{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

14dualmethods2

# 14dualmethods2 - EE236C(Spring 2008-09 14 Dual methods II...

This preview shows pages 1–6. Sign up to view the full content.

EE236C (Spring 2008-09) 14. Dual methods II single commodity network flow augmented Lagrangian method 14–1 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 0 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 14–2

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

View Full Document
Network flow optimization problem minimize n summationdisplay j =1 φ j ( x j ) subject to Ax + b = 0 φ ( x ) = n j =1 φ j ( x j ) is separable convex flow cost function convex, readily solved with standard methods dual decomposition yields decentralized solution method Dual methods II 14–3 Network flow dual Lagrangian L ( x, ν ) = φ ( x ) + ν T ( Ax + b ) = b T ν + n summationdisplay j =1 ( φ j ( x j ) + ( a T j ν ) x j ) a j is j th column of A we’ll interpret ν i as potential at node i y j = a T j ν is the potential difference across edge j (potential at start node minus potential at end node) dual problem: maximize g ( ν ) g ( ν ) = inf x L ( x, ν ) = b T ν n summationdisplay j =1 φ j ( a T j ν ) Dual methods II 14–4
Recovering primal from dual strictly convex φ j means unique minimizer x j ( y ) = argmin x j ( φ j ( x j ) yx j ) if φ j is differentiable, x j ( y ) = ( φ 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 ν : ( Ax ( y ) + b ) where y = A T ν subgradient is negative of flow conservation residual Dual methods II 14–5 Dual decomposition network flow algorithm given initial potential vector ν repeat 1. determine link flows from potential differences y = A T ν 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 ν i := ν i + ts i , i = 1 , . . . , p t is an appropriate step size Dual methods II 14–6

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

View Full Document
Dual decomposition network flow algorithm decentralized: flow calculated from potential difference across edge node potential updated from its own flow surplus g ( ν ) gives lower bound on p flow conservation Ax + b = 0 only holds in limit Dual methods II 14–7 Electrical network interpretation network flow optimality conditions (with differentiable φ j ) Ax + b = 0 , y + A T ν = 0 , y j = φ 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 ν = 0 ν j is node potential; y j = a T j ν is j th branch voltage current-voltage characterics: y j = φ j ( x j ) for example, φ j ( x j ) = R j x 2 j / 2 for linear resistor R j current and potentials in circuit are optimal flows and dual variables Dual methods II 14–8
Example: minimum queueing delay flow cost function φ j ( x j ) = x j c j x j , dom φ j = [0 , c j ) where c j > 0 are given link capacities

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 18

14dualmethods2 - EE236C(Spring 2008-09 14 Dual methods II...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online