Skip to main content
SpringerLink
Log in

New Complexity Analysis of the Primal—Dual Newton Method for Linear Optimization

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

Abstract

We deal with the primal–dual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the practical behavior of the algorithms and the theoretical performance results, in favor of the practical behavior. This is especially true for so-called large-update methods. We present some new analysis tools, based on a proximity measure introduced by Jansen et al., in 1994, that may help to close this gap. This proximity measure has not been used in the analysis of large-update methods before. The new analysis does not improve the known complexity results but provides a unified way for the analysis of both large-update and small-update methods.

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. E.D. Andersen, J. Gondzio, Cs. Mészáros and X. Xu, Implementation of interior point methods for large scale linear programming, in: Interior Point Methods of Mathematical Programming, ed. T. Terlaky (Kluwer Academic, Dordrecht, The Netherlands, 1996) pp. 189–252.

    Google Scholar 

  2. K.M. Anstreicher, J. Ji, F.A. Potra and Y. Ye, Probabilistic analysis of an infeasible-interior-point algorithm for linear programming, Mathematics of Operations Research 24 (1999) 176–192.

    Google Scholar 

  3. E. de Klerk, C. Roos and T. Terlaky, On primal-dual path-following algortihms for semidefinite programming, in: New Trends in Mathematical Programming, eds. F. Gianessi et al., Applied Optimization, Vol. 13 (Kluwer Academic, Dordrecht, The Netherlands, 1998) pp. 137–157.

    Google Scholar 

  4. E. de Klerk, Interior point methods for semidefinite programming, Ph.D. Thesis, Faculty of ITS/TWI, Delft University of Technology, The Netherlands (1997).

  5. A.V. Fiacco and G.P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968). Reprint: SIAM Classics in Applied Mathematics, Vol. 4 (SIAM Publications, Philadelphia, PA 19104–2688, USA, 1990).

    Google Scholar 

  6. R. Frisch, The logarithmic potential method for solving linear programming problems, Memorandum, University Institute of Economics, Oslo (1955).

  7. B. Jansen, C. Roos, T. Terlaky and J.-Ph. Vial, Primal-dual algorithms for linear programming based on the logarithmic barrier method, Journal of Optimization Theory and Applications 83 (1994) 1–26.

    Google Scholar 

  8. J. Jiang, A long step primal-dual path following method for semidefinite programming, Operations Research Letters 23(1–2) (1998) 53–61.

    Google Scholar 

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

    Google Scholar 

  10. 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) pp. 29–47.

    Google Scholar 

  11. M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science, Vol. 538 (Springer, Berlin, Germany, 1991).

    Google Scholar 

  12. F.A. Lootsma, Numerical Methods for Nonlinear Optimization (Academic Press, London, UK, 1972).

    Google Scholar 

  13. K.A. McShane, C.L. Monma and D.F. Shanno, An implementation of a primal-dual interior point method for linear programming, ORSA Journal on Computing 1 (1989) 70–83.

    Google Scholar 

  14. N. Megiddo, Pathways to the optimal set in linear programming, in: Progress in Mathematical Programming: Interior Point and Related Methods, ed. N. Megiddo (Springer, New York, 1989) pp. 131–158. Identical version in: Proceedings of the 6th Mathematical Programming Symposium of Japan, Nagoya, Japan (1986) pp. 1–35.

    Google Scholar 

  15. S. Mehrotra, On the implementation of a (primal-dual) interior point method, SIAM Journal on Optimization 2(4) (1992) 575–601.

    Google Scholar 

  16. S. Mehrotra and Y. Ye, On finding the optimal facet of linear programs, Mathematical Programming 62 (1993) 497–515.

    Google Scholar 

  17. S. Mizuno, An O(n3L) algorithm using a sequence for linear complementarity problems, Journal of the Operations Research Society of Japan 33 (1990) 66–75.

    Google Scholar 

  18. S. Mizuno, A primal-dual interior point method for linear programming, The Proceedings of the Institute of Statistical Mathematics 40(1) (1992) 27–44 (In Japanese).

    Google Scholar 

  19. S. Mizuno and A. Nagasawa, A primal-dual affine scaling potential reduction algorithm for linear programming, Mathematical Programming 62 (1993) 119–131.

    Google Scholar 

  20. 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 

  21. Y.E. Nesterov and M.J. Todd, Self-scaled barriers and interior-point methods for convex programming, Mathematics of Operations Research 22(1) (1997) 1–42.

    Google Scholar 

  22. C. Roos, T. Terlaky and J.-Ph. Vial, Theory and Algorithms for Linear Optimization. An Interior Approach (Wiley, Chichester, UK, 1997).

    Google Scholar 

  23. G. 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, Budapest, Hungary (September 1985), eds. A. Prekopa et al., Lecture Notes in Control and Information Sciences, Vol. 84 (Springer, Berlin, 1986) pp. 866–876.

    Google Scholar 

  24. M.J. Todd, A lower bound on the number of iterations of primal-dual interior-point methods for linear programming, in: Numerical Analysis 1993, eds. G.A. Watson and D.F. Griffiths, Pitman Research Notes in Mathematics, Vol. 303 (Longman Press, Harlow, (1994) pp. 237–259. See also [25].

    Google Scholar 

  25. M.J. Todd and Y. Ye, A lower bound on the number of iterations of long-step and polynomial interiorpoint linear programming algorithms, Annals of Operations Research 62 (1996) 233–252.

    Google Scholar 

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

    Google Scholar 

  27. Y. Ye, On the finite convergence of interior-point algorithms for linear programming, Mathematical Programming 57 (1992) 325–335.

    Google Scholar 

  28. Y. Ye, Interior Point Algorithms, Theory and Analysis (Wiley, Chichester, UK, 1997).

    Google Scholar 

  29. Y. Zhang and R.A. Tapia, Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited, Journal of Optimization Theory and Applications 73(2) (1992) 229–242.

    Google Scholar 

  30. G. Zhao, Large-step path-following primal-dual algorithms for linear programming, Research Report 613, National University of Singapore, Department of Mathematics, 10 Kent Ridge Crescent, Singapore 0511 (1994).

    Google Scholar 

  31. G.Y. Zhao, Interior point algorithms for linear complementarity problems based on large neighborhoods of the central path, SIAM Journal on Optimization 8 (1998) 397–413.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600, GA Delft, The Netherlands

    J. Peng

  2. Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600, GA Delft, The Netherlands

    C. Roos

  3. Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600, GA Delft, The Netherlands

    T. Terlaky

Authors
  1. J. Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Roos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Terlaky
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peng, J., Roos, C. & Terlaky, T. New Complexity Analysis of the Primal—Dual Newton Method for Linear Optimization. Annals of Operations Research 99, 23–39 (2000). https://doi.org/10.1023/A:1019280614748

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019280614748

Search

Navigation