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Degeneracy graphs: theory and application an updated survey

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Abstract

A survey of various aspects of the theory and application of degeneracy graphs (DGs for short) is given. The notion and some basic properties of DGs are introduced, cycling of the simplex method is discussed, the neighborhood problem is tackled, and the application of the so-called optimum DGs to particular problems which are connected with optimal degenerate solutions of a linear programming problem is presented. The impact of weakly redundant constraints on various postoptimal analyses under degeneracy is briefly described.

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References

  1. M. Akgül, A note on shadow prices in linear programminig, J. Oper. Res. Soc. 35 (1984) 425–431.

    Google Scholar 

  2. W. Altherr, An algorithm for enumerating all vertices of a convex polyhedron, Computing 15 (1975) 181–193.

    Article  Google Scholar 

  3. M. Altman, Optimum simplex methods and degeneracy in linear programming, Bull. de l'Acad. Polon. des Sci. 12 (1964) 217–225.

    Google Scholar 

  4. P. Armand, Programmation linéaire multicritère et problèmes de la dégénérescence, Thèse de Doctorat de l'Université de Limoges (1991).

  5. P. Armand, Bounds for the number of vertices of perturbed polyhedra, Ann. Oper. Res. (1993), this volume.

  6. D.C. Aucamp and D.I. Steinberg, On the nonequivalence of shadow prices and dual variables, Technical Report WUCS-79-11, Dept. of Computer Sciences, Washington University, St. Louis, MS (1979).

    Google Scholar 

  7. D.C. Aucamp and D.I. Steinberg, The computation of shadow prices in linear programming, J. Oper. Res. 33 (1982) 557–565.

    Google Scholar 

  8. D. Avis and V. Chvátal, Notes on Bland's pivoting rule, Math. Progr. Study 8 (1978) 24–34.

    Google Scholar 

  9. A.G. Azpeitia and D.J. Dickinson, A decision rule in the simplex method that avoids cycling, Numer. Mathematik 6 (1964) 329–331.

    Article  Google Scholar 

  10. M.L. Balinski, An algorithm for finding all vertices of convex polyhedral sets, J. Soc. Ind. Appl. Math. 9 (1961) 72–88.

    Article  Google Scholar 

  11. M.L. Balinski, Th.M. Liebling and A.-E. Nobs, On the average length of lexicographic paths, Math. Progr. 35 (1986) 362–364.

    Article  Google Scholar 

  12. B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer,Non-Linear Parametric Optimization (Akademie Verlag, 1982).

  13. R.S. Barr, F. Glover and D. Klingman, The alternating basis algorithm for assignment problems, Math. Progr. 13 (1977) 1–13.

    Article  Google Scholar 

  14. R.S. Barr, F. Glover and D. Klingman, The generalized alternating path algorithm for transportation problems, Eur. J. Oper. Res. 2 (1978) 137–144.

    Article  Google Scholar 

  15. E.M.L. Beale, Cycling in the dual simplex algorithm, Naval Res. Log. Quart. 2 (1955) 269–276.

    Article  Google Scholar 

  16. M. Bénichou, J.M. Gauthier, G. Hentges and G. Ribière, The efficient solution of large-scale linear programming problems — some algorithmic techniques and computational results, Math. Progr. 13 (1977) 280–322.

    Article  Google Scholar 

  17. R.G. Bland, New finite pivoting rules for the simplex method, Math. Oper. Res. 2 (1977) 103–107.

    Article  Google Scholar 

  18. N. Cameron, Stationarity in the simplex method, J. Austral. Math. Soc., Series A 43 (1987) 137–142.

    Article  Google Scholar 

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

    Article  Google Scholar 

  20. M.C. Cheng, New criteria for the simplex algorithm, Math. Progr. 19 (1980) 230–236.

    Article  Google Scholar 

  21. V. Chvátal,Linear Programming (Freeman, 1983).

  22. M. Cirina, Remarks on a recent simplex pivoting rule, in:Methods of Operations Research 55: Proc. 9th Symp. on Operations Research, Königsstein/Ts, ed. W. Domschke et al. (1985) pp. 187–199.

  23. W.H. Cunningham, Theoretical properties of the network simplex method, Math. Oper. Res. 4 (1979) 196–208.

    Article  Google Scholar 

  24. W.H. Cunningham and J.G. Klincewicz, On cycling in the network simplex method, Math. Progr. 26 (1983) 182–189.

    Article  Google Scholar 

  25. G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, 1963).

  26. G.B. Dantzig, Making a progress during a stall in the simplex method, Lin. Alg. Appl. 114/115 (1989) 81–96.

    Google Scholar 

  27. 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) 183–195.

    Google Scholar 

  28. M.E. Dyer and L.G. Proll, An algorithm for determining all extreme points of a convex polytope, Math. Progr. 12 (1977) 81–96.

    Article  Google Scholar 

  29. M.E. Dyer and L.G. Proll, An improved vertex enumeration algorithm, Eur. J. Oper. Res. 9 (1982) 359–368.

    Article  Google Scholar 

  30. J.G. Ecker and M. Kupferschmied,Introduction to Operations Research (Wiley, 1988).

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

    Article  Google Scholar 

  32. H.A. Eiselt and C.-L. Sandblom, External pivoting in the simplex algorithm, Statistica Neerlandica 39 (1985) 327–341.

    Article  Google Scholar 

  33. J.R. Evans and N.R. Baker, Degeneracy and the (mis)interpretation of sensitivity analysis in linear programming, Dec. Sci. 13 (1982) 348–354.

    Article  Google Scholar 

  34. R. Fletcher, Degeneracy in the presence of round-off errors, Techn. Rep. NA/89, Dept. of Math. Sci., University of Dundee (1989).

  35. B. Fleischmann, Duale und primale Schnittebeneverfahren in der ganzzahligen linearen Optimierung, Dissertation, Universität Hamburg (1970).

  36. T. Gal, Determination of all neighbours of a degenerate extreme point in polytopes, Working Paper No. 17b, FB Wirtschaftswissenschaft, FernUniversität Hagen (1978).

  37. T. Gal,Postoptimal Analyses, Parametric Programmiing, and Related Topics (McGraw-Hill, 1979).

  38. T. Gal, On the structure of the set bases of a degenerate point, J. Optim. Theory Appl. 45 (1985) 577–589.

    Article  Google Scholar 

  39. T. Gal, Shadow prices and sensitivity analysis in linear programming under degeneracy — State-of-the-art survey, OR Spektrum 8 (1986) 59–71.

    Article  Google Scholar 

  40. T. Gal, Weakly redundant constraints and their impact on postoptimal analyses, Eur. J. Oper. Res. (1992) 315–326.

  41. T. Gal, Degeneracy in mathematical programming and degeneracy graphs, ORiON 6 (1990) 3–36.

    Google Scholar 

  42. T. Gal and F. Geue, A new pivoting rule for solving various degeneracy problems, Oper. Res. Lett. 11 (1992) 23–32.

    Article  Google Scholar 

  43. T. Gal, H.-J. Kruse and P. Zörnig, Survey of solved and open problems in the degeneracy phenomenon, Math. Progr. 42 (1988) 125–133.

    Article  Google Scholar 

  44. S.I. Gass, Comments on the possibility of cycling with the simplex method, Oper. Res. 27 (1979) 848–852.

    Article  Google Scholar 

  45. S.I. Gass,Linear Programming: Methods and Applications, 3rd ed. (McGraw-Hill, 1985).

  46. S.I. Gass, private communication (1989).

  47. B.J. Gassner, Cycling in the transportation problem, Naval Res. Log. Quart. 11 (1964) 43–58.

    Article  Google Scholar 

  48. D.M. Gay, Electronic mail distribution of linear programming test problems, Math. Progr. Soc. COAL Newsletters (1985).

  49. F. Geue, Eine neue Pivot-Auswahlregel und die durch sie induzierten Teilgraphen des positiven Entartungsgraphen, Working Paper No. 141, FB Wirtschaftswissenschaft, FernUniversität Hagen (1989).

  50. F. Geue, Eckenabsuchende Verfahren unter Entartung: Theorie, Algorithmen und Vergleichstests (Vertex searching methods under degeneracy: Theory, algorithms and computational comparisons), Dissertation, FernUniversität Hagen (1993), to be published.

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

  52. P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, A practical anti-cycling procedure for linearly constrained optimization, Math. Progr. 45 (1989) 437–474.

    Article  Google Scholar 

  53. P.E. Gill, W. Murray and M.H. Wright,Numerical Linear Algebra and Optimization, Vol. 1 (Addison-Wesley, 1991).

  54. R. Götz, Verfahren zur Konstruktion von Beispielen zum Simplexzykeln und ihre Implementierung, Diplomarbeit, Fachbereich Wirtschaftswissenschaft, FernUniversität Hagen (1989).

  55. R. Götz, private communication (1991).

  56. H.J. Greenberg, An analysis of degeneracy, Naval Res. Log. Quart. 33 (1986) 635–655.

    Article  Google Scholar 

  57. B. Grünbaum,Convex Polytopes (Wiley, 1967).

  58. G. Hadley,Linear Programming (Addison-Wesley, 1975).

  59. P.M.J. Harris, Pivot selection methods of the DEVEX LP code, Math. Progr. 42 (1973) 135–145.

    Google Scholar 

  60. R.S. Hattersley and J. Wilson, A dual approach to primal degeneracy, Math. Progr. 72 (1988) 135–176.

    Article  Google Scholar 

  61. A.J. Hoffman, Cycling in the simplex algorithm, National Bureau of Standards Report No. 2974 (1953).

  62. M.H. Karwan, V. Lotfi, J. Telgen and S. Zionts,Redundancy in Mathematical Programming: A State-of-the-Art Survey, Lecture Notes in Economics and Mathematical Systems No. 206 (Springer, 1983).

  63. V. Klee and P. Kleinschmidt, Geometry of the Gass-Saaty parametric cost LP algorithm, Discr. Comp. Geom. 5 (1990) 13–26.

    Article  Google Scholar 

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

    Article  Google Scholar 

  65. G. Knolmayer,Programmierungsmodelle für die Produktionsprogrammplanung (Birkhäuser, 1980).

  66. G. Knolmayer, The effect of degeneracy on cost-coefficient ranges and an algorithm to resolve interpretation problems, Dec. Sci. 15 (1984) 14–21.

    Article  Google Scholar 

  67. T.C.T. Kotiah and D.I. Steinberg, Occurrences of cycling and other phenomena arising in a class of linear programming models, Commun. ACM 20 (1977) 107–112.

    Article  Google Scholar 

  68. T.C.T. Kotiah and D.I. Steinberg, On the possibility of cycling with the simplex method, Oper. Res. 26 (1978) 374–376.

    Article  Google Scholar 

  69. H.-J. Kruse,Degeneracy Graphs and the Neighbourhood Problem, Lecture Notes in Economics and Mathematical Systems No. 260 (Springer, 1986).

  70. H.-J. Kruse, Über spezielle Teilgraphen von Entartungsgraphen, Working Paper No. 121, Fachbereich Wirtschaftswissenschaft, FernUniversität Hagen (1987).

  71. H.-J. Kruse, On some properties of o-degeneracy graphs, Ann. Oper. Res. (1993), this volume.

  72. H.W. Kuhn, Loc. Cit. [51].

  73. T.L. Magnanti and J.B. Orlin, Parametric linear programming and anti-cycling pivoting rules, Math. Progr. 41 (1988) 317–325.

    Article  Google Scholar 

  74. A. Majthay, On degeneracy and cycling with the simplex method, Disc. Paper No. 41, Center for Econometrics and Decision Sciences, University of Florida (1981).

  75. K.T. Marshall and J.W. Suurballe, A note on cycling in the simplex method, Naval Res. Log. Quart. 16 (1969) 121–137.

    Google Scholar 

  76. T.H. Mattheiss, An algorithm for determining irrelevant constraints and all vertices in systems of linear inequalities, Oper. Res. 21 (1973) 247–260.

    Article  Google Scholar 

  77. T.H. Mattheiss and D.S. Rubin, A survey and comparison of methods for finding all vertices of convex polyhedral sets, Math. Oper. Res. 5 (1980) 167–185.

    Article  Google Scholar 

  78. N. Megiddo, A note on degeneracy in linear programming, Math. Progr. 35 (1986) 365–367.

    Article  Google Scholar 

  79. P.G. McKeown, Some computational results of using the Ahrens-Finke method for handling degeneracy in fixed charge transportation problems, Math. Progr. 15 (1978) 355–359.

    Article  Google Scholar 

  80. J. Mrazek, Zu speziellen Eigenschaften von Knoten eines positiven Entartungsgraphen, Working Paper No. 180, FB Wirtschaftswissenschaft, FernUniversität Hagen (1991).

  81. P. Nash,Algebraic Fundamentals of Linear Programming, Lecture Notes in Economics and Mathematical Systems No. 259 (Springer, 1985) pp. 37–52.

  82. M. Niggemeier, Entartungserscheinungen bei ganzzahligen Optimierungsproblemen — Eine Übersicht, Diplomarbeit, Fachbereich Wirtschaftswissenschaft, FernUniversität Hagen (1989).

  83. M. Niggemeier, Degeneracy in integer linear programming problems — A survey, Ann. Oper. Res. (1993), this volume.

  84. H.-J. Ollmert, Zur Theorie des Simplexverfahrens im ausgearteten Fall, Diplomarbeit, Fachbereich Wirtschaftswissenschaften, Universität Saarbrücken (1965).

  85. H.-J. Ollmert, Kreisende lineare Programme, in:Operations Research Verfahren VI, eds. R. Henn, H.P. Künzi and H. Schubert (Anton Hain, 1969).

  86. P.-Q. Pan, Practical finite pivoting rules for the simplex method, OR Spektrum 12 (1990) 219–225.

    Article  Google Scholar 

  87. C. van de Panne,Methods for Linear and Quadratic Programming (North-Holland, 1975).

  88. G. Piehler, Optimalbasenansatz zur Sensitivitätsanalyse bei linearer Programmierung unter Entartung, Working Paper No. 130, Fachbereich Wirtschaftswissenschaft, FernUniversität Hagen (1988).

  89. G. Piehler, Postoptimale Analyse für entartete lineare Optimierungsprobleme,Operations Research Proc. 1989, eds. K.-P. Kistner et al. (Springer, 1990) pp. 521–528.

  90. G. Piehler and H.-J. Kruse, Optimumgraphen zur Analyse linearer Optimierungsprobleme unter Entartung,Operations Research Proc. 1988, eds. D. Pressmar et al. (Springer, 1989) pp. 205–210.

  91. L.G. Proll, Goal aggregation via shadow prices — some counterexamples, Large Scale Syst. 12 (1987) 83–85.

    Google Scholar 

  92. C. Roos, The umbrella approach to linear programming, Proc. Math. Opt. 16 (1984) 67–79.

    Google Scholar 

  93. D.M. Ryan and M.R. Osborne, On the solution of highly degenerate linear programmes, Math. Progr. 41 (1988) 385–392.

    Article  Google Scholar 

  94. H.D. Sherali and S.E. Dickey, An extreme-point-ranking algorithm for the extreme point mathematical programming problem, Comp. Oper. Res. 13 (1986) 465–475.

    Article  Google Scholar 

  95. D. Solow,Linear Programming. An Introduction to Finite Improvement Algorithms (Elsevier, 1984).

  96. J.E. Strum, Note on two-sided shadow prices, J. Account. Res. 7 (1969) 160–162.

    Article  Google Scholar 

  97. J. Telgen, A note on a linear programming problem that cycled, COAL Newsletter 2 (1980) 8–11.

    Google Scholar 

  98. G.L. Thompson, F.M. Tonge and S. Zionts, Techniques for removing nonbinding constraints and extraneous variables from linear programming problems, Manag. Sci. 23 (1966) 588–608.

    Article  Google Scholar 

  99. M. Todd, Probabilistic models for linear programming, Math. Oper. Res. 16 (1991) 671–693.

    Article  Google Scholar 

  100. S.W. Wallace, Pivoting rules and redundancy schemes in extreme point enumeration, BIT 25 (1985) 274–280.

    Article  Google Scholar 

  101. A.C. Williams, Marginal values in linear programming, J. Soc. Ind. Appl. Math. 11 (1963) 82–94.

    Article  Google Scholar 

  102. P. Wolfe, A technique for resolving degeneracy in linear programming, J. Soc. Ind. Appl. Math. 11 (1963) 305–311.

    Article  Google Scholar 

  103. F.K. Wright, Measuring asset services: a linear programming approach, J. Account. Res. 6 (1968) 222–236.

    Article  Google Scholar 

  104. H.-J. Zimmermann and T. Gal, Redundanz und ihre Bedeutung für betriebliche Optimierungsentscheidungen, Z. für Betriebswirtschaft 45 (1975) 221–236.

    Google Scholar 

  105. P. Zörnig,Degeneracy Graphs and Simplex Cycling, Lecture Notes in Economics and Mathematical Systems No. 357 (Springer, 1991).

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

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    Tomas Gal

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Gal, T. Degeneracy graphs: theory and application an updated survey. Ann Oper Res 46, 81–105 (1993). https://doi.org/10.1007/BF02096258

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