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# 07 - Spring 2010 CS530 Analysis of Algorithms Homework 7...

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Spring 2010 CS530 – Analysis of Algorithms Homework 7 H OMEWORK 7, DUE M ARCH 24 You must prove your answer to every question. Problems with a ( * ) in place of a score may be a little too advanced, or too challenging to most students, so I do not assign a score to them. But I will still note if you solve them. Problem 1. (15pts) Prove the separating hyperplane theorem as formulated in class. [ Hint: To apply our version of the Farkas Lemma, it helps to turn the inequality v T x > z in the formulation into an inequality with . ] Solution. In class, we introduced the separating hyperplane theorem, here it is again. Let u 1 ,..., u m be some vectors in an n -dimensional space. Let L be the set of convex linear combinations of these points. Thus, a point v is in L if the following inequalities have a solution: j u i · y i = v , i y i = 1, y 0. (1) The theorem says that if v is not in L then there is a separating hyperplane between L and v , namely, the following set of inequalities has a solution for x : u T i x z ( i = 1,..., m ) , v T x > z .

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07 - Spring 2010 CS530 Analysis of Algorithms Homework 7...

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