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Unformatted text preview: z x 1 x 2 x 3 x 4 x 51 2 2 1 70 1 1 21 10 1 11 1 10 Now, all coeﬃcients in the objective function are positive, so we’ve found an optimal solution. The optimal value is z =70 at x 2 = 10 , x 3 = 10. Problem 3. The union of two polytopes is not a polytope. As a counterexample, { x  x ≥ 5 }∪{ x  x ≤ 1 } . Although each set is a polytope, the union is not a convex set, thus is not a polytope. As for the intersection of two polytopes, we think of two arbitrary polytopes, P 1 and P 2 . For each P i , we can ﬁnd A i , b i such that A 1 x ≥ b 1 and A 2 x ≥ b 2 . Since { x  A 1 x ≥ b 1 } ∩ { x  A 2 x ≥ b 2 } = { x  Ax ≥ b } , where A = " A 1 A 2 # , b = " b 1 b 2 # , the intersection is a polytope....
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 Spring '06
 UNKNOWN
 Linear Programming, Optimization, $0, $0.09, $0.03, Optimization Prof. Amin

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