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Some attempts to improve non-convex optimization algorithms for problems in networks
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Summer 2008 Internship Report
Advisor: Prof. Angela Y. ZhangThe Chinese University of Hong Kong, Hong Kong
Student: Pratik PoddarIndian Institute of Technology Bombay, India
Topic: NonConvex Optimization Problems in Networks
Introduction
3 topics to be discussed: 1) Polyblock Algorithm for Monotonic Optimization2) Network Utility Maximization3) Internet Congestion Control Problem
Basics of Optimization
● Standard Optimization problem● Linear Optimization problem● Convex Optimization problem● Monotonic Optimization problem
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.
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
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.
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.
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?
Polyblock Algorithm as in [1]
Implementation of Polyblock Algorithm● Consider the following optimization problem:
minimize x1 + x
2
such that (x13)3 + 9(x
23) ≥ 0
5x1 + 6x
2 – 36 ≤ 0
(x1,x
2) [0,6]∊ 2
Implementation of Polyblock Algorithm● Feasible region of the problem
Implementation of Polyblock Algorithm
Introduction
3 topics to be discussed:1) Polyblock Algorithm for Monotonic Optimization2) Network Utility Maximization3) Internet Congestion Control Problem
Network Utility Maximization● The framework of Network Utility Maximization
(NUM) has found many applications in network rate allocation algorithms and Internet Congestion Control Protocols.
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 U
s(x
s). 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.
Network Utility Maximization
Mathematically,
Network Utility Maximization● Concave Utilities Follows from Law of
Diminishing Marginal Utilities. Convex Optimization Problem.
● U(x) = log (1+x)
x
U(x)
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.
Network Utility Maximization● NonConcave Utilities – In multimedia applications
on Internet, the utilities are nonconcave. Nonconvex optimization problem.
● U(x) = (1 + eax+b) 1
x
U(x)
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.
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.
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.
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.
Idea and motivation
Idea and motivation
Idea and motivation
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.
Introduction
Not so much related topics:1) Polyblock Algorithm for Monotonic Optimization2) Network Utility Maximization3) Internet Congestion Control Problem
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.
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 be the desired rate of inelastic flows, m be the αnumber of inelastic flows and n be the number of elastic flows.
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 , then the fair share is used, αbut when the fair share is greater than , the αinelastic flow would still consume .α
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 n + (m1) <=1ε α
● Here represents the minimum rate admission εcontrol scheme tries to leave for elastic traffic. Depending upon , we can have two cases:α
Aggressive Admission Control● <<< – The arriving flow is admitted as long as it ε α
is possible to allocate to it the desired rate of , even αif this means all elastic flows have to run at their minimum rate of .ε
● So, an inelastic flow is admitted iff (m+1) ≤ 1 and αan elastic flow is always admitted.
Fair Admission Control
● = – The arriving flow is admitted as long as its ε αdesired rate is no greater than the prevailing fair share for each elastic flow.
● So, an inelastic flow is admitted iff (m+n+1) ≤ 1 αand 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.
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.
Thank You...........
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