Lecture 9
Unconstrained Optimization
Need to maximize a function f(x), where x is a scalar or a
vector
x = (x1, x2)
f(x) = -(x-a)2
f(x) = x
f(x) = -x12 - x22
Gradient Search Techniques
Suppose f(x) is differentiable, i.e.,  f(x) exists.
Then the series of updates xk+1 = xk +   f(xk)
reaches a local maxima starting from any initial value
x0, for all sufficiently small values of .
A function f(x) has a local maxima at xk if f(xk) is
less than or equal to f(y) for all y in some small
neighborhood of xk
Concave Function
Let C be a convex subset of Rn
A function f: CR is called concave if
f( x + (1-  )y)   f(x) + (1-  )f(y) for
all [0, 1] and x , y  C
f(x)
f( x + (1-  )y)
 f(x) + (1-  )f(y)
x
y
 x + (1-  )y
x
Strict Concave Function
Let C be a convex subset of Rn
A function f: CR is called strictly concave if
f( x + (1-  )y)  f(x) + (1-  )f(y) for
all [0, 1] and x , y  C
A straight line is concave, but not strictly concave.
Any local maxima is a global maxima for a concave function.
A strictly concave function has a unique global maxima
Constrained Optimization
Maximize a function f(x), where x must be in a given set S
Suppose, f(x) is linear in x.
Set S is specified by liner inequalities
Then this is a linear program
Network Flow
Maximize the total output flow from the source s,
every link (u, v)
0  xuv  Cuv
v: (v, u) Exvu = v: (u, v) Exuv
v: (v, s) Exvs - v: (s, v) Exsv = -d
v: (v, t) Exvt - v: (t, v) Extv = d
d such that
for
Network Flow as a Linear
Program
Here, x is a flow allocation.
f(x) is the net flow out of the source under flow allocation x
f(x) = v: (s, v) Exsv
- v: (v, s) Exvs
Set S is the set of feasible flows. Feasibility conditions
are
0  xuv  Cuv
v: (v, u) Exvu = v: (u, v) Exuv
Linear ProgramTechniques
Simplex
Exponential Complexity in the worst case
However, LP is polynomial complexity computable.
If a problem can be modelled as LP, then this shows that
the problem is polynomial complexity and not NP-hard.
Convex Optimization
Maximize a function f(x), where x must be in a given set S
Function f(x) is concave
Set S is convex
Get Rid of the constraints!
Consider a function g(x) such that g(x) is small if x is not in
S and large otherwise.
Now try to maximize f(x) + g(x) without any constraints
Now f(x) + g(x) can be maximized by gradient search
techniques if f(x) + g(x) is differentiable.
Let us be more specific about set S.
A vector x belongs to set S if gi(x)  0, i=1,…M
We would like to maximize f(x) -  i gi(x)
What should be the value of ?
q() = maxx (f(x) -  i gi(x))
min0 q()
The output of the above minimization is the maximum
value of f(x) subject to x S
Primal-Dual Approach
Primal
Maximize f(x) subject to gi(x)  0, i=1,…M
Dual
q() = maxx (f(x) -  i gi(x))
min0 q()
Advantages of Dual Approach
For the convex programming we are considering,
primal = dual
(may not hold if the feasible set is nonconvex).
Dual is always convex and hence has a unique
global minima.
If the dual is differentiable, then it can be solved by gradient
search techniques.
The dual is differentiable if the objective function f(x) is
strictly concave.
Flow Control
Sessions are normally greedy.
Would like to send as much traffic as possible
However, this would congest the network.
Need to regulate the flow of the sessions.
Flow of a session must depend on the bandwidth
requirements and the revenues paid by the sessions.
Every session has a utility function.
Utility is a function of the bandwidth. It reflects the value
attached to the bandwidth by a session.
The objective of the network is to allocate bandwidths to
maximize the sum of the utilities of the sessions.
The underlying assumption is that the network charges
the users in accordance of the declared utility functions.
Utility Maximization
Utility of session i is Ui(x)
Let there be N sessions.
Every session has a predetermined path.
Maximize  Ui(ri) subject to
i:session i traverses link lri  Cl
This has been studied by several researchers:
F. Kelly, ``Charging and Rate Control for Elastic
Traffic’’,European Transactions on Telecommunications,
vol. 8, No. 1, 1997, pp 33-37
F. Kelly, A. Maulloo, D. Tan, ``Rate Control for
Communications Networks: Shadow Prices,
Proportional Fairness and Stability’’, Journal of
Operations Research Society, vol. 49, No. 3, 1998, pp
237-52
S. Low and D. Lapsley, ``Optimization Flow Control
I: Basic Algorithm and Convergence,’’ IEEE/ACM
Transactions on Networking, vol. 7, No. 6, Dec. 1999
R. La, V. Anantharam, ``Charge Sensitive TCP and Rate
Control in the Internet,’’Proceedings of INFOCOM 2000,
March 2000
S. Kunniyur, R. Srikant, ``End-to-End Congestion Control
Schemes: Utility Functions, Random Losses and ECN
Marks’’, Proceedings of INFOCOM 2000, March 2000
Dual based Approach
Primal
Maximize  Ui(ri) subject to
i:session i traverses link lri  Cl
Dual
L(r, p) = i Ui(ri) - lpl(i:session i traverses link lri - Cl)
= i (Ui(ri) - ril is on session i path pl) + lpl Cl
D( p) = max r L (r, p)
= max r i (Ui(ri) - ril is on session i path pl) + lpl Cl
= i (Ui(ri(p)) - ri (p)l is on session i path pl) + lpl Cl
Ui’(ri (p)) = l is on session i path pl
The optimum is attained at minp0 D( p)
Assumption: Utility functions are strictly concave.
Then it turns out that the objective function  Ui(ri) is
strictly concave. It follows that the dual D(p) is
differentiable.
Hence, we can use gradient search to attain minp0 D( p)
pk+1 = (pk -  D( p) )+ , where  is sufficiently small
D( p) = i (Ui(ri(p)) - ri (p)l is on session i path pl) + lpl Cl
D( p) /  pl = i  ri (p) /  pl (Ui’(ri(p)) - l is on session i path pl)
+ Cl - i: session i traverses link l ri (p)
 D( p)/  pl = Cl - i: session i traverses link l ri (p)
Hence, pk+1l = (pkl - (Cl - i: session i traverses link l ri (p)))+
Also, Ui’(ri (p)) = l is on session i path pl
That is, ri (p) = Ui’-1(l is on session i path pl)
Initially, choose p0l = 0 for all links l.
Update session rates as ri (0) = Ui’-1(0), for all sessions i
Update link prices as p1l = (Cl - i: session i traverses link l ri (p))+
Update session rates again and subsequently link rates etc.
Finally, the dual minimum is attained, and the rates
converge to those which attain the maximum utilities.
Intuitive Explanation
pl is the link price.
Link price update:
pk+1l = (pkl - (Cl - i: session i traverses link l ri (p)))+
New link price = Old link price - (Link Capacity –
Sum of rates of sessions traversing the link)
Link price increases if there is congestion, and decreases if the
link bandwidth is underutilized.
Session rate update:
Session rate = Ui’-1(path price for session i)
Since utility functions are strictly concave, derivatives of
utility functions are decreasing.
It follows that rate of a session increases if path price decreases for
the session and decreases if session price increases.
Path price increases if there is congestion in the path, and
decreases if there is underutilization.
So session rate increases if resources are under-utilized
in its path, and decreases if there is congestion in its
path
Distributed Implementation
Link price update requires only the sum of the session
rates in the link.
Session rate update requires only the sum of the
link prices in its path.
So, a session source sends a control packet with 0 price
value.
Every link increments this value by its link price
By the time the control packet reaches the receiver, the
price value in the packet equals the path price for the
session.
Session rate is updated based on the price in the control
packet.
Receiver communicates the rate to the session source.
Links can learn the session rate from the control packet
traversing towards the source from the receiver,
And subsequently use these rates to update the link price.
Alternatively, source sends at the rate directed by the source.
Links measure the session rates from the number of incoming
packets.
Scheduling may play a role in the convergence for the
latter!
Random Early Marking
A heuristic implementation of the previous optimization
algorithm with one bit marking.
Athuraliya, Low and Lapsley, GLOBECOM 1999
In current internet, there is a proposition to include one bit in
the header of every packet for congestion notification (ECN:
explicit congestion notification)
If a router is congested, then it marks the bit for every
packet traversing the router.
When the marked packet reaches the receiver, it
knows that the path is congested and asks the
source to reduce the transmission rate.
REM suggests that a link l mark a packet of any
session with probability 1-exp(-pl), where pl is the
link price.
Probability that a packet is marked is 1 - l(1 – (1-exp(-pl)))
(assuming that packet markings in
different links are independent events)
Probability that a packet is marked for a session i is
1 - exp (-l is on session i pathpl)
This probability can be estimated from the fraction
of marked packets reaching the receiver.
Clearly, path price for a session can be estimated
from this probability.
Path price of a session = -ln(1 – fraction of marked
packets)
Links estimate session rates from the packet arrival
rates,
Links compute the link prices from these session
rates
Use the link prices to probabilistically mark packets
Receiver estimates the path price from fraction
of marked packets
Update session rates on the basis of the
estimated path price
Communicate session rates to the source.
Source sends packets accordingly.
Estimation error possible.
However, simulation results indicate that the actual rates
oscillate in a neighborhood of the optimum rates.
No convergence proof.