462774_751308825_776166850

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A SIMPLE ALGEBRAIC PROOF OF FARKAS'S LEMMA AND RELATED THEOREMS C. G. BROYDEN* A pprof is given of Farkas's lemma ba\eti on a neu theorem pertaining to orthogor/al matrices. it is claimed that this theorem i.; (lightly more general than Tucker's theorel 1. ~vhich concerns skew-symmetric matrice\ and which may itself be derived simply from t I e neu theorem. Farkas's lemma and other theorem\ of the alternative then follow trivial/ly from Tricker's theorem. I Kryorzlr: Orthogonal matrices: Cayley transforms: linear programming: duality ! 1 INTRODUCTION I Farkas', Lemma jq one of the principal foundations of the theory of linear inequalitief. It may be stated thus: Theorem 1.1 (Farkac', lemma). Let A E R"'"" arzd b E R"' both Be urbitmry. Then either (a) 3x > 0 5uch that As = b. or (h) 3: such thnt~~: 5 0 md br: > 0. I I Its importance stems from the fact that it is a typical theorem of the alternative that represents a large class of such theorems, theorems that constitute constructive optimality conditions for several important optimization problems (for more background information on duality, theorems of the alternative and other relevant matters see e.g. Gale [I 91 or " Co~sespond~ng Author. Downloaded By: [Brown University] At: 07:28 2 November 2009
of this theorem in 1261 is based on simple oblique projections. but this

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