# The Concept of Degeneracy

Chapter

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## Abstract

Consider the system of linear inequalities The condition (2.2) ensures that the solution set is nonempty and thus always a polyhedral set

$$Ax\le b,x\ge 0,{{(A\in I{{R}^{mxn}},b\in I{{R}^{m}})}^{1)}}$$

(2.1)

$$b\ge 0.$$

(2.2)

^{2)}$$X=\{x\in I{{R}^{n}}|Ax\le b,x\ge 0\}$$

(2.3)

^{3)}having at least one vertex. Denote by X the solution set of the canonical form of (2.1)^{4)}. Each basis B of the enlarged matrix P =(AüI) is also called a basis of (2.1) or of the canonical form. Tab. 2.1 shows the pivot tableau which is uniquely assigned to a basis B (and vice versa)^{5)}.## Keywords

Canonical Form Basic Solution Linear Inequality Neighbourhood Problem Basic Feasible Solution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- Cf. Appendix, Al.Google Scholar
- Cf. Appendix, A2.Google Scholar
- Cf. Appendix, A3.Google Scholar
- Cf. Appendix, A5 and A6.Google Scholar
- Cf. Appendix, All and Tab. Al.Google Scholar
- It should be noted that this definition of the degeneracy degree differs slightly from the corresponding definition in GAL [1978a, p. 8]. When using GAL’s definition a basic solution with the degeneracy degree
*a*according to the above definition has the degeneracy degree a + 1.Google Scholar - Cf. Appendix, A9.Google Scholar
- Cf. Appendix, A14.Google Scholar
- The question concerning the number of different bases assigned to a degenerate vertex (or complete basic feasible solution) is treated in detail in Chapter 4.Google Scholar
- Cf. Appendix, Fig. Al.Google Scholar
- Two of these cases are illustrated in Fig. 2.2.Google Scholar
- For the concept of (weak) redundancy see Appendix, A7. This concept is dealt with in greater detail by TELGEN [1979] and KARWAN et al. [1983, pp. 14–21].Google Scholar
- BRADLEY/BROWN/GRAVES [1983] for example are concerned with the problem of redundancy in large-scale problems. A general survey of large-scale problems is given by DANTZIG/DEMPSTER/KALLIO [1981].Google Scholar
- Causes and consequences of redundancy in optimization problems are summerized in KARWAN et al. [1983, pp. 1–6]. An economic interpretation of.redundancy is proposed by ZIMMERMANN/GAL [1975].Google Scholar
- A comprehensive survey of procedures for determining redundant restrictions is given by KARWAN et al. [ 1983 ]. Moreover, this book contains a comparison with respect to the efficiency of the procedures proposed by GAL [1981; 1983], MATTHEISS [1973; 1983], RUBIN [1983], TELGEN [1983], and ZIONTS/WALLENIUS [1983].Google Scholar
- Cf. the “classical” example of degeneracy presented by NELSON [1957].Google Scholar
- The corresponding system of linear inequalities can be seen from Tab. 3.1.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1986