Skip to main content
SpringerLink
Log in

Polynomiality of infeasible-interior-point algorithms for linear programming

  • Published:
Mathematical Programming Submit manuscript

Abstract

Kojima, Megiddo, and Mizuno investigate an infeasible-interior-point algorithm for solving a primal—dual pair of linear programming problems and they demonstrate its global convergence. Their algorithm finds approximate optimal solutions of the pair if both problems have interior points, and they detect infeasibility when the sequence of iterates diverges. Zhang proves polynomial-time convergence of an infeasible-interior-point algorithm under the assumption that both primal and dual problems have feasible points. In this paper, we show that a modification of the Kojima—Megiddo—Mizuno algorithm “solves” the pair of problems in polynomial time without assuming the existence of the LP solution. Furthermore, we develop anO(nL)-iteration complexity result for a variant of the 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. M. Kojima, N. Megiddo and S. Mizuno, “A primal—dual, infeasible-interior-point algorithm for linear programming,”Mathematical Programming 61 (1993) 263–280.

    Google Scholar 

  2. M. Kojima, S. Mizuno and A. Yoshise, “A primal—dual interior-point algorithm for linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior-Point and Related Methods (Springer-Verlag, New York, 1989) 29–47.

    Google Scholar 

  3. M. Kojima, S. Mizuno and A. Yoshise, “A polynomial-time algorithm for a class of linear complementarity problems,”Mathematical Programming 44 (1989) 1–26.

    Google Scholar 

  4. I.J. Lustig, “Feasibility issues in a primal—dual interior-point method for linear programming,”Mathematical Programming 49 (1990/91) 145–162.

    Google Scholar 

  5. I.J. Lustig, R.E. Marsten and D.F. Shanno, “Computational experience with a primal—dual interior-point method for linear programming,”Linear Algebra and Its Applications 152 (1991) 191–222.

    Google Scholar 

  6. R. Marsten, R. Subramanian, M. Saltzman, I.J. Lustig and D. Shanno, “Interior-point methods for linear programming: Just call Newton, Lagrange, and Fiacco and McCormick!,”Interfaces 20 (1990) 105–116.

    Google Scholar 

  7. N. Megiddo, “Pathways to the optimal set in linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior-Point and Related Methods (Springer-Verlag, New York, 1989) 131–158.

    Google Scholar 

  8. S. Mizuno, M.J. Todd and Y. Ye, “On adaptive-step primal—dual interior-point algorithms for linear programming,”Mathematics of Operations Research 18 (1993) 964–981.

    Google Scholar 

  9. R.D.C. Monteiro and I. Adler, “Interior-path following primal—dual algorithms. Part I: linear programming,”Mathematical Programming 44 (1989) 27–41.

    Google Scholar 

  10. R.D.C. Monteiro and I. Adler, “Interior path following primal—dual algorithms. Part II: convex quadratic programming,”Mathematical Programming 44 (1989) 43–66.

    Google Scholar 

  11. F.A. Potra, “An infeasible interior-point predictor—corrector algorithm for linear programming,” Report No. 26, The University of Iowa, Department of Mathematics (Iowa City, Iowa, 1992).

    Google Scholar 

  12. K. Tanabe, “Centered Newton method for mathematical programming,” in: M. Iri and K. Yajima, eds.,System Modeling and Optimization (Springer-Verlag, New York, 1988) 197–206.

    Google Scholar 

  13. K. Tanabe, “Centered Newton method for linear programming: Interior and ‘exterior’ point method’ (in Japanese),” in: K. Tone, ed.,New Methods for Linear Programming 3, The Institute of Statistical Mathematics (Tokyo, Japan, 1990) 98–100.

    Google Scholar 

  14. Y. Zhang, “On the convergence of a class of infeasible interior-point algorithms for the horizontal linear complementarity problem,”Siam Journal on Optimization 4 (1994) 208–227.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, 106, Tokyo, Japan

    Shinji Mizuno

Authors
  1. Shinji Mizuno
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The original title was “Polynomiality of the Kojima—Megiddo—Mizuno infeasible-interior-point algorithm for linear programming”.

Research performed while visiting Cornell University (April 1992 – January 1993) as an Overseas Research Scholar of the Ministry of Science, Education and Culture of Japan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mizuno, S. Polynomiality of infeasible-interior-point algorithms for linear programming. Mathematical Programming 67, 109–119 (1994). https://doi.org/10.1007/BF01582216

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01582216

Keywords

Search

Navigation