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The Concept of Degeneracy

  • H.-J. Kruse
Chapter
  • 25 Downloads
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 260)

Abstract

Consider the system of linear inequalities
$$Ax\le b,x\ge 0,{{(A\in I{{R}^{mxn}},b\in I{{R}^{m}})}^{1)}}$$
(2.1)
$$b\ge 0.$$
(2.2)
The condition (2.2) ensures that the solution set2)
$$X=\{x\in I{{R}^{n}}|Ax\le b,x\ge 0\}$$
(2.3)
is nonempty and thus always a polyhedral set3) 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

  1. Cf. Appendix, Al.Google Scholar
  2. Cf. Appendix, A2.Google Scholar
  3. Cf. Appendix, A3.Google Scholar
  4. Cf. Appendix, A5 and A6.Google Scholar
  5. Cf. Appendix, All and Tab. Al.Google Scholar
  6. 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
  7. Cf. Appendix, A9.Google Scholar
  8. Cf. Appendix, A14.Google Scholar
  9. 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
  10. Cf. Appendix, Fig. Al.Google Scholar
  11. Two of these cases are illustrated in Fig. 2.2.Google Scholar
  12. 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
  13. 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
  14. 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
  15. 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
  16. Cf. the “classical” example of degeneracy presented by NELSON [1957].Google Scholar
  17. The corresponding system of linear inequalities can be seen from Tab. 3.1.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • H.-J. Kruse
    • 1
  1. 1.Fachbereich WirtschaftswissenschaftFernuniversität HagenHagen 1Germany

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