J., Grosse, E., Rowan, T.: The netlib mathematical software repository. Corporation for National Research Initiatives (1995)

  5. Hung, P., Ye, Y.: An asymptotical \(O(\sqrt{n}L)\)-iteration path-following linear programming algorithm that use wide neighborhoods. SIAM J. Optim. 6, 570–586 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jansen, B., Roos, C., Terlaky, T.: Improved complexity using higher-order correctors for primal-dual Dikin affine scaling. Math. Program. 76, 117–130 (1996)

    MathSciNet  MATH  Google Scholar 

  7. Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–393 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Lustig, I.J., Marsten, R.E., Shanno, D.F.: On implementing Mehrotra’s predictor-corrector interior-point method for linear programming. SIAM J. Optim. 2, 435–449 (1992)

    Article  MathSciNet  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. Mizuno, S., Todd, M.J., Ye, Y.: On adaptive-step primal-dual interior-point algorithms for linear programming. Math. Oper. Res. 18, 964–981 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mehrotra, S.: On the implementation of a primal-dual interior point method. SIAM J. Optim. 2, 575–601 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Monteiro, R.D., Adler, I., Resende, M.G.: A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension. Math. Oper. Res. 15, 191–214 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nesterov, Y.: Long-step strategies in interior-point primal-dual methods. Math. Program. 76, 47–94 (1996)

    MathSciNet  MATH  Google Scholar 

  15. Potra, F.: An infeasible-interior-point predictor-corrector algorithm for linear programming. SIAM J. Optim. 6, 19–32 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Roos, C., Terlarky, T., Vial, J.P.: Theory and algorithms for linear optimization: an interior-point approach. Springer, New York (2006)

    Google Scholar 

  17. Tanabe, K.: Centered newton method for linear programming: interior and ‘exterior’ point method. New Methods Linear Program. 3, 98–100 (1990). (in janpanese)

    Google Scholar 

  18. Tian, D.G.: An entire space polynomial-time algorithm for linear programming. J. Glob. Optim. 58, 109–135 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tian, D.G.: An exterior point polynomial-time algorithm for convex quadratic programming. Comput. Optim. Appl., 1–28 (2014)

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

    Book  MATH  Google Scholar 

  21. Yang, Y.: Arc-search path-following interior-point algorithms for linear programming. Optim. Online (2009)

  22. Yang, Y.: A polynomial arc-search interior-point algorithm for convex quadratic programming. Eur. J. Oper. Res. 215, 25–38 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yang, Y.: A polynomial arc-search interior-point algorithm for linear programming. J. Optim. Theory Appl. 158, 859–873 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, Y.: Arc-Search Infeasible Interior-Point Algorithm for Linear Programming. arXiv preprint arXiv:1406.4539 (2014)

  25. Zhang, Y., Zhang, D.: On polynomiality of the Mehrotra-type predictor-corrector interior-point algorithms. Math. Program. 68, 303–318 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, People’s Republic of China

    Ximei Yang & Yinkui Zhang

  2. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, People’s Republic of China

    Hongwei Liu

Authors
  1. Ximei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yinkui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hongwei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ximei Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, X., Zhang, Y. & Liu, H. A wide neighborhood infeasible-interior-point method with arc-search for linear programming. J. Appl. Math. Comput. 51, 209–225 (2016). https://doi.org/10.1007/s12190-015-0900-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-015-0900-z

Keywords

Mathematics Subject Classification

Search

Navigation