Skip to main content
SpringerLink
Log in

On some properties of o-degeneracy graphs

  • Part II: Specific Papers
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Degenerate optima in linear programming problems lead in a canonical way to so-called o-degeneracy graphs as subgraphs of degeneracy graphs induced by the set of optimal bases. Fundamental questions about the structure of o-degeneracy graphs suggest the closer inspection of some properties of these graphs, such as, for example, the connectivity and the complexity. Finally, some open questions are pointed out.

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. M. Agkül, A note on shadow prices in linear programming, J. Oper. Res. Soc. 35(1984)425.

    Google Scholar 

  2. D.C. Aucamp and D.I. Steinberg, The computation of shadow prices in linear programming, J. Oper. Res. Soc. 33(1982)557.

    Google Scholar 

  3. E.M.L. Beale, Cycling in the dual simplex algorithm, Naval Res. Log. Quarterly 2(1955)269.

    Google Scholar 

  4. R.G. Bland, New finite pivoting rules for the simplex method, Math. Oper. Res. 2(1977)103.

    Article  Google Scholar 

  5. A. Charnes, Optimality and degeneracy in linear programming, Econometrica 20(1952)160.

    Google Scholar 

  6. G.B. Dantzig, A. Orden and P. Wolfe, The generalized simplex method for minimizing a linear form under linear inequality restraints, Pacific J. Math. 5 (1955) 183.

    Google Scholar 

  7. J.R. Evans and N.R. Baker, Degeneracy and the (mis-)interpretation of sensitivity analysis in linear programming, Dec. Sci. 13(1982)348.

    Google Scholar 

  8. T. Gal, On the structure of the set bases of a degenerate point, J. Optim. Theory Appl. 45(1985)577.

    Article  Google Scholar 

  9. T. Gal, Shadow prices and sensitivity analysis in linear programming under degeneracy — State-of-the-art-survey, OR Spektrum 8(1986)59.

    Article  Google Scholar 

  10. T. Gal, Degeneracy problems in mathematical programming and degeneracy graphs, ORION 6(1991)3.

    Google Scholar 

  11. T. Gal, Degeneracy graphs: Theory and application — A survey, Ann. Oper. Res. 46 (1993), this volume.

  12. T. Gal and F. Geue, A new pivoting rule for solving various degeneracy problems, Oper. Res. Lett. 11(1992)23.

    Article  Google Scholar 

  13. T. Gal, H.-J. Kruse and P. Zörnig, Survey of solved and open problems in the degeneracy phenomenon, Math. Progr. B 42(1988)125.

    Article  Google Scholar 

  14. F. Geue, An improved N-tree method to determine all neighbors of a degenerate vertex, Ann. Oper. Res. 47 (1993), this volume.

  15. H.J. Greenberg, An analysis of degeneracy, Naval Res. Log. Quarterly 33(1986)635.

    Google Scholar 

  16. A.J. Hoffman, Cycling in the simplex algorithm, National Bureau of Standards Report 2974 (1953).

  17. G. Knolmayer, The effects of degeneracy on cost-coefficient ranges and an algorithm to resolve interpretation problems, Dec. Sci. 15(1984)14.

    Google Scholar 

  18. G. Knolmayer and M. Klöber, Heuristics for determining a minimal set of informative nodes in the optimum degeneracy graph, in:Operations Research Proceedings 1990, eds. W. Bühler et al. (Springer, 1992).

  19. H.-J. Kruse,Degeneracy Graphs and the Neighbourhood Problem (Springer, 1986).

  20. H.-J. Kruse, Über spezielle Teilgraphen von Entartungsgraphen, Discussion paper No. 121, Fachbereich Wirtschaftswissenschaften, FernUniversität, Hagen, Germany (1987).

    Google Scholar 

  21. G. Piehler, Some new aspects of the theory of o-degeneracy graphs.

  22. P. Zörnig,Degeneracy Graphs and Simplex Cycling (Springer, 1991).

  23. P. Zörnig, A general theory of degeneracy graphs, Ann. Oper. Res. (1993), this volume.

Download references

Author information

Authors and Affiliations

  1. Entscheidungsmodelle und Operations Research, Schering AG, Berlin, Germany

    Hermann-Josef Kruse

Authors
  1. Hermann-Josef Kruse
    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

Kruse, HJ. On some properties of o-degeneracy graphs. Ann Oper Res 46, 393–408 (1993). https://doi.org/10.1007/BF02023106

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02023106

Keywords

Search

Navigation