Lecture16

Lecture16 - C&O 355 Mathematical Programming Fall 2010 Lecture 16 N Harvey Topics Semidefinite Programs(SDP Vector Programs(VP Quadratic Integer

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Mathematical Programming Fall 2010 Lecture 16 N. Harvey
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Topics Semidefinite Programs (SDP) Vector Programs (VP) Quadratic Integer Programs (QIP) Finding a cut from the SDP solution Analyzing the cut
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U U The Max Cut Problem Our first foray into combinatorial optimization Let G=(V,E) be a graph with n vertices. For U µ V, let ± (U) = { {u,v} : u 2 U, v U } Find a set U µ V such that | ± (U)| is maximized. This is a computationally hard problem: it cannot be solved exactly. (Unless P = NP) Our only hope: find a nearly-optimal solution, i.e., a big cut that might not be maximum. Philosophy: How can our powerful continuous optimization tools help to solve combinatorial problems?
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My Example Data Here is (a portion of) the adjacency matrix of a graph with 750 vertices, 3604 edges Probably cannot find the max cut before the end of the universe Can we find a big cut in this example?
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Semidefinite Programs Where x 2 R n is a vector and n = d(d+1)/2 A is a m x n matrix, c 2 R n and b 2 R m X is a d x d symmetric matrix, and x is the vector corresponding to X. There are infinitely many constraints!
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Vectorizing a 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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PSD matrices ´ Vectors in R d Key Observation: PSD matrices correspond directly to vectors and their dot-products . : Given vectors v 1 ,…,v d in R d , let V be the d x d matrix whose i th column is v i . Let X = V
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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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Lecture16 - C&O 355 Mathematical Programming Fall 2010 Lecture 16 N Harvey Topics Semidefinite Programs(SDP Vector Programs(VP Quadratic Integer

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