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Survey of solved and open problems in the degeneracy phenomenon

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References

  1. M. Akgül, “A note on shadow prices in linear programming,”Journal of ORS 35 (1984) 425–431.

    Google Scholar 

  2. D.C. Aucamp, D.I. Steinberg, “The computation of shadow prices in linear programming,”J. of ORS 33 (1982) 557–565.

    Google Scholar 

  3. D. Avis and V. Chvatal, “Notes on Bland's pivoting rule,”Mathematical Programming 8 (1978) 24–34.

    Google Scholar 

  4. E.M.L. Beale, “Cycling in the dual simplex algorithm,”Naval Research Logical Quarterly 2 (1955) 269–276.

    Google Scholar 

  5. R.G. Bland, “New finite pivoting rules for the simplex method,”Mathematics of Operations Research 2 (1977) 103–107.

    Google Scholar 

  6. A. Charnes “Optimality and degeneracy in linear programming,”Econometrica 20 (1952) 160–170.

    Google Scholar 

  7. G.B. Dantzig, A. Orden and P. Wolfe “The generalized simplex method for minimizing a linear form under linear inequalities,”Pacific Journal of Mathematics 5 (1955) 183–195.

    Google Scholar 

  8. A. Eilon and R. Flavell, “Note on many-sided shadow prices,”OMEGA 2 (1974) 821–823.

    Google Scholar 

  9. J.R. Evans and N.R. Baker “Degeneracy and the (mis-) interpretation of sensitivity analysis in linear programming,”Decision Sciences 13 (1982) 348–354.

    Google Scholar 

  10. T. Gal, “Determination of all neighbours of a degenerate extreme point in polytopes,” Discussion Paper No. 17B, FernUniversität Hagen, 1978.

  11. T. Gal,Postoptimal Analyses, Parametric Programming, and Related Topics (McGraw-Hill, New York 1979).

    Google Scholar 

  12. T. Gal, “On the structure of the set bases of a degenerate point,”JOTA 45 (1985) 577–589.

    Google Scholar 

  13. T. Gal, “Shadow prices and sensitivity analysis in linear programming under degeneracy-A state-of-the art survey,”OR Spektrum 8 (1986) 59–71.

    Google Scholar 

  14. T. Gal, “Degeneracy and redundancy in linear programming,” in preparation, Fernuniversität Hagen, 1987.

  15. T. Gal and H.-J. Kruse, “Ein Verfahren zur Lösung des Nachbarschaftsproblems,” Operations Research Proceedings 1984 (Springer Verlag 1985) pp. 447–454.

  16. T. Gal and H.-J. Kruse, “An improved method to solve the neighbourhood problem under degeneracy,” Disc. Paper, FernUniversität Hagen, 1986.

  17. T. Gal, H.-J. Kruse and P. Zörnig, “New developments in the area of degeneracy graphs,” Presented at the Joint National TIMS/ORSA Meeting, Los Angeles, April 1985.

  18. S.I. Gass, “Comments on the possibility of cycling with the simplex method,”Operations Research 27 (1979) 848–852.

    Google Scholar 

  19. H.-J. Greenberg, “An analysis of degeneracy,”Naval Research Logical Quarterly 33 (1986) 635–655.

    Google Scholar 

  20. B. Grünbaum,Convex Polytopes (J. Wiley, London-New York 1969).

    Google Scholar 

  21. A.J. Hoffman, “Cycling in the simplex algorithm,” National Bureau of Standards Rep. No. 2974, 1953

  22. M.H. Karwan, V. Lofti, J. Telgen and S. Zionts, “Redundancy in mathematical programming-A state-of-the-art survey,” Springer Verlag, Berlin-Heidelberg-New York-Tokyo 1983.

    Google Scholar 

  23. G. Knolmayer, “How many sided are shadow prices at degenerate primal optima?“OMEGA 4 (1976) 493–494.

    Google Scholar 

  24. G. Knolmayer, “The effects of degeneracy on cost-coefficient ranges and an algorithm to resolve interpretation problems,”Decision Sciences 15 (1984) 14–21.

    Google Scholar 

  25. T.C.T. Kotiah and D.I. Steinberg, “Occurrences of cycling and other phenomena arising in a class of linear programming models,”Communications of the Association for Computing Machinery 20 (1977) 107–112

    Google Scholar 

  26. T.C.T. Kotiah and D.I. Steinberg, “On the possibility of cycling with the simplex method,”Operations Research 26 (1978) 374–376.

    Google Scholar 

  27. H.-J. Kruse, “Degeneracy graphs and the neighbourhood problem,” Lecture notes in economics and mathematical systems No. 260 (Springer Verlag, Berlin-Heidelberg-New York-Tokyo 1986).

    Google Scholar 

  28. H.-J. Kruse, “Über spezielle Teilgraphen von Entartungsgraphen,” Discussion Paper No 121, FernUniversität Hagen, 1987.

  29. A. Majthay, “On degeneracy and cycling with the simplex method,” Discussion Paper No 41, Center for Econometrics and Decision Sciences, University of Florida 1981.

  30. G. Piehler, “Notes on determination of shadow prices and sensitivity analysis under degeneracy,” in preparation, Fernuniversität Hagen, 1987.

  31. J.E. Strum, “Note on two-sided shadow prices,”Journal of Accounting Research 7 (1969) 160–162.

    Google Scholar 

  32. J. Telgen, “A note on a linear programming problem that cycled,”COAL Newsletters 2 (1980) 8–11.

    Google Scholar 

  33. P. Wolfe, “A technique for resolving degeneracy in linear programming,”Journal of SIAM 11 (1963) 205–211.

    Google Scholar 

  34. P. Zörnig, “Strukturuntersuchungen an 2 ×n-Entartungsgraphen,” Discussion Paper No. 87, Fern-universität Hagen 1985.

  35. P. Zörnig, “On cycling of the simplex-algorithm in linear programming,” in preparation, Fern-universität Hagen, 1988.

Additional references

  1. M.L. Balinski, Th.M. Liebling and A.-E. Nobs, “On the average length of lexicographic paths,”Mathematical Programming 35 (1986) 362–364.

    Google Scholar 

  2. R.J. Caron, J.F. McDonald and C.M. Ponic, “A degenerate point strategy for the classification of linear constraints as redundant or necessary,” Windsor Mathematical Reports WMR 85–09, June 1987.

  3. N. Megiddo, “A note on degeneracy in linear programming,”Mathematical Programming 35 (1986) 365–367.

    Google Scholar 

  4. T. Gal, “Degeneracy graphs-Theory and application. a state-of-the-art survey,” Discussion Paper No. 126, Fernuniversität Hagen, April 1988.

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Partially supported by the DFG Grant No. 335/1-1-II A3.

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Gal, T., Kruse, HJ. & Zörnig, P. Survey of solved and open problems in the degeneracy phenomenon. Mathematical Programming 42, 125–133 (1988). https://doi.org/10.1007/BF01589397

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