Some attempts to improve non-convex optimization algorithms for problems in networks

1. Summer 2008 Internship
Report
Advisor: Prof. Angela Y. Zhang
The Chinese University of Hong Kong, Hong Kong
Student: Pratik Poddar
Indian Institute of Technology Bombay, India
Topic: Non­Convex Optimization
Problems in Networks

2. Introduction
3 topics to be discussed:
1) Polyblock Algorithm for Monotonic Optimization
2) Network Utility Maximization
3) Internet Congestion Control Problem

3. Basics of Optimization
●
Standard Optimization problem
●
Linear Optimization problem
●
Convex Optimization problem
●
Monotonic Optimization problem

4. Polyblock Algorithm
●
We have had two major events in the history of
optimization theory.
●
The first was linear programming and simplex
method in late 1940s­ early 1950s.
●
The second was convex optimization and interior
point method in late 1980s­ early 1990s.

5. Polyblock Algorithm
●
Convex optimization problems are known to be
solved, very reliably and efficiently.
●
"..in fact, the great watershed in optimization isn't
between linearity and nonlinearity, but convexity
and nonconvexity" ­ R. Tyrrell Rockafellar, in
SIAM Review, 1993

6. Polyblock Algorithm
●
Current research in optimization is mainly to have
that third event ­ Solving non­convex optimization
efficiently. Although solving convex optimization
problems is easy and non­convex optimization
problems is hard, but a variety of approaches have
been proposed to solve non­convex optimization
problems.

7. Polyblock Algorithm
●
In 2000, H. Tuy proposed an algorithm to solve
optimization problems involving d.i functions under
monotonic constraints.
●
This algorithm (Polyblock Algorithm) was inspired
by the idea of Polyhedral Outer Approximation
Method for maximizing a quasi­convex function
over a convex set.

8. Polyblock Algorithm
●
What is a polyblock?
●
Then what is the difference between a polyblock and
a polyhedron?
●
What are its properties?
●
How is polyblock algorithm implemented?

9. Polyblock Algorithm as in
[1]

10. Implementation of Polyblock
Algorithm
●
Consider the following optimization problem:
minimize x1 + x2
such that (x1­3) + 9(x2­3) ≥ 0
3
5x1 + 6x2 – 36 ≤ 0
(x1,x2) ∊ [0,6]2

11. Implementation of Polyblock
Algorithm
●
Feasible region of the problem

12. Implementation of Polyblock
Algorithm

13. Introduction
3 topics to be discussed:
1) Polyblock Algorithm for Monotonic Optimization
2) Network Utility Maximization
3) Internet Congestion Control Problem

14. Network Utility
Maximization
●
The framework of Network Utility Maximization
(NUM) has found many applications in network rate
allocation algorithms and Internet Congestion
Control Protocols.

15. Network Utility
Maximization
●
Problem: Consider a network with L links, each with
a fixed capacity cl bps, and S sources (i.e. end
users), each transmitting at the rate of xs bps. Each
source s uses the set L(s) of links in its path and has
a utility function Us(xs). Each link l is shared by a set
S(l) of sources. So, Network Utility Maximization is
basically the problem of maximizing the total utility
of the system over source rates subject to congestion
constraints for all links.

16. Network Utility
Maximization
Mathematically,

17. Network Utility
Maximization
●
Concave Utilities ­ Follows from Law of
Diminishing Marginal Utilities. Convex
Optimization Problem.
●
U(x) = log (1+x)
U(x)
x

18. NUM for Concave Utilities
●
The problem of Network Utility Maximization in
case of concave utilities is essentially a convex
optimization problem which is solvable efficiently
and exactly.

19. Network Utility
Maximization
●
Non­Concave Utilities – In multimedia applications
on Internet, the utilities are non­concave. Non­
convex optimization problem.
●
U(x) = (1 + e­ax+b) ­1
U(x)
x

20. NUM for Non-Concave
Utilities
●
The problem is a non­convex optimization problem.
Three ways have been suggested to solve it.
●
In [3], a 'self­regulation' heuristic is proposed,
however it converges only to a sub­optimal solution.
●
In [4], a set of sufficient and necessary conditions is
presented under which the canonical distributed
algorithm converges to a global optimal solution.
However, these conditions may not hold in most
cases.

21. NUM for Non-Concave
Utilities
●
In [2], Using a family of convex SDP relaxations
based on the sum­of­squares method and
Positivestellensatz Theorem in real algebraic
geometry, a centralized computational method to
bound the total network utility in polynomial time is
proposed.
●
This is effectively a centralized method to compute
the global optimum when the utilities can be
transformed into polynomial utilities.

22. NUM for Non-Concave
Utilities
●
In summary, currently there is no theoretically
polynomial­time algorithm (distributed or
centralised) known for non­concave utility
maximization.
●
We worked to find ways to convexify the above
problem.

23. Idea and motivation
●
The set may not be a convex set but if it can be
broken into a constant number of convex sets, we
can solve the problem in polynomial time.

24. Idea and motivation

25. Idea and motivation

26. Idea and motivation

27. Motivation
●
By this method, we can solve NUM problem in
polynomial time. NUM finds applications in
network rate allocation algorithms and Internet
Congestion Control Protocol.

28. Introduction
Not so much related topics:
1) Polyblock Algorithm for Monotonic Optimization
2) Network Utility Maximization
3) Internet Congestion Control Problem

29. Internet Congestion Control
●
Internet relies on congestion control implemented in
the end­systems to prevent offered load exceeding
network capacity, as well as allocate network
resources to different users and applications.
●
In the past, the applications (email, file transfer) had
concave utilities (i.e were elastic). As number of
multimedia applications are increasing, there are
various talks on different congestion controls.

30. Internet Congestion Control
●
In [5], It has been argued that fairness congestion
control does not maximize the network's utility.
Infact, Admission control is shown to be better
control (in terms of both elastic and inelastic
utilities) than Fair Congestion Control in a
simplified case.
●
Let α be the desired rate of inelastic flows, m be the
number of inelastic flows and n be the number of
elastic flows.

31. Fair Congestion Control
●
Perform TCP­friendly congestion control. We model
it as the same fair congestion control as adopted for
elastic flows, with a slight difference. When the fair
share is smaller than α, then the fair share is used,
but when the fair share is greater than α, the
inelastic flow would still consume α.

32. Admission Control
●
Perform admission control but no congestion control
once admitted. Assume the network already has n
elastic flows and m inelastic flows, a new inelastic
flow is admitted iff nε + (m­1)α <=1
●
Here ε represents the minimum rate admission
control scheme tries to leave for elastic traffic.
Depending upon α, we can have two cases:

33. Aggressive Admission
Control
●
ε <<< α – The arriving flow is admitted as long as it
is possible to allocate to it the desired rate of α, even
if this means all elastic flows have to run at their
minimum rate of ε.
●
So, an inelastic flow is admitted iff (m+1)α ≤ 1 and
an elastic flow is always admitted.

34. Fair Admission Control
●
ε = α – The arriving flow is admitted as long as its
desired rate is no greater than the prevailing fair
share for each elastic flow.
●
So, an inelastic flow is admitted iff (m+n+1)α ≤ 1
and an elastic flow is always admitted.
●
In [5], it is proved that Fair Admission control is
better than both Aggressive Admission contol and
Fair Congestion Control.

35. Idea
●
Solving the optimization problem using the
polyblock algorithm would help us to prove (or
disprove) that admission control is better than fair
congestion control.
●
Status: Coding to check it under progress.

36. Thank You...........

37. Bibliography
●
[1] H. Tuy, ”Monotonic Optimization: Problems and Solution Approaches”,
SIAM Journal on Optimization, 11:2(2000), 464­494
●
[2] M. Fazel, M. Chiang, ”Network Utility Maximization With Nonconcave
Utilities Using Sum­of­Squares Method”, Proc. IEEE CDC, December 2005
●
[3] J.W.Lee, R.R. Mazumdar, N. Shroff, ”Non­convex optimization and rate
control for multi­class services in the Internet”, Proc. IEEE Infocom, March
2004
●
[4] M. Chiang, S. Zhang, P. Hande, ”Distributed rate allocation for inelastic
flows: Optimization framework, optimality conditions, and optimal
algorithms”, Proc. IEEE Infocom, March 2005
●
[5] D. M. Chiu, A. ­S. W. Tam, ”Fairness of traffic controls for inelastic
flows in the Internet”, Comput. Netw. (2007), doi:10.1016/j.comnet.
2006.12.2006