Access this article
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
M. Akgül, “A note on shadow prices in linear programming,”Journal of ORS 35 (1984) 425–431.
D.C. Aucamp, D.I. Steinberg, “The computation of shadow prices in linear programming,”J. of ORS 33 (1982) 557–565.
D. Avis and V. Chvatal, “Notes on Bland's pivoting rule,”Mathematical Programming 8 (1978) 24–34.
E.M.L. Beale, “Cycling in the dual simplex algorithm,”Naval Research Logical Quarterly 2 (1955) 269–276.
R.G. Bland, “New finite pivoting rules for the simplex method,”Mathematics of Operations Research 2 (1977) 103–107.
A. Charnes “Optimality and degeneracy in linear programming,”Econometrica 20 (1952) 160–170.
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.
A. Eilon and R. Flavell, “Note on many-sided shadow prices,”OMEGA 2 (1974) 821–823.
J.R. Evans and N.R. Baker “Degeneracy and the (mis-) interpretation of sensitivity analysis in linear programming,”Decision Sciences 13 (1982) 348–354.
T. Gal, “Determination of all neighbours of a degenerate extreme point in polytopes,” Discussion Paper No. 17B, FernUniversität Hagen, 1978.
T. Gal,Postoptimal Analyses, Parametric Programming, and Related Topics (McGraw-Hill, New York 1979).
T. Gal, “On the structure of the set bases of a degenerate point,”JOTA 45 (1985) 577–589.
T. Gal, “Shadow prices and sensitivity analysis in linear programming under degeneracy-A state-of-the art survey,”OR Spektrum 8 (1986) 59–71.
T. Gal, “Degeneracy and redundancy in linear programming,” in preparation, Fernuniversität Hagen, 1987.
T. Gal and H.-J. Kruse, “Ein Verfahren zur Lösung des Nachbarschaftsproblems,” Operations Research Proceedings 1984 (Springer Verlag 1985) pp. 447–454.
T. Gal and H.-J. Kruse, “An improved method to solve the neighbourhood problem under degeneracy,” Disc. Paper, FernUniversität Hagen, 1986.
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.
S.I. Gass, “Comments on the possibility of cycling with the simplex method,”Operations Research 27 (1979) 848–852.
H.-J. Greenberg, “An analysis of degeneracy,”Naval Research Logical Quarterly 33 (1986) 635–655.
B. Grünbaum,Convex Polytopes (J. Wiley, London-New York 1969).
A.J. Hoffman, “Cycling in the simplex algorithm,” National Bureau of Standards Rep. No. 2974, 1953
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.
G. Knolmayer, “How many sided are shadow prices at degenerate primal optima?“OMEGA 4 (1976) 493–494.
G. Knolmayer, “The effects of degeneracy on cost-coefficient ranges and an algorithm to resolve interpretation problems,”Decision Sciences 15 (1984) 14–21.
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
T.C.T. Kotiah and D.I. Steinberg, “On the possibility of cycling with the simplex method,”Operations Research 26 (1978) 374–376.
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).
H.-J. Kruse, “Über spezielle Teilgraphen von Entartungsgraphen,” Discussion Paper No 121, FernUniversität Hagen, 1987.
A. Majthay, “On degeneracy and cycling with the simplex method,” Discussion Paper No 41, Center for Econometrics and Decision Sciences, University of Florida 1981.
G. Piehler, “Notes on determination of shadow prices and sensitivity analysis under degeneracy,” in preparation, Fernuniversität Hagen, 1987.
J.E. Strum, “Note on two-sided shadow prices,”Journal of Accounting Research 7 (1969) 160–162.
J. Telgen, “A note on a linear programming problem that cycled,”COAL Newsletters 2 (1980) 8–11.
P. Wolfe, “A technique for resolving degeneracy in linear programming,”Journal of SIAM 11 (1963) 205–211.
P. Zörnig, “Strukturuntersuchungen an 2 ×n-Entartungsgraphen,” Discussion Paper No. 87, Fern-universität Hagen 1985.
P. Zörnig, “On cycling of the simplex-algorithm in linear programming,” in preparation, Fern-universität Hagen, 1988.
Additional references
M.L. Balinski, Th.M. Liebling and A.-E. Nobs, “On the average length of lexicographic paths,”Mathematical Programming 35 (1986) 362–364.
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.
N. Megiddo, “A note on degeneracy in linear programming,”Mathematical Programming 35 (1986) 365–367.
T. Gal, “Degeneracy graphs-Theory and application. a state-of-the-art survey,” Discussion Paper No. 126, Fernuniversität Hagen, April 1988.
Author information
Authors and Affiliations
Additional information
Partially supported by the DFG Grant No. 335/1-1-II A3.
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01589397