Lecture7 - • Let E be an ellipsoid centered at z • Let...

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Mathematical Programming Fall 2010 Lecture 7 N. Harvey
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Covering Hemispheres by Ellipsoids Let B = { unit ball }. Let H u = { x : x T u ¸ 0 }, where k u k =1. Find a small ellipsoid B’ that covers B Å H . B u
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Rank-1 Updates Def: Let z be a column vector and ® a scalar. A matrix of the form is called a rank-1 update matrix . Claim 1: Suppose ® -1/z T z. Then where ¯ = - ® /(1+ ® z T z). Claim 2: If ® ¸ -1/z T z then is PSD. If ® >-1/z T z then is PD. Claim 3:
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Main Theorem: Let B = { x : k x 1 } and H u = { x : x T u ¸ 0 }, where k u k =1. Let and . Let B’ = E( M, b ). Then: 1) B Å H u µ B’. 2) Remark: This notation only makes sense if M is positive definite. Claim 2 on rank-1 updates shows that it is, assuming n ¸ 2.
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Covering Half-ellipsoids by Ellipsoids
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Unformatted text preview: • Let E be an ellipsoid centered at z • Let H a = { x : a T x ¸ a T z } • Find a small ellipsoid E’ that covers E Å H a E E’ z H a Use our solution for hemispheres! Goal Find an affine map f and choose u such that: f ( B ) = E and f ( H u ) = H a Define E’ = f ( B’ ). Claim: E’ is an ellipsoid. Claim: E Å H a µ E’ . E E’ z B H a Choosing u E E’ z B H a • Assume E=E(N,z) and consider the map f (x) = N 1/2 x+ z. In Lecture 6 we showed that E = f ( B ). • Now choose u such that f ( H u ) = H a . H a = { x : a T (x-z) ¸ 0 } H u = { x : u T x ¸ 0 } ) take u = N 1/2 a...
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This note was uploaded on 06/16/2011 for the course CO 355 taught by Professor Harvey during the Winter '10 term at Waterloo.

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Lecture7 - • Let E be an ellipsoid centered at z • Let...

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