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Unformatted text preview: ( A ) < n (where n is the number of variables) then P has no extreme points. (2) EXTRA CREDIT (HARDER). Show that if rank ( A ) = n then P has at least one extreme point. Exercise 4. Let A be an m × n matrix and let d and b be m dimensional vectors. Consider the following statements, (S1) There is no n-dimensional non-negative vector x such that d ≤ Ax ≤ b , (S2) There are m-dimensional non-negative vectors z and y such that y T A ≥ z T A and y T b < z T d . In this question you have to, (1) Give a self contained proof that (S2) implies (S1). (2) Using the duality theorem and duality theory, give a proof that (S1) implies (S2). HINT: Use the same strategy as for the proof of Farkas’ Lemma. 1...
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This note was uploaded on 07/28/2009 for the course CO 350 taught by Professor S.furino,b.guenin during the Winter '07 term at Waterloo.
- Winter '07