Abstract
Based on the concept of degeneracy graphs, theoretical and algorithmic aspects of the neighborhood-problem are dealt with. It is shown that any subgraph of a positive degeneracy graph which is induced by a set of nodes feasible with respect to an arbitrary lexicographic pivot selection will supply sufficient information. A special lexicographic pivot selection strategy is presented which leads to an improved version of the N-tree method. The increase in efficiency is illustrated by test results.
Similar content being viewed by others
References
P. Armand, Bounds on the number of vertices of perturbed polyhedra, Ann. Oper. Res. 47 (1993), this volume.
P. Armand, Combinatorial behaviour of perturbed polyhedra in linear programming, C. R. Académie des Sciences, to appear.
P. Armand and C. Malivert, Determination of the efficient set in multiobjective linear programming, J. Opt. Theory Apl. 70 (1991) 467–489.
M.L. Balinski, An algorithm for finding all vertices of convex polyhedral sets, SIAM J. Appl Math. 9 (1961) 72–88.
E.M.L. Beale, Cycling in the dual simplex algorithm, Naval Res. Log. Quarterly 2 (1955) 269–276.
R.G. Bland, New finite pivoting rules for the simplex method, Math. Oper. Res. 2 (1977) 103–107.
A. Brøndsted,An Introduction to Convex Polytopes (Springer, New York/Heidelberg/Berlin, 1983).
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 inequality restraints, Pacific J. Math. 5 (1955) 183–195.
M.E. Dyer and L.G. Proll, An algorithm for determining all extreme points of a convex polytope,. Math. Prog. 12 (1977) 81–96.
M.E. Dyer and L.G. Proll, An improved vertex enumeration algorithm, Eur. J. Oper. Res. 9 (1982) 359–368.
T. Gal, Determination of all neighbors of a degenerate extreme point in polytopes, Discussion paper Nr. 17b, Department of Economics, FernUniversität-Gesamthochschule, Hagen, Germany (1978).
T. Gal, On the structure of the set bases of a degenerate point, J. Optim. Theory Appl. 45 (1985) 577–589.
T. Gal, Weakly redundant constraints and their impact on postoptimal analyses in LP, Eur. J. Oper. Res. 60 (1992) 315–326.
T. Gal, Degeneracy graphs: Theory and application — An updated survey, Ann. Oper. Res. 46 (1993), this volume.
T. Gal, Determining the set of all efficient solutions of an LVMP under degeneracy,Proc. 10th Int. Conf. on MCDM, Taipeh (Taiwan) (July 1992), to appear with Springer.
T. Gal and F. Geue, A new pivoting rule for solving various degeneracy problems, Oper. Res. Lett. 11 (1992) 23–32.
F. Geue, Eine neue Pivotauswahlregel und die durch sie induzierten Teilgraphen des positiven Entartungsgraphen, Discussion paper Nr. 141, Department of Economics, FernUniversität — Gesamthochschule, Hagen, Germany (1989).
F. Geue, Eckenabsuchende Verfahren unter Entartung: Theorie, Algorithmen und Vergleichstests, Dissertation, Fernuniversität — Gesamthochschule, Hagen, Germany (1992).
M.C. Golumbic,Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980).
M. Gondran and M. Minoux,Graphes et Algorithmes (Editions Eyrolles, Paris, 1979).
G. Hadley,Linear Programming (Addison-Wesley, Reading, 1974).
A.J. Hoffmann, Cycling in the simplex algorithm, National Bureau of Standards Report 2974.
M.H. Karwan, V. Lotfi, J. Telgen and St. Zionts,Redundancy in Mathematical Programming (Springer, Berlin/Heidelberg/New York/Tokyo, 1983).
H.-J. Kruse,Degeneracy Graphs and the Neighbourhood Problem, Lecture Notes in Economics and Mathematical Systems 260 (Springer, Berlin/Heidelberg/New York/Tokyo, 1986).
K.T. Marshall and J.W. Suurballe, A note on cycling in the simplex method, Naval Res. Log. Quarterly 16 (1969) 121–137.
T.H. Mattheiss, An algorithm for determining irrelevant constraints and all vertices in systems of linear inequalities, Oper. Res. 21 (1973) 247–260.
B.K. Schmidt and T.H. Mattheiss, The probability that a random polytope is bounded, Math. Oper. Res. 2 (1977) 292–296.
S.W. Wallace, Pivoting rules and redundancy schemes in extreme point enumeration, BIT 25 (1985) 274–280.
P. Wolfe, A technique for resolving degeneracy in linear programming, SIAM J. Appl. Math. 11 (1963) 305–311.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Geue, F. An improved N-tree algorithm for the enumeration of all neighbors of a degenerate vertex. Ann Oper Res 46, 361–391 (1993). https://doi.org/10.1007/BF02023105
Issue Date:
DOI: https://doi.org/10.1007/BF02023105