Lecture24

# Lecture24 - C&O 355 Lecture 24 N Harvey Topics Semidefinite Programs(SDP Vector Programs(VP Quadratic Integer Programs(QIP QIP SDP for Max Cut

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Lecture 24 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
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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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 T V. Then X is PSD and X i,j = v i T v j 8 i,j. : Given a d x d PSD matrix X, find spectral decomposition X = UDU T , and let V = D 1/2 U. To get vectors in R d , let v i = i th column of V. Then X = V T V ) X i,j = v i T v j 8 i,j.
Vector Programs A Semidefinite Program: Equivalent definition as “vector program”

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Our usual Integer Program Quadratic Integer Program Integer Programs There are no efficient, general- purpose algorithms for solving IPs, assuming P NP. Let’s make things even harder: Quadratic Objective Function & Quadratic Constraints! Could also use {0,1} here. {-1,1} is more convenient.
Quadratic Integer Program Vector Programs give a natural relaxation: Why is this a relaxation? (QIP) (VP) If we added constraint v i 2 {(- 1,0,…,0),(1,0,…,0) } 8 i, then VP is equivalent to QIP

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QIP for Max Cut Let G=(V,E) be a graph with n vertices. For U
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## This note was uploaded on 06/16/2011 for the course C 355 taught by Professor Harvey during the Fall '09 term at Waterloo.

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Lecture24 - C&O 355 Lecture 24 N Harvey Topics Semidefinite Programs(SDP Vector Programs(VP Quadratic Integer Programs(QIP QIP SDP for Max Cut

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