L9_prob_more

L9_prob_more - Lecture 9: Convex Problems February 19, 2007...

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Lecture 9: Convex Problems February 19, 2007
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Lecture 9 Outline Geometric Programming Semidefinite Programming Preliminary for Duality Theory Convex Optimization 1
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Lecture 9 Geometric Programming A Geometric Program is a problem of the following form minimize f ( x ) subject to g j ( x ) 1 , j = 1 ,...,m h r ( x ) = 1 , r = 1 ,...,p with each g i being posynomial and each h j being monomial A geometric program transforms to a convex problem minimize ln K 0 X k =1 e a T 0 k y + b 0 k subject to ln K j X k =1 e a T jk y + b jk 0 , j = 1 ,...,m g T r y + d r = 0 , r = 1 ,...,p Convex Optimization 2
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Lecture 9 Minimizing Spectral Radius of a Nonnegative Matrix Perron-Frobenius eigenvalue λ pf ( A ) Exists for matrices A R n × n with nonnegative entries A real (positive) eigenvalue of A equal to spectral radius max i | λ i ( A ) | Determines asymptotic growth (decay) rate of A k : A k λ k pf as k → ∞ Alternatively characterized by: λ pf ( A ) = inf { λ | Av ± λv for some v ² 0 } Minimizing spectral radius of a matrix of posynomials Minimize λ pf ( A ( w )) , where the entries [ A ( w )] ij are posynomials of w The problem is equivalent to the following geometric program: minimize λ subject to n j =1 [ A ( w )] ij v j λv i 1 , i = 1 ,...,n, v ² 0 variables λ , v , and w [where w may be subject to additional constraints] Convex Optimization 3
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Lecture 9 Consensus Example A simple case of Concensus Problem: There are n nodes in a fully connected network Initally, each node i has an estimate x i (0) , a scalar The nodes communicate their estimates at time instances t = 1 , 2 ,... Each node independently updates its estimates by combining the received information with its own information in some fashion The issues are: Will they reach a concensus, i.e., “agree” on a common value ¯ x ? If they do, how long it will take?
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L9_prob_more - Lecture 9: Convex Problems February 19, 2007...

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