Encyclopedia of Operations Research and Management Science

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

Farkas' Lemma

  • Saul I. Gass
  • Carl M. Harris
Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_330

Given a matrix Av and a column vector b,v one and only one of the following two alternatives holds. Either: (1) there exists a column vector xv ⩾ 0 with Axv = b,v or (2) there exists an unrestricted row vector yv for which yAv ⩾ 0 and ybv < 0. This lemma can be proved by defining appropriate primal and dual linear-programming problems and applying the duality theorem.  Gordan's theorem; Strong duality theorem;  Theorem of the alternatives.

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Saul I. Gass
    • 1
  • Carl M. Harris
    • 2
  1. 1.Robert H. Smith School of BusinessUniversity of MarylandCollege PartUSA
  2. 2.School of Information Technology & EngineeringGeorge Mason UniversityFairfaxUSA