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Unformatted text preview: Let P = { x R n : Ax b } where A R m n and b R m . (a) Consider a point x P . Split the constraints Ax b into two parts A x b and A 1 x b 1 where A x = b and A 1 x < b 1 . Show that, if rank( A ) = n , then x is an extreme point of P . (Your proof should be direct; do not try to reduce the problem to standard equality form.) (b) Show that either P has an extreme point or there exists x , d R n such that { x + td : t R } P . (That is, either P has an extreme point or it contains a line.) Hint: Choose x P attaining equality in as many constraints as possible. If x is not an extreme point, then, by the result in ( b ), you can nd an appropriate vector d ....
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 Spring '07
 S.Furino,B.Guenin

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