Skip to main content
SpringerLink
Log in

Improving the rate of convergence of interior point methods for linear programming

  • Published:
Mathematical Programming Submit manuscript

Abstract

This paper proposes a procedure for improving the rate of convergence of interior point methods for linear programming. If (x k) is the sequence generated by an interior point method, the procedure derives an auxiliary sequence (\(\bar x^k\)). Under the suitable assumptions it is shown that the sequence (\(\bar x^k\)) converges superlinearly faster to the solution than (x k). Application of the procedure to the projective and afflne scaling algorithms is discussed and some computational illustration is provided.

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. I. Adler and R.C. Monteiro, “Limiting behaviour of the affine scaling continuous trajectories for linear programming problems,”Mathematical Programming 50 (1991) 29–51.

    Google Scholar 

  2. M.D. Asic, V.V. Kovacevic-Vujcic and M.D. Radosavljevic-Nikolic, “Asymptotic behaviour of Karmarkar's method for linear programming,”Mathematical Programming 46 (1990) 173–190.

    Google Scholar 

  3. M.D. Asic, V.V. Kovacevic-Vujcic and M.D. Radosavljevic-Nikolic, “A note on limiting behaviour of the projective and the affine rescaling algorithms,” J.C. Lagarias and M.J. Todd, eds.,Mathematical Developments Arising from Linear Programming, AMS Series Contemporary Mathematics (AMS, Providence, RI, 1990) pp. 151–157.

    Google Scholar 

  4. E.R. Barnes, “A variation on Karmarkar's algorithm for solving linear programming problems,”Mathematical Programming 36 (1986) 174–182.

    Google Scholar 

  5. D.A. Bayer and J.C. Lagarias, “The nonlinear geometry of linear programming. I. Affine and projective rescaling trajectories,”Transactions of the American Mathematical Society 314 (1989) 499–526.

    Google Scholar 

  6. D.A. Bayer and J.C. Lagarias, “The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories,”Transactions of the American Mathematical Society 314 (1989) 527–581.

    Google Scholar 

  7. I.I. Dikin, “Iterative solution of problems of linear and quadratic programming,”Soviet Mathematics Doklady 8 (1967) 674–675.

    Google Scholar 

  8. C.C. Gonzaga, “An algorithm for solving linear programming problems in O(n 3 L) operations,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior Point and Related Methods (Springer, New York, 1988) pp. 1–28.

    Google Scholar 

  9. M. Iri and H. Imai, “A multiplicative barrier function method for linear programming,”Algorithmica 1 (1986) 373–395.

    Google Scholar 

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

    Google Scholar 

  11. V.V. Kovacevic-Vujcic, “Ubrzanje konvergencije metode kaznenih funkcija za linearno programiranje,” in:Proceedings of the 16th Yugoslav Symposium on Operations Research (Kupari, Yugoslvia, 1989) pp. 283–286.

    Google Scholar 

  12. J.C. Lagarias, “The nonlinear geometry of linear programming. III. Projective Legendre transform coordinates and Hilbert geometry,”Transactions of the American Mathematical Society 320 (1990) 193–225.

    Google Scholar 

  13. N. Megiddo and M. Shub, “Boundary behaviour of interior point algorithms for linear programming,”Mathematics of Operations Research 14 (1989) 97–146.

    Google Scholar 

  14. R.D.C. Monteiro, “Convergence and boundary behaviour of the projective scaling trajectories for linear programming,” to appear in:Journal of Complexity.

  15. M.D. Radosavljevic-Nikolic, “Asymptotic behaviour of the affine variant of Karmarkar's algorithm for linear programming,”XX Conference “Mathematical Optimization,”Humboldt-Universität zu Berlin (Berlin, 1988).

    Google Scholar 

  16. J. Renegar and M. Shub, “Simplified complexity analysis for Newton LP methods,” Technical Report 807, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, 1988).

    Google Scholar 

  17. M. Shub, “On the asymptotic behaviour of the projective rescaling algorithm for linear programming,”Journal of Complexity 3 (1987) 258–269.

    Google Scholar 

  18. T. Tsuchiya and K. Tanabe, “Local convergence properties of new methods in linear programming,”Journal of the Operations Research Society of Japan 33(1) (1990) 22–45.

    Google Scholar 

  19. T. Tsuchiya, “Global convergence of the affine scaling methods for degenerate linear programming problems,”Mathematical Programming (Series B) 52 (1991) 377–404, this issue.

    Google Scholar 

  20. C. Witzgall, P. Boggs and P. Domich, “On center trajectories and their relatives in linear programming,” J.C. Lagarias and M.J. Todd, eds.,Mathematical Developments Arising from Linear Programming, AMS Series Contemporary Mathematics (AMS, Providence, RI, 1990) pp. 161–187.

    Google Scholar 

  21. Y. Zhang, R.A. Tapia and J.E. Dennis, “On the superlinear and quadratic convergence of primal—dual interior point linear programming algorithms,” Technical Report TR90-6, Rice University (Houston, TX, 1990).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory for Operations Research, Faculty of Organizational Sciences, Belgrade University, Belgrade, Yugoslavia

    Vera V. Kovacevic-Vujcic

Authors
  1. Vera V. Kovacevic-Vujcic
    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

Kovacevic-Vujcic, V.V. Improving the rate of convergence of interior point methods for linear programming. Mathematical Programming 52, 467–479 (1991). https://doi.org/10.1007/BF01582901

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01582901

Key words

Search

Navigation