Skip to main content
SpringerLink
Log in

A lower bound on the number of iterations of long-step primal-dual linear programming algorithms

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Recently, Todd has analyzed in detail the primal-dual affine-scaling method for linear programming, which is close to what is implemented in practice, and proved that it may take at leastn 1/3 iterations to improve the initial duality gap by a constant factor. He also showed that this lower bound holds for some polynomial variants of primal-dual interior-point methods, which restrict all iterates to certain neighborhoods of the central path. In this paper, we further extend his result to long-step primal-dual variants that restrict the iterates to a wider neighborhood. This neigh-borhood seems the least restrictive one to guarantee polynomiality for primal-dual path-following methods, and the variants are also even closer to what is implemented in practice.

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. K.M. Anstreicher, On the performance of Karmarkar's algorithm over a sequence of iterations, SIAM J. Optim. 1(1991)22.

    Google Scholar 

  2. K.M. Anstreicher, J. Ji, F. Potra and Y. Ye, Average performance of a self-dual interior-point algorithm for linear programming, in:Complexity in Numerical Optimization, ed. P. Pardalos (World Scientific, New Jersey, 1993) p. 1.

    Google Scholar 

  3. D. Bertsimas and X. Luo, On the worst case complexity of potential reduction algorithms for linear programming, Working Paper 3558-93, Sloan School of Management, MIT, Cambridge, MA 02139, USA (1993).

    Google Scholar 

  4. D. Goldfarb and M.J. Todd, Linear programming, in:Optimization, Volume 1 ofHandbooks in Operations Research and Management Science, ed. G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd (North-Holland, Amsterdam, The Netherlands, 1989) p. 73.

    Google Scholar 

  5. C.C. Gonzaga, Path following methods for linear programming, SIAM Rev. 34(1992)167.

    Google Scholar 

  6. J. Ji and Y. Ye, A complexity analysis for interior-point algorithms based on Karmarkar's potential function, SIAM J. Optim. 4(1994)512.

    Google Scholar 

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

    Google Scholar 

  8. M. Kojima, N. Megiddo and S. Mizuno, A primal-dual infeasible-interior-point algorithm for linear programming, Math. Progr. 61(1993)263.

    Google Scholar 

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

    Google Scholar 

  10. M. Kojima, S. Mizuno and A. Yoshise, A polynomial-time algorithm for a class of linear complementarity problems, Math. Progr. 44(1989)1.

    Google Scholar 

  11. M. Kojima, S. Mizuno and A. Yoshise, An\(O(\sqrt {nL} )\) iteration potential reduction algorithm for linear complementarity problems, Math. Progr. 50(1991)331.

    Google Scholar 

  12. I.J. Lustig, R.E. Marsten and D.F. Shanno, Computational experience with a primal-dual interior point method for linear programming, Lin. Alg. Appl. 152(1991):191.

    Google Scholar 

  13. I.J. Lustig, R.E. Marsten and D.F. Shanno, Computational experience with a globally convergent primal-dual predictor-corrector algorithm for linear programming, Math. Progr 66(1994)123.

    Google Scholar 

  14. S. Mehrotra and Y. Ye, On finding an interior point on the optimal face of linear programs, Math. Progr. 62(1993)497.

    Google Scholar 

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

    Google Scholar 

  16. R.D.C. Monteiro and I. Adler, Interior path following primal-dual algorithms: Part I: Linear programming, Math. Progr. 44(1989)27.

    Google Scholar 

  17. A.S. Nemirovsky, An algorithm of the Karmarkar type, Tekhnicheskaya Kibernetika 1(1987)105, translated in: Sov. J. Comp. Syst. Sci. 25(5)(1987)61.

    Google Scholar 

  18. M.J.D. Powell, On the number of iterations of Karmarkar's algorithm for linear programming, Math. Progr. 62(1993)153.

    Google Scholar 

  19. G. Sonnevend, J. Stoer and G. Zhao, On the complexity of following the central path of linear programs by linear extrapolation II, Math. Progr. 52(1991)527.

    Google Scholar 

  20. M.J. Todd, Recent developments and new directions in linear programming, in:Mathematical Programming: Recent Developments and Applications, ed. M. Iri and K. Tanabe (Kluwer Academic, Dordrecht The Netherlands, 1989), p. 109.

    Google Scholar 

  21. M.J. Todd, A lower bound on the number of iterations of primal-dual interior-point methods for linear programming, in:Numerical Analysis 1993, Volume 303 ofPitman Research Notes in Mathematics, ed. G.A. Watson and D.F. Griffiths (Longman, Burnt Hill, UK, 1994) p. 89.

    Google Scholar 

  22. Y. Ye, Toward probabilistic analysis of interior-point algorithms for linear programming, Math. Oper. Res. 19(1994)38.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Operations Research and Industrial Engineering, Cornell University, 14853, Ithaca, NY, USA

    Michael J. Todd

  2. Department of Management Sciences, The University of Iowa, 52242, Iowa City, IA, USA

    Yinyu Ye

Authors
  1. Michael J. Todd
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yinyu Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by NSF, AFOSR and ONR through NSF Grant DMS-8920550.

This author is supported in part by NSF Grant DDM-9207347. Part of thiw work was done while the author was on a sabbatical leave from the University of Iowa and visiting the Cornell Theory Center, Cornell University, Ithaca, NY 14853, supported in part by the Cornell Center for Applied Mathematics and by the Advanced Computing Research Institute, a unit of the Cornell Theory Center, which receives major funding from the National Science Foundation and IBM Corporation, with additional support from New York State and members of its Corporate Research Institute.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Todd, M.J., Ye, Y. A lower bound on the number of iterations of long-step primal-dual linear programming algorithms. Ann Oper Res 62, 233–252 (1996). https://doi.org/10.1007/BF02206818

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02206818

Keywords

Search

Navigation