Title: EE 685 presentation
1EE 685 presentation
- Optimization Flow Control,
- I Basic Algorithm and Convergence
- By Steven Low and David Lapsley
- Asynchronous Distributed Algorithm Proof
2Objective 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
3Problem 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 ? S Us(xs)
over xs subject to capacity constraints.
4Problem 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
5Centralized 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.
6The 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
7The optimization problem Primal problem
8The 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
9The 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
10The 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 xx(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
11Notations and assumptions
s number of flows/sources
R11 R12 ... R1s
R21 R22 ... R2s
.... .... ... ...
Rl1 Rl2 ... Rls
l number of links
R
- R is the routing matrix where Rls1 if l ? L(s)
or s ? 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
12Routing matrix
DESTINATION NODES
s1 s2
l1 1 0
l2 1 0
l3 1 0
l4 1 1
l5 0 1
l6 0 1
link l4
R
l3
l5
l2
L(s1)l1,l2,l3,l4
l1
l6
S
..........
SOURCE NODES
s1
s2
s3
ss
13Concave utility function its derivative and
inverse
TYPICAL CONCAVE UTILITY FUNCTION
Inverse of derivative of utility rate x
derivative of utility U(x)
U(x)
rate x
14Source 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.
15Synchronous Distributed Algorithmbased 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)
16Synchronous 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.
17Synchronous Distributed AlgorithmGeneric 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) - In each iteration sources s solve (3)
independently and communicate their results xs(p)
to links on their path (L(s)). - Links l then update their prices pl according to
(9) and communicate their new prices to sources s - 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)
18Synchronous Distributed Algorithm
19THEOREM 1
20THEOREM 1 (proof)
- LEMMA 1 Under C1, D(p) is convex, lower bounded
and continuously differentiable
21THEOREM 1 (proof)
- LEMMA 2 Under C1, The Hessian of D is given by
?2D(p)RB(p)RT
22THEOREM 1 (proof)
23THEOREM 1 (proof)
- LEMMA 3 Under C1-C2, ?D is Lipschitz continuous
with - ? ?D(q) - ?D(p) ?2 aLS ?q-p ? 2 for all p,q
0
24THEOREM 1 (proof)
- LEMMA 3 Under C1-C2, ?D is Lipschitz continous
with - ? ?D(q) - ?(p) ?2 aLS ?q-p ? 2 for all p,q
0
25THEOREM 1 (proof)
26Fundamental assumptions C1,C2 and C3
assumptions for the utility functions
- Here ms and Ms are minimum and maximum
transmission rates for source s
27Asynchronous 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.
28Asynchronous 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.
29Asynchronous Distributed Algorithm
30Asynchronous Distributed AlgorithmProof
31Asynchronous Distributed AlgorithmProof (lemma 4)
32Asynchronous Distributed AlgorithmProof (lemma 4)
- Lemma 4 a) comes from applying the projection
theorem to the scalar pl(t1) pl(t)-??l(t) - Lemma 4 b) follows from
33Asynchronous Distributed AlgorithmProof (lemma 4)
34Asynchronous Distributed AlgorithmProof (lemma 4)
35Asynchronous Distributed AlgorithmProof (lemma 4)
- The next lemma (LEMMA 5) bounds the error in
gradient estimation in terms of the successive
price change p(t) p(t1)-p(t).
36Asynchronous Distributed AlgorithmProof (lemma 5)
37Asynchronous Distributed AlgorithmProof (lemma 5)
38Asynchronous Distributed AlgorithmProof (lemma 5)
39Asynchronous Distributed AlgorithmProof (lemma 6)
40Asynchronous Distributed AlgorithmProof (lemma 6)
41Asynchronous Distributed AlgorithmProof (lemma 6)
42Asynchronous Distributed AlgorithmTheorem 2
43Asynchronous Distributed AlgorithmTheorem 2
44Asynchronous Distributed AlgorithmTheorem 2