Outline
Tokyo Institute of Technology
Tokyo Institute of Technology
Network Optimization and Control
Srinivas Shakkottai and R. Srikant
Foundations and Trends in Networking,
Vol.2,No.3,2007.
Primal-Dual Algorithm for Utility Maximization
• Ｂackground
• Primal Algorithm
• Dual Algorithm
• Primal-Dual Algorithm
• Power network
Yosuke Hoshi
FL 10 – 06 – 01
7th,June,2010
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Fujita Laboratory
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Background - Network
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Background – Lagrangian
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n3
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Network
Concave function
: traffic source
n1
n4
Strictly concave function
: rate of source
route r’s transmission
link1
x2
s : x1
a strictly concave function has
a unique maximum
: the utility that the source
obtains from transmitting
data on route r at rate xr
link2
n2
example
Lagrangian
Utility Maximization in Networks
Utility
Constraints
Convex function
each source is associated
with a route r
link3
x3
: link 1
: link 2
Lagrangian
Utility
Constraints
strictly concave
: link 3
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Utility
constraints
strictly concave
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Background – Karush-Kuhn-Tucker heorem
x2
convex
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Outline
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example
min
constraints
2
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(1,2)
strictly convex
x1
Karush-Kuhn-Tucker theorem
• Ｂackground
• Primal Algorithm
• Dual Algorithm
• Primal-Dual Algorithm
• Power network
the optimal value and constraint’s grad is Linear independence,
then there exists constants such that
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Primal Algorithm - example
Primal Algorithm - example
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Bl = xr2
Primal Algorithm - rate-update functions at source side
example
strictly
concave
Utility
strictly concave
convex
∗
1
U r = log xr
∗
2
cl
when V(xr) is maximum at point x and x ,
link Penalty
xr
V ( xr )
: link 1
Penalty
: link 2
not to exceed link capacity cl
V (x )
Utility
xr
cl
:Utility
Primal algorithm
( )
V x∗
:Penalty
k r (xr ) : non-negative,increasing and continuous
We want to maximize
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Primal Algorithm - example
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Primal Algorithm – Penalty function
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Primal algorithm
Network utility maximization problem
: utility (strictly concave)
k1(x1) = 0.01
k2(x2) = 0.01
: the arrival rate on a link l
Penalty function
Bl(.) increases when the arrival rate on a link l
approaches the link capacity cl
x1 = 0.3827
x2 = 0.5412
congestion
price function
: increasing, continuous
function
: convex function
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y
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convex
cl
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Outline
Primal Algorithm
strictly concave
Bl ( y )
fl ( y )
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• Ｂackground
• Primal Algorithm
• Dual Algorithm
• Primal-Dual Algorithm
• Power network
strictly concave
the condition of miximization
drive x towards the solution of
Primal algorithm
: non-negative,increasing
and continuous
drive xr towards the direction of ascent
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Dual Algorithm - example
Dual Algorithm - example
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Lagrange dual
Dual Algorithm - price-update functions on each of links
example
Utility
The Dual problem
Link capacity
: link 1
: link 2
calculate
Lagrange dual
: Lagrange multiplier
constraints
We need to descend down to gradient
Dual algorithm
!
The Dual problem
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Dual Algorithm - example
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Dual Algorithm
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Dual algorithm
resource allocation problem
utility
constraints
Each link has a capacity
c1 = 1.0
c2 = 1.5
ω1 = 1.0
ω2 = 3.0
The Lagrange dual
costraints
: Lagrange multiplier
x1 =
x2 =
p1 =
p2 =
0.3750
1.1250
-1.2723e-013
2.6667
The Dual problem
price
To find the direction of a gradient descent, we need to know
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Outline
Dual Algorithm
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Before derive
, we calculate
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and
Q
• Ｂackground
• Primal Algorithm
• Dual Algorithm
• Primal-Dual Algorithm
• Power network
The price of route r
Q
The load on link l
We need to descend down to gradient
!
Dual algorithm
pl increases when the arrival rate is larger than the capacity
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Primal-Dual Algorithm
Primal-Dual Algorithm
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Primal
rate-update functions
at source side
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Dual
price-update functions
on each of links
Primal-Dual Algorithm - flow base functions
⎧a beginning node b( f )
each flow has ⎨a ending node
e( f )
⎩
Exact form of the update were different in the two cases!
in wireless network,
interference among various
links necessitates scheduling
of links
scheduling
: time of rate
Rind (i ) : inflow rate (destination d at node i)
n3
x3
n4
x2
b( f )
Rij
n1
s : x1
n2
Rind (i )
i
d
Rout
(i )
43
42
24
14
d
Rout
(i ) : outflow rate (destination d at node i)
d
e( f )
time
the arrival rate is less than the departure rate
Primal-Dual Algorithm
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Primal-Dual Algorithm example
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Primal-Dual Algorithm example
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Utility
Constraints
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the congestion control problem
1→3
the scheduling problem
3→2
2→3
Primal-Dual Algorithm
the primal algorithm
Lagrange function
the dual algorithm
at each ⎧ node i
⎨
⎩ destination d
at each source f
Utility
constraints
Decoupled
calculated at each time instant by solving the scheduling problem
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Primal-Dual Algorithm
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Primal-Dual Algorithm
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Lagrange function
the optimization problem
each flow constraints
the congestion control problem
the scheduling problem
i→d
Primal-Dual Algorithm to solve the congestion control problem
the primal algorithm
Lagrange function
Decoupled
xf’s terms
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at each source f
the dual algorithm
at each ⎧ node i
⎨
⎩ destination d
pid : Lagrange multiplier
calculated at each time instant by solving the scheduling problem
R’s terms
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Power network
Outline
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purpose
to supply each building demands
for energy ri
• Ｂackground
• Primal Algorithm
• Dual Algorithm
• Primal-Dual Algorithm
• Power network
constraints
battery
Utility function
purpose
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Power network
battery
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Power network
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constraints
energy to building
line to charge battery
a→e
b→f
the congestion control problem
c→g
battery to building
energy to charge battery
e→a
d→e
f→b
d→f
g→c
d→g
the scheduling problem
Lagrange function
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