Skip to main content
SpringerLink
Log in

Automatic identification of embedded network rows in large-scale optimization models

  • Published:
Mathematical Programming Submit manuscript

Abstract

The solution of a large-scale linear, integer, or mixed integer programming problem is often facilitated by the exploitation of special structure in the model. This paper presents heuristic algorithms for identifying embedded network rows within the coefficient matrix of such models. The problem of identifying a maximum-size embedded pure network is shown to be among the class of NP-hard problems. The polynomially-bounded, efficient algorithms presented here do not guarantee network sets of maximum size. However, upper bounds on the size of the maximum network set are developed and used to show that our algorithms identify embedded networks of close to maximum size. Computational tests with large-scale, real-world models are presented.

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. R. Bixby and W. Cunningham, “Converting linear programs to network problems“,Mathematics of Operations Research 5 (1980) 321–357.

    MATH  MathSciNet  Google Scholar 

  2. A. Brearly, G. Mitra and H. Williams. Analysis of mathematical programming models prior to applying the simplex algorithm”,Mathematical Programming 8 (1975) 54–83.

    Article  MathSciNet  Google Scholar 

  3. G. Brown and D. Thomen, “Automatic identification of generalized upper bounds in large-scale optimization models”Management Science 26 (1980) 1166–1184.

    MATH  MathSciNet  Google Scholar 

  4. G. Brown and R. McBride, “Extracting embedded generalized network problems from general LP problems”, paper presented at ORSA/TIMS (Houston, TX, September 1981).

  5. G. Brown and W. Wright, “Automatic identification of embedded structure in large-scale optimization models”, in: H. Greenberg and J. Maybee, eds,Computer-assisted analysis and model simplification (Academic Press, New York, 1981) (proceedings from Boulder, CO, March 28, 1980) pp. 369–388.

    Google Scholar 

  6. G. Dantzig and R. Van Slyke, “Generalized upper bounding techniques”,Journal of Computer and System Sciences 1 (1967) 213–226.

    MATH  MathSciNet  Google Scholar 

  7. M. Garey and D. Johnson,Computers and intractability: A guide to the theory of NP-completeness (W.H. Freeman and Company, San Francisco, 1979).

    Google Scholar 

  8. G. Graves and R. McBride, “The factorization approach to large-scale linear programming”,Mathematical Programming 10 (1976) 91–110.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Graves and T. Van Roy, “Decomposition for large-scale linear and mixed integer linear programming”, UCLA Technical Report (Los Angeles, CA, November 1979).

  10. H. Greenberg and D. Rarick, “Determining GUB sets via an invert agenda algorithm”Mathematical Programming 7 (1974) 240–244.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. McBride. “Linear programming with linked lists and automatic guberization”, Working Paper No. 8175, University of Southern California, School of Business (Los Angeles, CA, July 1975).

    Google Scholar 

  12. R. McBride, “Solving linear programs with network factorization”, Working Paper, University of Southern California, School of Business (Los Angeles, CA, September, 1982).

    Google Scholar 

  13. V. Klee, “Combinatorial optimization: What is the state of the art”.Mathematics of Operations Research 5 (1980) 1–26.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Musalem, “Converting linear models to network models” Ph.D. Dissertation, UCLA (Los Angeles, CA, January 1980).

    Google Scholar 

  15. L. Schrage, “Some comments on hidden structure in linear programs”, in: H. Greenberg and J. Maybee, eds.,Computer-assisted analysis and model simplification (Academic Press, New York, 1981) (proceedings from Boulder, CO, March 28, 1980) pp. 389–395.

    Google Scholar 

  16. S. Senju and Y. Toyoda, “An approach to linear programming with 0–1 variables”.Management Science 15 (1968) B196-B207.

    Google Scholar 

  17. D. Thomen, “Automatic factorization of generalized upper bounds in large-scale optimization problems”, M.S. Thesis, Naval Postgraduate School (Monterey, CA September, 1979).

    Google Scholar 

  18. W. Wright, “Automatic identification of network rows in large-scale optimization models,” M.S. Thesis, Naval Postgraduate School (Monterey, CA, September 1980).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Naval Postgraduate School, 93940, Monterey, CA, USA

    Gerald G. Brown & William G. Wright

Authors
  1. Gerald G. Brown
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William G. Wright
    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

Brown, G.G., Wright, W.G. Automatic identification of embedded network rows in large-scale optimization models. Mathematical Programming 29, 41–56 (1984). https://doi.org/10.1007/BF02591728

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02591728

Key words

Search

Navigation