Encyclopedia of Operations Research and Management Science

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

Farkas' Lemma

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.

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© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.Robert H. Smith School of BusinessUniversity of MarylandCollege PartUSA
  2. 2.School of Information Technology & EngineeringGeorge Mason UniversityFairfaxUSA