a8 - Let P = { x R n : Ax b } where A R m n and b R m . (a)...

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CO350 Assignment 8 Due: July 6 at 2pm. Problem 1: Consider the linear system ( Ax = b, x 0) where A = ± 1 2 1 - 1 1 4 1 - 2 ² and b = ± 3 7 ² . (a) Formulate the auxilliary problem ( P 0 ) (b) Solve ( P 0 ) using the Simplex Method. (c) Let B be the optimal basis. Compute y = ( A T B ) - 1 c B . (Recall that y is the optimal solution to the dual of ( P 0 ).) Show that y satisfies the Farkas’ Lemma. Problem 2: Let R + denote the set of non-negative real numbers. Given three points x 1 , x 2 , x 3 R n , we define conv( x 1 , x 2 , x 3 ) = { λ 1 x 1 + λ 2 x 2 + λ 3 x 3 : λ R 3 + , λ 1 + λ 2 + λ 3 = 1 } . (a) Describe the set conv( x 1 , x 2 , x 3 ) geometrically. (b) Prove that conv( x 1 , x 2 , x 3 ) is convex. (c) Show that a set P R n is convex if and only if for each x 1 , x 2 , x 3 P we have conv( x 1 , x 2 , x 3 ) P . (d) (bonus question) Let P R n and let k 2 be an integer. For x 1 , x 2 , . . . , x k R n we define conv( x 1 , x 2 , . . . , x k ) to be the set of all points λ 1 x 1 + λ 2 x 2 + ··· + λ k x k where λ R k + and λ 1 + λ 2 + ··· + λ k = 1. Prove that, if x 1 , x 2 , . . . , x k P , then conv( x 1 , x 2 , . . . , x k ) P . Problem 3:
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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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