Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

Degeneracy graphs

  • Tomas Gal
Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_224

INTRODUCTION

For a given linear-programming problem, primal degeneracy means that a basic feasible solution has at least one basic variable equal to zero. The problem is dual degenerate if a nonbasic variable has its reduced cost equal to zero (the condition for a multiple optimal solution to exist). Primal degeneracy may arise when there are some (weakly) redundant constraints (Karwan et al., 1983) or the structure of the corresponding convex polyhedral feasible set causes an extreme point to become overdetermined.

In nonlinear programming, such points are sometimes called singularities (Guddat et al., 1990). Here, constraint redundancy is equivalent to the failure of the linear independence constraint qualification of the binding constraint gradients, which, in general, leads to the nonuniqueness of optimal Lagrange multipliers (Fiacco and Liu, 1993).

We focus here on primal degeneracy in the linear case: it is associated with multiple optimal bases and it allows for basis cycling to...

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References

  1. [1]
    Charnes, A. (1952). “Optimality and degeneracy in linear programming.” Econometrica 20, 160–170.Google Scholar
  2. [2]
    Dantzig, G.B. (1963). Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey.Google Scholar
  3. [3]
    Fiacco, A.V. and Liu, J. (1993). “Degeneracy in NLP and the development of results motivated by its presence.” In: T. Gal (ed.): Degeneracy in Optimization Problems. Annals of OR, 46/47, 61–80. Google Scholar
  4. [4]
    Gal, T. (1985). “On the structure of the set bases of a degenerate point.” Jl. Optim. Theory Appl., 45, 577–589.Google Scholar
  5. [5]
    Gal, T. (1986). “Shadow prices and sensitivity analysis in LP under degeneracy — State-of-the-art survey.” OR Spektrum, 8, 59–71.Google Scholar
  6. [6]
    Gal, T. (1992). “Weakly redundant constraints and their impact on postoptimal analysis in LP.” Euro. Jl. OR, 60, 315–336.Google Scholar
  7. [7]
    Gal, T. (1993). “Selected bibliography on degeneracy.” In: T. Gal (ed.): Degeneracy in optimization problems. Annals of OR, 46/47, 1–7. Google Scholar
  8. [8]
    Gal, T. (1995). Postoptimal analyses, parametric programming, and related topics. W. de Gruyter, Berlin, New York.Google Scholar
  9. [9]
    Gal, T. (1997). “Linear programming 2: Degeneracy graphs.” In: T. Gal and H.J. Greenberg (eds.). Advances in Sensitivity analysis and parametric programming. Kluwer Acad. Publishers, Dordrecht.Google Scholar
  10. [10]
    Gal, T. and Geue, F. (1992). “A new pivoting rule for solving various degeneracy problems.” OR Letters, 11, 23–32.Google Scholar
  11. [11]
    Geue, F. (1993). “An improved N-tree algorithm for the enumeration of all neighbors of a degenerate vertex.” In: T. Gal (ed.). Degeneracy in optimization problems. Annals of OR, 46/47, 361–392. Google Scholar
  12. [12]
    Guddat, J.F., Guerra Vasquez, F., and Th. Jongen, H. (1991). Parametric Optimization: Singularities, Path Following and Jumps.” R.G. Teubner and J. Wiley, New York.Google Scholar
  13. [13]
    Karwan, M.H., Lotfi, F., Telgen, J., and Zionts, S. eds. (1983). “Redundancy in mathematical programming: A state-of-the-art survey.” Lecture Notes in Econ. and Math.. Systems 206, Springer Verlag, Berlin.Google Scholar
  14. [14]
    Kruse, H.-J. (1986). “Degeneracy graphs and the neighborhood problem.” Lecture Notes in Econ. and Math. Systems 260, Springer Verlag, Berlin.Google Scholar
  15. [15]
    Kruse, H.-J. (1993). “On some properties of σ -degeneracy graphs.” In: T. Gal (ed.). Degeneracy in optimization problems. Annals of OR, 46/47, 393–408. Google Scholar
  16. [16]
    Niggemeier, M. (1993). “Degeneracy in integer linear optimization problems: A selected bibliography.” In: T. Gal (ed.). Degeneracy in optimization problems. Annals of OR, 46/47, 195–202. Google Scholar
  17. [17]
    Zörnig, P. (1993). “A theory of degeneracy graphs.” In: T. Gal (ed.). Degeneracy in optimization problems. Annals of OR, 46/47, 541–556. Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Tomas Gal
    • 1
  1. 1.Fern UniversitätHagenGermany