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


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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© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Tomas Gal
    • 1
  1. 1.Fern UniversitätHagenGermany