{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture9

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

This preview shows pages 1–10. Sign up to view the full content.

C&O 355 Mathematical Programming Fall 2010 Lecture 9 N. Harvey

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Topics Semi-Definite Programs (SDP) Solving SDPs by the Ellipsoid Method Finding vectors with constrained distances
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ?
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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 =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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