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Unformatted text preview: EE236C (Spring 200809) 14. Dual methods II single commodity network flow augmented Lagrangian method 141 Single commodity network flow network connected, directed graph with n links, p nodes node incidence matrix A R p n is A ij = 1 arc j enters i 1 arc j leaves node i otherwise flow vector and external sources variable x j denotes flow (traffic) on arc j given external source (or sink) flow b i at node i , 1 T b = 0 flow conservation: Ax + b = 0 Dual methods II 142 Network flow optimization problem minimize n summationdisplay j =1 j ( x j ) subject to Ax + b = 0 ( x ) = n j =1 j ( x j ) is separable convex flow cost function convex, readily solved with standard methods dual decomposition yields decentralized solution method Dual methods II 143 Network flow dual Lagrangian L ( x, ) = ( x ) + T ( Ax + b ) = b T + n summationdisplay j =1 ( j ( x j ) + ( a T j ) x j ) a j is j th column of A well interpret i as potential at node i y j = a T j is the potential difference across edge j (potential at start node minus potential at end node) dual problem: maximize g ( ) g ( ) = inf x L ( x, ) = b T n summationdisplay j =1 j ( a T j ) Dual methods II 144 Recovering primal from dual strictly convex j means unique minimizer x j ( y ) = argmin x j ( j ( x j ) yx j ) if j is differentiable, x j ( y ) = ( j ) 1 ( y ) (inverse of derivative function) optimal flows, from optimal potentials, are x j ( y j ) where y = A T v subgradient of negative dual function g at : ( Ax ( y ) + b ) where y = A T subgradient is negative of flow conservation residual Dual methods II 145 Dual decomposition network flow algorithm given initial potential vector repeat 1. determine link flows from potential differences y = A T x j := x j ( y j ) , j = 1 , . . . , n 2. compute flow surplus at each node s i := a T i x + b i , i = 1 , . . . , p 3. update node potentials i := i + ts i , i = 1 , . . . , p t is an appropriate step size Dual methods II 146 Dual decomposition network flow algorithm decentralized: flow calculated from potential difference across edge node potential updated from its own flow surplus g ( ) gives lower bound on p flow conservation Ax + b = 0 only holds in limit Dual methods II 147 Electrical network interpretation network flow optimality conditions (with differentiable j ) Ax + b = 0 , y + A T = 0 , y j = j ( x j ) , j = 1 , . . . , n network with node incidence matrix A , nonlinear resistors in branches Kirchhoff current law (KCL) : Ax + b = 0 x j is the current flow in branch j ; b i is external current injected at node i Kirchhoff voltage law (KVL) : y + A T = 0 j is node potential; y j = a T j is j th branch voltage currentvoltage characterics:...
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This note was uploaded on 01/25/2010 for the course EE 236 taught by Professor Staff during the Spring '08 term at UCLA.
 Spring '08
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