Skip to main content
SpringerLink
Log in

Two computationally efficient polynomial-iteration infeasible interior-point algorithms for linear programming

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Since the beginning of the development of interior-point methods, there exists a puzzling gap between the results in theory and the observations in numerical experience, i.e., algorithms with good polynomial bounds are not computationally efficient and algorithms demonstrated efficiency in computation do not have a good or any polynomial bound. Todd raised a question in 2002: “Can we find a theoretically and practically efficient way to reoptimize?” This paper is an effort to close the gap. We propose two arc-search infeasible interior-point algorithms with infeasible central path neighborhood wider than all existing infeasible interior-point algorithms that are proved to be convergent. We show that the first algorithm is polynomial and its simplified version has a complexity bound equal to the best known complexity bound for all (feasible or infeasible) interior-point algorithms. We demonstrate the computational efficiency of the proposed algorithms by testing all Netlib linear programming problems in standard form and comparing the numerical results to those obtained by Mehrotra’s predictor-corrector algorithm and a recently developed more efficient arc-search algorithm (the convergence of these two algorithms is unknown). We conclude that the newly proposed algorithms are not only polynomial but also computationally competitive compared to both Mehrotra’s predictor-corrector algorithm and the efficient arc-search algorithm.

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Lustig, I., Marsten, R., Shannon, D.: Computational experience with a primal-dual interior-point method for linear programming. Linear Algebra Appl. 152, 191–222 (1991)

    Article  MathSciNet  Google Scholar 

  4. Lustig, I., Marsten, R., Shannon, D.: On implementing Mehrotra’s predictor-corrector interior-point method for linear programming. SIAM J. Optim. 2, 432–449 (1992)

    Article  MathSciNet  Google Scholar 

  5. Monteiro, R., Adler, I., Resende, M.: 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  Google Scholar 

  6. Kojima, M., Mizuno, S., Yoshise, A.: A polynomial-time algorithm for a class of linear complementarity problem. Math. Program. 44, 1–26 (1989)

    Article  MathSciNet  Google Scholar 

  7. Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior-point algorithm for linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior-point and Related Methods. Springer-Verlag, New York (1989)

    Chapter  Google Scholar 

  8. Cartis, C.: Some disadvantages of a Mehrotra-type primal-dual corrector interior-point algorithm for linear programming. Appl. Numer. Math. 59, 1110–1119 (2009)

    Article  MathSciNet  Google Scholar 

  9. Klee, V., Minty, G.: How good is the simplex algorithm? In: Shisha, O. (ed.) Inequalities, vol. III, pp 159–175, Academic Press (1972)

  10. Khachiyan, L.: A polynomial algorithm in linear programming. Doklady Akademiia Nauk SSSR 224, 1093–1096 (1979)

    MathSciNet  MATH  Google Scholar 

  11. Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 375–395 (1984)

    Article  MathSciNet  Google Scholar 

  12. Todd, M.J.: The many facets of linear programming. Math. Progr. Ser B 91, 417–436 (2002)

    Article  MathSciNet  Google Scholar 

  13. Yang, Y.: A polynomial arc-search interior-point algorithm for linear programming. J. Optim. Theory Appl. 158(3), 859–873 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Cartis, C., Gould, N.I.M.: Finding a point in the relative interior of a polyhedron. Technical Report NA-07/01, Computing Laboratory Oxford University (2007)

  16. Yang, Y.: CurveLP-a MATLAB implementation of an infeasible interior-point algorithm for linear programming. Numer. Algorithms 74(4), 967–996 (2017)

    Article  MathSciNet  Google Scholar 

  17. 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  Google Scholar 

  18. Mizuno, M.: Polynomiality of infeasible interior-point algorithms for linear programming, vol. 67, pp. 109–119 (1994)

    Article  MathSciNet  Google Scholar 

  19. Miao, J.: Two infeasible interior-point predict-corrector algorithms for linear programming. SIAM J. Optim. 6(3), 587–599 (1996)

    Article  MathSciNet  Google Scholar 

  20. Yang, Y., Yamashita, M.: An arc-search O(nL) infeasible-interior-point algorithm for linear programming. Optim. Lett. https://doi.org/10.1007/s11590-017-1142-9

    Article  MathSciNet  Google Scholar 

  21. Kojima, M.: Basic lemmas in polynomial-time infeasible interior-point methods for linear programming. Ann. Oper. Res. 62, 1–28 (1996)

    Article  MathSciNet  Google Scholar 

  22. Andersen, E.D.: Finding all linearly dependent rows in large-scale linear programming. Optim. Methods Softw. 6, 219–227 (1995)

    Article  Google Scholar 

  23. Yang, Y.: Arc-search path-following interior-point algorithms for linear programming, Optimization Online. http://www.optimization-online.org/DB_HTML/2009/08/2375.html (2009)

  24. Yang, Y.: An Efficient Polynomial Interior-Point Algorithm for Linear Programming, arXiv:1304.3677 [math.OC] (2013)

  25. Kojima, M., Megiddo, N., Mizuno, S.: A primal-dual infeasible interior-point algorithm for linear programming. Math. Program. Series A 61, 261–280 (1993)

    Article  MathSciNet  Google Scholar 

  26. Czyzyk, J., Mehrotra, S., Wagner, M., Wright, S.J.: PCx User Guide (version 1.1). Technical Report OTC 96/01 Optimization Technology Center (1997)

  27. Zhang, Y.: Solving large-scale linear programs by interior-point methods under the Matlab environment. Technical Report TR96-01, Department of Mathematics and Statistics University of Maryland (1996)

  28. Ng, E., Peyton, B.W.: Block sparse Cholesky algorithm on advanced uniprocessor computers. SIAM J. Sci. Comput. 14, 1034–1056 (1993)

    Article  MathSciNet  Google Scholar 

  29. Liu, J.W.: Modification of the minimum degree algorithm by multiple elimination. ACM Trans. Math. Softw. 11, 141–153 (1985)

    Article  MathSciNet  Google Scholar 

  30. Guler, O., den Hertog, D., Roos, C., Terlaky, T., Tsuchiya, T.: Degeneracy in interior-point methods for linear programming: A survey. Ann. Oper. Res. 46, 107–138 (1993)

    Article  MathSciNet  Google Scholar 

  31. Gill, P.E., Murray, W., Saunders, M.A., Tomlin, J.A., Wright, M.H.: On projected Newton barrier methods for linear programming and an equivalence of Karmarkar’s projective method. Math. Program. 36, 183–209 (1986)

    Article  MathSciNet  Google Scholar 

  32. Ekefer, J.: Sequential minimax search for a maximum. Proc. Amer. Math. Soc. 4, 502–506 (1953)

    Article  MathSciNet  Google Scholar 

  33. Luenberger, D.: Linear and Nonlinear Programming, 2nd edn. Addison-Wesley Publishing Company, Menlo Park (1984)

  34. Tits, A.L., Yang, Y.: Globally convergent algorithms for robust pole assignment by state feedback. IEEE Trans. Autom. Control 41, 1432–1452 (1996)

    Article  MathSciNet  Google Scholar 

  35. Dolan, E.D., More, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author would like to thank Dr. Chris Hoxie, in the Office of Research at US NRC, for providing a computational environment for this research.

Author information

Authors and Affiliations

  1. Office of Research, US NRC, Two White Flint North 11545 Rockville Pike, Rockville, MD, 20852-2738, USA

    Y. Yang

Authors
  1. Y. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, Y. Two computationally efficient polynomial-iteration infeasible interior-point algorithms for linear programming. Numer Algor 79, 957–992 (2018). https://doi.org/10.1007/s11075-018-0469-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-018-0469-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation