462774_751308825_776166850

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PLEASE SCROLL DOWN FOR ARTICLE This article was downloaded by: [Brown University] On: 2 November 2009 Access details: Access Details: [subscription number 784168975] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Optimization Methods and Software Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713645924 A simple algebraic proof of Farkas's lemma and related theorems C. G. Broyden a a Facoltá di Science MM.FF.NN, Universitá di Bologna, Cesena, FO, Italy Online Publication Date: 01 January 1998 To cite this Article Broyden, C. G.(1998)'A simple algebraic proof of Farkas's lemma and related theorems',Optimization Methods and Software,8:3,185 — 199 To link to this Article: DOI: 10.1080/10556789808805676 URL: http://dx.doi.org/10.1080/10556789808805676 Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
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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
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of this theorem in 1261 is based on simple oblique projections. but this
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