Skip to main content
SpringerLink
Log in

A computational study of redundancy in randomly generated polytopes

Eine Rechenstudie der Redundanz zufällig erzeugter Polytope

  • Published:
Computing Aims and scope Submit manuscript

Abstract

In this paper we first survey a number of principles used to randomly generate linear programming problems. Three major classes of LP generators are identified and on that basis five generators for further study are defined. In a series of tests the average degree of redundancy of problems generated by these generators is determined. The evaluation of the results also develops an indicator for the expected degree of redundancy of a problem.

Zusammenfassung

Dieser Beitrag gibt einen Überblick über eine Reihe von Methoden, mit denen lineare Optimierungsprobleme zufällig erzeugt werden können. Drei Hauptgruppen von LP Generatoren werden identifiziert, und auf dieser Basis werden fünf Zufallsgeneratoren bestimmt. In einer Testreihe wird der durchschnittliche Redundanzgrad der durch diese Generatoren erzeugten Probleme festgestellt. Die Ergebnisse ermöglichen zudem, einen Indikator zur Bestimmung des erwarteten Redundanzgrades eines Problems zu bestimmen.

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. Abel, P.: On the choice of the pivot columns of the simplex method: Gradient criteria. Computing38, 13–21 (1987).

    Google Scholar 

  2. Avis, D., Chvátal, V.: Notes on Bland's pivoting rule. Mathematical Programming Study8, 24–34 (1978).

    Google Scholar 

  3. Bixby, R. E., Wagner, D. K.: A note on detecting simple redundancies in linear systems. Operations Research Letters6, 15–17 (1987).

    Google Scholar 

  4. Borgwardt, K.-H.: Some distribution-independent results about the asymptotic order of the average number of pivot steps of the simplex method. Mathematics of Operations Research7, 441–462 (1982).

    Google Scholar 

  5. Borgwardt, K.-H., Damm, R., Donig, R., Joas, G.: Empirical studies on the average efficiency of simplex-variants under rotation-symmetry, Report 240, Inst. of Math, Univ. Augsburg, Germany (1990). ORSA Journal of Computing (to appear).

    Google Scholar 

  6. Brearly, A. L., Mitra, G., Williams, H. P.: Analysis of mathematical programming problems prior to applying the simplex algorithm. Mathematical Programming8, 53–83 (1975).

    Google Scholar 

  7. Cheng, M. C.: General criteria for redundant and nonredundant linear inequalities. J. Optimization Theory and Applications53, 37–42 (1987).

    Google Scholar 

  8. Dunham, J. R., Kelly, D. G., Tolle, J. W.: Some experimental results concerning the expected number of pivots for solving randomly generated linear programs. Technical Report 77-16, University of North Carolina at Chapel Hill, North Carolina, USA (1977).

    Google Scholar 

  9. Eiselt, H. A., Sandblom, C.-L.: External pivoting in the simplex algorithm. Statistica Neerlandica39, 327–341 (1985).

    Google Scholar 

  10. Eiselt, H. A., Sandblom, C.-L.: Experiments with external pivoting. Computers & Operations Research17, 325–332 (1990).

    Google Scholar 

  11. Eiselt, H. A., Sandblom, C.-L., DeMarr, R.: Computational experience with external pivoting. Committee on Algorithms (COAL) Newsletter12, 16–20 (1985).

    Google Scholar 

  12. Fathi, Y., Murty, K. G.: Computational behavior of a feasible direction method for linear programming. European Journal of Operational Research40, 322–328 (1989).

    Google Scholar 

  13. Feller, W.: An introduction to probability theory and its applications, Vol. 2, 2nd ed. New York: John Wiley & Sons 1971.

    Google Scholar 

  14. Goldfarb, D., Reid, J. K.: A practicable steepest-edge simplex algorithm. Mathematical Programing12, 361–371 (1977).

    Google Scholar 

  15. Karwan, M. H., Lotfi, V., Telgen, J., Zionts, S.: Redundancy in mathematical programming. Berlin: Springer 1983 (Lecture Notes in Economics and Mathematical Systems,206).

    Google Scholar 

  16. Kuhn, H. W., Quandt, R. E.: An experimental study of the simplex method. In: Proceedings of Symposia in Applied Mathematics, vol, XV, Experimental Arthmetic, High Speed Computing and Mathematics, 107–124; American Mathematical Society, Providence, Rhode Island, US (1963).

    Google Scholar 

  17. Liebling, T. M.: On the number of iterations of the simplex method. Operations Research Verfahren17, 248–264 (1972).

    Google Scholar 

  18. Lustig, I. J.: An analysis of an available set of linear programming test problems. Computers & Operations Research16, 173–184 (1989).

    Google Scholar 

  19. May, J. H., Smith, R. L.: Random polytopes: Their definition, generation, and aggregate properties. Mathematical programming.24, 39–54 (1982).

    Google Scholar 

  20. Mitra, G., Tamiz, M., Yadegar, J.: Experimental investigation of an interior search method within a simplex framework, Technical Report TR/06/86, revised February 1987, Department of Mathematics and Statistics, Brunel University, Great Britain (1987).

    Google Scholar 

  21. Muller, R. K., Cooper, L.: A comparison of the primal simplex and primal-dual algorithms for linear programming. Communications of the ACM8, 682–686 (1965).

    Google Scholar 

  22. Murty, K. G.: The gravitational method for linear programming. Opsearch23, 206–214 (1986).

    Google Scholar 

  23. Murty, K. G., Fathi, Y.: A feasible direction method for linear programming. Operations Research Letters3, 121–127 (1984).

    Google Scholar 

  24. Punnen, A. P.: Minimax optimization. Ph. D. Thesis, Department of Mathematics, Indian Institute of Technology, Kanpur, India (1988).

    Google Scholar 

  25. Ravindran, A.: A comparison of the primal simplex and complementary pivot methods for linear programming. Naval Research Logistics Quarterly20, 95–100 (1973).

    Google Scholar 

  26. Schmidt, B. K., Mattheiss T. H.: The probability that a random polytope is bounded. Mathematics of Operations Research2, 292–296 (1977).

    Google Scholar 

  27. Shamir, R.: The efficiency of the simplex method: A survey. Management Science33, 301–334 (1987).

    Google Scholar 

  28. Sherali, H. D., Soyster, A. L., Baines, S. G.: Nonadjacent extreme point methods for solving linear programs. Naval Research Logistics Quarterly30, 145–161 (1983).

    Google Scholar 

  29. Telgen, J.: Redundancy and linear programs. Mathematical Centre Tracts #137, Amsterdam, 1981.

  30. Van Dam, W. B., Telgen, J.: Some computational experiments with a primal-dual surrogate simplex algorithm. Report 7828/0. Erasmus University, Rotterdam, The Netherlands 1978.

    Google Scholar 

  31. Van Dam, W. B., Frenk, J. B. G., Telgen, J.: Randomly generated polytopes for testing mathematical programming algorithms. Mathematical Programming26, 172–181 (1983).

    Google Scholar 

  32. Wolfe, P., Cutler, L.: Experiments in linear programming. In: Graves, R. L., Wolfe, P. (eds.) Advances in mathematical programming, 177–200. New York: McGraw-Hill 1963.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of New Brunswick, Fredericton, Canada

    H. A. Eiselt

  2. Technical University of Nova Scotia, Halifax, Canada

    C. L. Sandblom

Authors
  1. H. A. Eiselt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. L. Sandblom
    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

Eiselt, H.A., Sandblom, C.L. A computational study of redundancy in randomly generated polytopes. Computing 49, 315–327 (1993). https://doi.org/10.1007/BF02248692

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02248692

AMS Subject Classifications

Key words

Search

Navigation