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Lecture9 - C&O 355 Mathematical Programming Fall 2010...

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C&O 355 Mathematical Programming Fall 2010 Lecture 9 N. Harvey
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Topics Semi-Definite Programs (SDP) Solving SDPs by the Ellipsoid Method Finding vectors with constrained distances
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LP is great, but… Some problems cannot be handled by LPs Example: Find vectors v 1 ,…,v 10 2 R 10 s.t. All vectors have unit length: k v i k = 1 8 i Distance between vectors is: k v i -v j k2 [1/3, 5/3] 8 i j Sum-of-squared distances is maximized Not obvious how to solve it. Not obvious that LP can help. This problem is child’s play with SDPs
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How can we make LPs more general? An (equational form) LP looks like: In English: Find a non-negative vector x Subject to some linear constraints on its entries While maximizing a linear function of its entries What object is “more general” than vectors? How about matrices ?
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Generalizing LPs: Attempt #1 How about this? In English: Find a “non-negative matrix” X Subject to some linear constraints on its entries While maximizing a linear function of its entries Does this make sense? Not quite… What is a “non-negative matrix”? Objective function c T X is not a scalar AX is not a vector
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What is a “non-negative matrix”? Let’s define “non-negative matrix” by symmetric, positive semi-definite matrix ”. So our “variables” are the entries of an n x n symmetric matrix . The constraint “X ¸ 0” is replaced by “ X is PSD Note : The constraint “X is PSD” is quite weird. We’ll get back to this issue.
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Vectorizing the Matrix A d x d matrix can be viewed as a vector in R d 2 . (Just write down the entries in some order.) A d x d symmetric matrix can be viewed as a vector in R d(d+1)/2 . Our notation: X is a d x d symmetric matrix, and x is the corresponding vector. 1 2 3 4 1 2 3 4 1 2 2 4 1 2 4 X = x =
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Semi-Definite Programs Where x 2 R n is a vector A is a m x n matrix, c 2 R n and b 2 R m X is a d x d symmetric matrix, where n = d(d+1)/2, and x is vector corresponding to X. In English: Find a symmetric, positive semidefinite matrix X Subject to some linear constraints on its entries While maximizing a linear function of its entries This constraint looks suspiciously non-linear
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