EE 685 presentation
Optimization Flow Control,
I: Basic Algorithm and Convergence
By Steven Low and David Lapsley
Objective of the paper

Propose an optimization approach for flow control on a network
whose resources are shared by a set of S of sources

Maximization of aggregate source utility over transmission rates is
aimed

Sources select transmission rates that maximize their benefit (utility –
bandwidth cost)

Synchronous and asynchronous distributed algorithms for
converging optimal behavior in static environment is presented
Problem Framework
The problem is formulated for
 A network that consists of a set L of unidirectional links of capacities cl,
where l is element of L.

The network is shared by a set S of sources, where source s is
characterized by a utility function Us(xs) that is concave increasing in its
transmission rate xs
.

The goal is to calculate source rates that maximize the sum of the
utilities ∑s ϵ S Us(xs) over xs subject to capacity constraints.
Problem Framework
DESTINATION NODES
link l4 : S(l4)={s1,s3}
l3
l5
l2
L(s1)={l1,l2,l3,l4}
l1
l6
S
SOURCE
NODES
..........
s1
s2
s3
ss
Centralized optimization : Why not

Centralized optimization of source rates is possible in theory but not
feasible and practical in real networks as



Knowledge of all utility functions are required
Resource usage is coupled due to shared links. So all the sources should be
coordinated simultaneously
Therefore a distributed and decentralized approach is needed.
The value of the optimization frame-work
presented

It is not always critical or feasible to attain exact optimality in a flow
control problem

However the optimal framework acts as a guideline to shape the
network dynamics to a desirable operating point where source
utilities and resource costs are taken into consideration

Optimization frameworks may be used to refine and ameliorate
practical flow control schemes
The optimization problem :
Primal problem
The optimization problem :
Lagrangian for primal problem



pl represents Lagrange multipliers utilized in standard convex optimization
method
By using this approach coupled link capacity constraints are integrated to the
objective function
Notice separability in terms of xs so maximizing lagrangian function as aggregate
of different xs related terms means gives the same result as summing up
maximum of each individual xs related term. Therefore we have
The optimization problem :
Dual problem




Here pl is the price per unit bandwidth at link l.
ps is the total price per unit bandwidth for all links in the path of s
The dual problem has been defined as minimization of D(p) (upper bound of
Lagrangian function) for non-negative bandwidth prices.
Each source can independently solve maximization problem in (3) for a given p
The optimization problem :
Concavity and duality gap

For each p, a unique maximizer denoted by xs(p) exists since Us (source utility
function) is strictly concave

Concavity of Us and linear constraints for the primal problem guarantees that
there is no duality gap and dual optimal prices exist in the form of Lagrange
Multipliers

Once p* is obtained by solving the dual problem, the primal optimal source rates
x*=x(p*) can be computed by individual sources s by solving (3)

Therefore given p* individual sources can solve (3) without any coordination
(key concept for distributed algorithm)

So p* acts as a coordination signal that aligns individual and joint optimality for
flow control problem
Notations and assumptions :
s number of flows/sources
R=
R11
R12
...
R1s
R21
R22
...
R2s
....
....
...
...
Rl1
Rl2
...
Rls
l number
of links

R is the routing matrix where Rls=1 if l ϵ L(s) or s ϵ L(s)

For each source s, pTR is the path bandwidth price that source s faces which is
equal to ps

Let xs(p) be the unique maximizer for (3) then xs(p) could be written as follows :
Routing matrix
DESTINATION
NODES
link l4
l3
R=
l5
l2
L(s1)={l1,l2,l3,l4}
l1
l6
S
SOURCE
NODES
..........
s1
s2
s3
ss
s1
s2
l1
1
0
l2
1
0
l3
1
0
l4
1
1
l5
0
1
l6
0
1
Concave utility function:
its derivative and inverse
TYPICAL CONCAVE UTILITY FUNCTION
İnverse of derivative of utility
rate x
derivative of utility
U’(x)
rate x
U’(x)
Source rate as demand function

The above figure depict xs(p) as a possible solution of (6)

Similar to inverse of U’ figure in previous slide the rate is obtained as a
decreasing function of U’-1(rate)

This means that xs(p) acts as a demand function seen in Microeconomics.
Fundamental assumptions :
C1,C2 and C3 assumptions for the utility functions

Here ms and Ms are minimum and maximum transmission rates for source s
Synchronous Distributed Algorithm
based on gradient projection applied to dual problem

The dual problem is solved via gradient projection method where link prices are
adjusted in the opposite direction of gradient of D(p)
Synchronous Distributed Algorithm
based on gradient projection appliedd to dual problem

Equation (9) shows that the price of a link l is updated based on how much
demand exceeds supply.
Synchronous Distributed Algorithm
Generic outline of the algorithm

Given aggregate source rate that goes through link l, the adjustment algorithm
(9) is completely distributed

Therefore network links l and sources s could be treated as processors in a
distributed computation system to solve the dual problem at (5)

1.
In each iteration sources s solve (3) independently and communicate
their results xs(p) to links on their path (L(s)).
2.
Links l then update their prices pl according to (9) and communicate
their new prices to sources s
3.
The cycle repeats goes back to 1 with updated p values
It is possible to prove that under C1 and C2 conditions this algorithm
converges to a stable and optimal x* (optimal source rates) and p* (optimal
bandwidth prices) for static network conditions (THEOREM 1)
Synchronous Distributed Algorithm
THEOREM 1
THEOREM 1 (proof)

LEMMA 1: Under C1, D(p) is convex, lower bounded and continuously
differentiable
THEOREM 1 (proof)

LEMMA 2: Under C1, The Hessian of D is given by 2D(p)=RB(p)RT
THEOREM 1 (proof)
THEOREM 1 (proof)

LEMMA 3: Under C1-C2, D is Lipschitz continuous with
‖ D(q) - D(p) ‖2 ≤ αLS ‖q-p ‖ 2 for all p,q ≥ 0
THEOREM 1 (proof)

LEMMA 3: Under C1-C2, D is Lipschitz continous with
‖ D(q) - (p) ‖2 ≤ αLS ‖q-p ‖ 2 for all p,q ≥ 0
THEOREM 1 (proof)
Asynchronous Distributed Algorithm
why asynchronous model is needed

The synchronous model of the last section assumes that updates at the
sources and the links are synchronized

In realistic large network scenarios, synchronous updates might not be possible
as

Sources may be located at different distances from the network links

.Network states (prices in our case) may be probed by different sources
at different rates, e.g., the Resource Management

Feedbacks may reach different sources after different, and variable,
delays.

These complications make our distributed computation system consisting of
links and sources asynchronous.

The communication delays may be substantial and time-varying.
Asynchronous Distributed Algorithm
Generic outline of the algorithm

The main approach of interdependent update of source rates and bandwidth
prices iteratively is followed in asynchronous version of the algorithm as well

For bandwidth price updates the links use an estimate of the gradient based on
past source rates at link l

Two type of policies are applied

Latest data only: Only the last received rate is used

Latest average: Only the average of latest k received rates is used

The convergence of both synchronous and asyncronous algorithms depend on
sufficiently small step size for (7) and (9)

Convergence for asynchronous version of the algorithm could be proven as
long as assumptions C1 and C3 hold.
Asynchronous Distributed Algorithm
Fairness, quasi-stationarity and pricing
Homogeneous sources case
with equal user utility functions
Single link path case
(with C4 condition)
Single link path case
(proof of theorem 4)