Farkas.lemma - Farkas Lemma The following are equivalent...

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Farkas’ Lemma The following are equivalent statements of Farkas' Lemma. FARKAS’ LEMMA (1). Let A be an m × n matrix. Then A x = b, x 0 has a solution if and only if b is in the cone generated by the columns a 1 , … , a n of A, i.e., b { λ 1 a 1 + … + λ n a n | λ 1 , … , λ n 0}. PROOF. The vector x = (x 1 , … , x n ) 0 is a solution of A x = b is and only if a ij x j = b i for 1 i m, or equivalently, if and only if x 1 a 1 + … + x n a n = b. Q.E.D. FARKAS’ LEMMA (2). A x = b has a solution x 0 if and only if y A 0 implies y b 0. PROOF. Let A x = b have a solution x 0 ; then y A x = y b . If y A 0 , we get that y b 0 since x 0 . Conversely, suppose that y A 0 implies that y b 0. Let C be the set of all vectors C = { λ 1 a 1 + … + λ n a n | λ i 0}. Here a i is the ith column of A. Note that C is a closed cone in n. This requires some additional arguments which seem to be important but not available from the text. We supply the arguments separately below. Here C may be written as C = {A x x 0 }. We show that b C. By part a), we will then have that A x = b has a solution x 0 . Suppose that b C. Then there is a vector y such that lub{ y c | c C} < y b . Since λ c C for every λ 0 and for every c C, we must have that lub{ y c | c C} 0. In particular, we have that
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This note was uploaded on 11/03/2009 for the course CS 149 taught by Professor Meinolfsellman during the Spring '09 term at Sanford-Brown College.

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Farkas.lemma - Farkas Lemma The following are equivalent...

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