[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Primal-dual algorithms for linear programming based on the logarithmic barrier method

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we deal with primal-dual interior point methods for solving the linear programming problem. We present a short-step and a long-step path-following primal-dual method and derive polynomial-time bounds for both methods. The iteration bounds are as usual in the existing literature, namely\(O(\sqrt n L)\) iterations for the short-step variant andO(nL) for the long-step variant. In the analysis of both variants, we use a new proximity measure, which is closely related to the Euclidean norm of the scaled search direction vectors. The analysis of the long-step method depends strongly on the fact that the usual search directions form a descent direction for the so-called primal-dual logarithmic barrier function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Karmarkar, N. K.,A New Polynomial-Time Algorithm for Linear Programming, Combinatorica, Vol. 4, pp. 373–395, 1984.

    Google Scholar 

  2. Gill, P. E., Murray, W., Saunders, M. A., Tomlin, J. A., andWright, M. H.,On Projected Newton Barrier Methods for Linear Programming and an Equivalence to Karmarkar's Projective Method, Mathematical Programming, Vol. 36, pp. 183–209, 1986.

    Google Scholar 

  3. Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Technique, John Wiley and Sons, New York, New York, 1968.

    Google Scholar 

  4. Gonzaga, C. C.,An Algorithm for Solving Linear Programming Problems in O(n 3 L)Operations, Progress in Mathematical Programming, Interior Point and Related Methods, Edited by N. Megiddo, Springer Verlag, New Yor, New York, pp. 1–28, 1989.

    Google Scholar 

  5. Gonzaga, C. C.,Large-Step Path-Following Methods for Linear Programming, Part 1: Barrier Function Method, SIAM Journal on Optimization, Vol. 1, pp. 268–279, 1991.

    Google Scholar 

  6. Roos, C., andVial, J. P.,Long Steps with the Lagarithmic Penalty Barrier Function in Linear programming, Economic Decision Making: Games, Economics, and Optimization, Edited by J. Gabszevwicz, J. F. Richard, and L. Wolsey, Elsevier Science Publishers, Amsterdam, Holland, pp. 433–441, 1990.

    Google Scholar 

  7. Roos, C., andVial, J. P.,A Polynomial Method of Approximate Centers for Linear Programming, Mathematical Programming, Vol. 54, pp. 295–305, 1992.

    Google Scholar 

  8. Anstreicher, K. M.,On Long-Step Path-Following and SUMT for Linear and Quadratic Programming, Working Paper, Department of Operations Research, Yale University, New Haven, Connecticut, 1990.

    Google Scholar 

  9. Mylander, W. C., Holmes, R. L., andMcCormick, G. P.,A Guide to SUMT, Version 4: The Computer Program Implementing the Sequential Unconstrained Minimization Technique for Nonlinear Programming, Research Paper RAC-P-63, Research Analysis Corporation, McLean, Virginia, 1971.

    Google Scholar 

  10. McCormick, G. P.,The Projective SUMT Method for Convex Programming, Mathematics of Operations Research, Vol. 14, pp. 203–223, 1989.

    Google Scholar 

  11. Kojima, M., Mizuno, S., andYoshise, A.,A Primal-Dual Interior-Point Algorithm for Linear Programming, Progress in Mathematical Programming, Interior Point and Related Methods, Edited by N. Megiddo, Springer Verlag, New York, New York, pp. 29–47, 1989.

    Google Scholar 

  12. Monteiro, R. D. C., andAdler, I.,Interior Path-Following Primal-Dual Algorithms, Part 1: Linear Programming, Mathematical Programming, Vol. 44, pp. 27–41, 1989.

    Google Scholar 

  13. Kojima, M., Mizuno, S., andYoshise, A., An\(O(\sqrt n L)\) Iteration Potential Reduction Algorithm for Linear Complementarity Problems, Mathematical Programming, Vol. 50, pp. 331–342, 1991.

    Google Scholar 

  14. Kranich, E.,Interior-Point Methods for Mathematical Programming: A Bibliography, Diskussionsbeitrag No. 171, Fern Universität Hagen, Hagen, Germany, 1991.

    Google Scholar 

  15. Lustig, I. J., Marsten, R. E., andShanno, D. F.,Computational Experience with a Primal-Dual Interior-Point Method for Linear Programming, Linear Algebra and Its Applications, Vol. 152, pp. 191–222, 1991.

    Google Scholar 

  16. McShane, K. A., Monma, C. L., andShanno, D. F.,An Implementation of a Primal-Dual Interior-Point Method for Linear Programming, ORSA Journal on Computing, Vol. 1, pp. 70–83, 1989.

    Google Scholar 

  17. Mehrotra, S.,On the Implementation of a Primal-Dual Interior-Point Method, SIAM Journal on Optimization, Vol. 2, pp. 575–601, 1992.

    Google Scholar 

  18. Kojima, M., Mizuno, S., andYoshise, A.,A Primal-Dual Interior-Point Algorithm for a Class of Linear Complementarity Problems, Mathematical Programming, Vol. 44, pp. 1–26, 1989.

    Google Scholar 

  19. Ding, J., andLi, T. Y.,An Algorithm Based on Weighted Logarithmic Barrier Functions for Linear Complementarity Problems, Arabian Journal for Science and Engineering, Vol. 15, pp. 679–685, 1990.

    Google Scholar 

  20. Den Hertog, D., Roos, C., andVial, J. P.,A Complexity Reduction for the Long-Step Path-Following Algorithm for Linear Programming, SIAM Journal on Optimization, Vol. 2, pp. 71–87, 1992.

    Google Scholar 

  21. Gonzaga, C. C.,Path-Following Methods for Linear Programming, SIAM Review, Vol. 34, pp. 167–224, 1992.

    Google Scholar 

  22. Den Hertog, D.,Interior-Point Approach to Linear, Quadratic, and Convex Programming: Algorithms and Complexity, Kluwer Publishers, Dordrecht, The Netherlands, 1993.

    Google Scholar 

  23. Nesterov, Y. E., andNemirovsky, A. S.,Interior-Point Polynomial Methods in Convex Programming: Theory and Algorithms, SIAM Publications, Philadelphia, Pennsylvania, 1994.

    Google Scholar 

  24. Monteiro, R. D. C., andAdler, I.,Interior Path-Following Primal-Dual Algorithms, Part 2: Convex Quadratic Programming, Mathematical Programming, Vol. 44, pp. 43–66, 1989.

    Google Scholar 

  25. Ling, P. D., Private Communication, University of East Anglia, Norwich, England, 1992.

  26. Papadimitriou, C. H., andSteiglitz, K.,Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, New Jersey, 1982.

    Google Scholar 

Download references

Author information

Author notes

  1. T. Terlaky (Assistant Professor)

    Present address: Eötvös University, Budapest, Hungary

Authors and Affiliations

  1. Faculty of Technical Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands

    B. Jansen (PhD Student), C. Roos (Assistant Professor) & T. Terlaky (Assistant Professor)

  2. Department of Commercial and Industrial Economics, University of Geneva, Geneva, Switzerland

    J. P. Vial Professor

Authors
  1. B. Jansen
    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

  4. J. P. Vial Professor
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by L. C. W. Dixon

This work was supported by a research grant from Shell, by the Dutch Organization for Scientific Research (NWO) Grant 611-304-028, by the Hungarian National Research Foundation Grant OTKA-2116, and by the Swiss National Foundation for Scientific Research Grant 12-26434.89.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jansen, B., Roos, C., Terlaky, T. et al. Primal-dual algorithms for linear programming based on the logarithmic barrier method. J Optim Theory Appl 83, 1–26 (1994). https://doi.org/10.1007/BF02191759

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02191759

Key Words

Search

Navigation