# hwk8 - by induction to show that any convex combination of...

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CS 525 - Fall 2011 - Homework 8 * assigned 11/16/11 - due 11/30/11 1. Which of the following sets are convex? (a) { x R : x 2 = 1 } . (b) { x R : x 2 1 } . (c) { x R : x 2 1 } . 2. Let A be an m × n matrix. Verify that the set { x R n : Ax = b, x 0 } is a convex set. 3. A point x R n is a convex combination of points x 1 ,...,x m R n if there are scalars λ 1 ,...,λ m such that x = m X k =1 λ k x k m X k =1 λ k = 1 λ k 0 Show that a set C is convex if and only if every convex combination of a a ﬁnite number of points from C is contained in C . Hint: Note that a convex combination of two points is a line segment. Use proof
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Unformatted text preview: by induction to show that any convex combination of m points lies in C provided that any convex combination of m-1 points lies in C . 4. Prove that x = (1 / 2 , 1) is a global solution of minimize 2 x 2 1 + 4 x 1 x 2 + 5 x 2 2-6 x 1-12 x 2 subject to-1 ≤ x 1 ≤ 1-1 ≤ x 2 ≤ 1 Hint: Use the optimality conditions in Section 7.1 5. Do exercise 7-2-1. * Hard copy to be submitted in class on the due date. No late homework accepted....
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