Introduction to Network
Mathematics (1)
- Optimization techniques
Yuedong Xu
10/08/2012
Purpose
• Many networking/system problems boil
down to optimization problems
– Bandwidth allocation
– ISP traffic engineering
– Route selection
– Cache placement
– Server sleep/wakeup
– Wireless channel assignment
– … and so on (numerous)
Purpose
• Optimization as a tool
– shedding light on how good we can achieve
– guiding design of distributed algorithms
(other than pure heuristics)
– providing a bottom-up approach to reverseengineer existing systems
Outline
•
•
•
•
•
•
Toy Examples
Overview
Convex Optimization
Linear Programming
Linear Integer Programming
Summary
Toys
• Toy 1: Find x to minimize f(x) := x2
.
x*=0 if no restriction on x
.
x*=2 if 2≤x≤4
Toys
• Toy 2: Find the minimum
minimize


subject to x  0 ,
10
i 1
log( x i  a i )

Optimal solution ?
10
i 1
xi  1
Toys
• Toy 3: Find global minimum
local minimum
global minimum
Outline
•
•
•
•
•
Toy Examples
Overview
Convex Optimization
Linear Programming
Linear Integer Programming
Overview
Ingredients!
• Objective Function
– A function to be minimized or maximized
• Unknowns or Variables
– Affect value of objective function
• Constraints
– Restrict unknowns to take on certain values
but exclude others
Overview
• Formulation:
Minimize f0(x)
subject to
Objective function
fi(x) ≤ 0; i=1,…,m Inequality constraint
hi(x) = 0; i=1,…,p Equality constraint
x: Decision variables
Overview
• Optimization tree
Overview
• Our coverage
Nonlinear
Programs
Convex
Programs
Linear
Programs
(Polynomial)
Flow
and
Matching
Integer
Programs
(NP-Hard)
Outline
•
•
•
•
•
Toy Examples
Overview
Convex Optimization
Linear Programming
Linear Integer Programming
Convex Optimization
• Concepts
– Convex combination:
x, y  Rn
01
z = x +(1)y
A convex combination of x, y.
A strict convex combination
of x, y if   0, 1.
Convex Optimization
• Concepts
– Convex set:
S
n
R
convex
is convex if it contains all convex
combinations of pairs x, y  S.
nonconvex
Convex Optimization
• Concepts
– Convex set: more complicated case
The intersection of any number of convex sets is convex.
Convex Optimization
• Concepts
– Convex function:(大陆教材定义与国外相反！)
S  Rn
c:S  R
a convex set
a convex function if
c(x +(1)y)  c(x) + (1)c(y), 0    1
c
c(x) + (1)c(y)
c(y)
c(x)
x
c(x +(1)y)
x +(1)y
y
Convex Optimization
• Concepts – convex functions
• Exponential: eax is convex on R
• Powers:
xa is convex on R+ when a≥1 or
a≤0, and concave for 0≤a≤1
• Logarithm: log x is concave on R+
• Jensen’s inequality:
– if f() is convex, then f(E[x]) <= E[f(x)]
– You can also check it by taking 2nd order derivatives
We are lucky to find that many networking problems have convex obj.s
Convex Optimization
• Concepts
– Convex Optimization:
Minimize f0(x)
subject to
fi(x) ≤ 0; i=1,…,m
hi(x) = 0; i=1,…,p
Convex
Convex
Linear/Affine
– A fundamental property: local optimum =
global optimum
Convex Optimization
• Method
– You may have used
• Gradient method
• Newton method
to find optima where constraints are real explicit
ranges (e.g. 0≤x≤10, 0≤y≤20, -∞≤z≤∞ ……).
– Today, we are talking about more
generalized constrained optimization
Convex Optimization
How to solve a constrained optimization problem?
• Enumeration? Maybe for small, finite feasible
set
• Use constraints to reduce number of
variables? Works occasionally
• Lagrange multiplier method – a general
method (harder to understand)
Convex Optimization
Example:
Minimize
subject to
x2 + y2
x+y=1
Lagrange multiplier method:
• Change problem to an unconstrained problem:
L(x, y, p) = x2 + y2 + p(1-x-y)
• Think of p as “price”, (1-x-y) as amount of “violation”
of the constraint
• Minimize L(x,y,p) over all x and y, keeping p fixed
• Obtain x*(p), y*(p)
• Then choose p to make sure constraint met
• Magically, x*(p*) and y*(p*) is the solution to the
original problem!
Convex Optimization
Example:
Minimize x2 + y2
subject to x + y = 1
Lagrangian:
L(x, y, p) = x2 + y2 + p(1-x-y)
• Setting dL/dx and dL/dy to 0, we get
x = y = p/2
• Since x+y=1, we get p=1
• Get the same solution by substituting y=1-x
Convex Optimization
• General case:
minimize
f0(x)
subject to fi(x) ≤ 0, i = 1, …, m
(Ignore equality constraints for now)
Optimal value denoted by f*
Lagrangian:
L(x, p) = f0(x) + p1f1(x) + … + pmfm(x)
Define
g(p) = infx (f0(x) + p1f1(x) + … + pmfm(x))
If pi≥0 for all i, and x feasible, then g(p) ≤ f*
Convex Optimization
• Revisit earlier example
L(x,p) = x2 + y2 + p(1-x-y)
x* = y* = p/2
g(p) = p(1-p/2)
This is a concave function, with g(p*) = 1/2
• We know f* is 1/2, and g(p) is a lower bound
for f* with different values of p – the
tightest lower bound occurs at p=p*.
Convex Optimization
• Duality
• For each p, g(p) gives a lower bound for f*
• We want to find as tight a lower bound as possible:
maximize g(p)
subject to p≥0
a) This is called the (Lagrange) “dual” problem, original
problem the “primal” problem
b) Let the optimal value of dual problem be denoted d*.
We always have d* ≤ f* (called “weak duality”)
c) If d* = f*, then we have “strong duality”. The
difference (f*-d*) is called the “duality gap”
Convex Optimization
• Price Interpretation
• Solving the constrained problem is
equivalent to obeying hard resource limits
• Imaging the resource limits can be violated;
you can pay a marginal price per unit
amount of violation (or receive an amount if
the limit not met)
• When duality gap is nonzero, you can
benefit from the 2nd scenario, even if the
price are set in unfavorable terms
Convex Optimization
• Duality in algorithms
An iterative algorithm produces at iteration j
• A primal feasible x(j)
• A dual feasible p(j)
With f0(x(j))-g(p(j)) -> 0 as j -> infinity
The optimal solution f* is in the interval
[g(p(j)), f0(x(j))]
Convex Optimization
• Complementary Slackness
To make duality gap zero, we need to have
pi fi(x) = 0
for all i
This means
pi > 0 => fi(x*) = 0
fi(x*) < 0 => pi = 0
If “price” is positive, then the corresponding
constraint is limiting
If a constraint is not active, then the “price”
must be zero
Convex Optimization
• KKT Optimality Conditions
• Satisfy primal and dual constraints
fi(x*) ≤0, pi* ≥0
• Satisfy complementary slackness
pi* fi(x*) = 0
• Stationary at optimal
f0’(x*) + Σi pi* fi’(x*) = 0
If primal problem is convex and KKT conditions met,
then x* and p* are dual optimal with zero duality gap
KKT = Karush-Kuhn-Tucker
Convex Optimization
• Method
– Take home messages
• Convexity is important (non-convex problems
are very difficult)
• Primal problem is not solved directly due to
complicated constraints
• Using Lagrangian dual approach to obtain dual
function
• KKT condition is used to guarantee the strong
duality for convex problem
Convex Optimization
• Application: TCP Flow Control
– What’s in your mind about TCP?
• Socket programming?
• Sliding window?
• AIMD congestion control?
• Retransmission?
• Anything else?
Convex Optimization
• Application: TCP Flow Control
– Our questions:
• It seems that TCP works as Internet scales.
Why?
• How does TCP allocate bandwidth to flows
traversing the same bottlenecks?
• Is TCP optimal in terms of resource allocation?
Convex Optimization
• Application: TCP Flow Control
Network
Links l of capacities cl
Sources i
L(s) - links used by source i (routing)
Ui(xi) - utility if source rate = xi
i). The larger rate xi, the more happiness;
ii). The increment of happiness is shrinking as xi
increases.
Convex Optimization
• Application: TCP Flow Control
– A simple network
x1
x1  x 3  c 2
x1  x 2  c1
c1
c2
x2
x3
Convex Optimization
• Application: TCP Flow Control
– Primal problem
max
xi  0
subject to
 U (x )
i
i
i
yl  cl ,
l L
Convex Optimization
• Application: TCP Flow Control
– Lagrangian dual problem
Dual
function
:
D ( x, p ) 
 U s ( x s )   pl (cl  x )
l
s
Dual
problem :
min
p0

D ( p )   max
 xs 0
l
l 
U
(
x
)

p
(
c

x
 s s  l l )
s
l

You can solve it using KKT conditions!
You need whole information!
Convex Optimization
• Application: TCP Flow Control
– Question:
Is distributed flow control possible without
knowledge of the network?
Convex Optimization
• Application: TCP Flow Control
– Primal-dual approach:
• Source updates sending rate xi
• Link generates congestion signal
– Source’s action:
• If congestion severe, then reduce xi
• If no congestion, then increase xi
– Congestion measures:
• Packet loss (TCP Reno)
• RTT
(TCP Vegas)
Convex Optimization
• Application: TCP Flow Control
– Gradient based Primal-dual algorithm:
Source updates rate given congestion signal
source :
'1
x i ( t  1)  U i ( q i ( t ))
Total prices of a flow
along its route
Link updates congestion signal
link :
p l ( t  1)  [ p l ( t )   l ( y l ( t )  c l )]

Price of congestion at a link: packet drop prob., etc.
Convex Optimization
• Application: TCP Flow Control
– Relation to real TCP
Convex Optimization
• Application: Fairness concerns
– Many concepts regarding fairness
• Max-min, proportional fairness, etc.
• (add some examples to explain the concepts)
Convex Optimization
• Application: Fairness concerns
– How does fairness relate to optimization?
Reflected by utility function
U s x   ws
x
1 
1
U s  x   w s log  x 
if   1,
otherwise
– Esp. max-min fairness as alpha infinity
Convex Optimization
• Application: TCP Flow Control
Take home messages:
– TCP is solving a distributed optimization
problem!
– Primal-dual algorithm can converge!
– Packet drop, and RTT are carrying “price of
congestion”!
– Fairness reflected by utility function
Outline
•
•
•
•
•
•
Toy Examples
Overview
Convex Optimization
Linear Programming
Linear Integer Programming
Summary
Linear Programming
• A subset of convex programming
• Goal: maximize or minimize a linear
function on a domain defined by linear
inequalities and equations
• Special properties
Linear Programming
• Formulation
– General form
Max or min
c1 x1
Subject to
a11 x1
a11 x1
 c2 x2
 a12 x 2
 a12 x 2
 cn xn


 a1 n x n
 a1 n x n
 
  
   b1
  b1
 
 
 
 amn xn
 amn xn
 
  
   b m
  b m
 
 
……

a m 1 x1
a m 1 x1
 am 2 x2
 am 2 x2


Linear Programming
• Formulation
– General form
Max or min
cA1 xlinear
 c 2objective
x2
 function
 cn xn
1
Subject to
a11 x1  a12 x 2

 a1 n x n
aA
x1 of
 alinear
x

 a1 n x n
constraints
11 set
12 2
……
• linear inequalities or
linear
a•
x
 aequations
x

m1 1
m2 2
a m 1 x1  a m 2 x 2 
 amn xn
 amn xn
 
  
   b1
  b1
 
 
 
 
  
   b m
  b m
 
 
Linear Programming
• Formulation
– Canonical form
Min
subject to
“greater than”
only
 c2 x2

 cn xn
a11 x1
 a12 x 2

 a1 n x n
 b1
a 21 x1
 a 22 x 2

 a2 n xn
 b2
a m 1 x1
 am 2 x2

 amn xn
 bm
c1 x1
Non-negative
x1 , x 2 ,
, xn  0
Linear Programming
• Example
y
Maximize
3x

2y
Subject to
x

3y

12
x

y

8
2x

y

10
x

0
y

0
8
7
6
x

 12
y
3
x
y
1
0
5

y
4

8
2x
3
2
1
x
1
2
3
4
5
6
7
8
Linear Programming
• Example
3x

2y
Subject to  x

3y

12
x

y

8
2x

y

10
x

0
y

0
8
7
6
x
19
y
y
2
8
y
2
15
y
2
3x
3x
1

2
2
3x
3
22
y 8
2 y
x
3x
3x
4
y
1
0
3x
5
2

 12
y
3
2x
Maximize
y
6
y
0
1
x
2
3
4
5
6
7
8
Linear Programming
• Example
– Observations:
• Optimum is at the corner!
– Question:
• When can optimum be
a point not at the corner?
• How to find optimum in a
more general LP problem?
feasible region
Linear Programming
(Dantzig 1951) Simplex method
•For USA Air Force
•Very efficient in practice
• Exponential time in worst case
(Khachiyan 1979) Ellipsoid method
• Not efficient in practice
• Polynomial time in worst case
Linear Programming
• Simplex method
– Iterative method to find optimum
– Finding optimum by moving around the
corner of the convex polytope in cost
descent sense.
basic
feasible at a corner
solution
– Local optimum = global optimum (convexity)
Linear Programming
• Application
– Max flow problem
Outline
•
•
•
•
•
•
Toy Examples
Overview
Convex Optimization
Linear Programming
Linear Integer Programming
Summary
Linear Integer Programming
• Linear programming is powerful in many
areas
• But ………
Linear Integer Programming
• Random variables are integers
– Building a link between two nodes, 0 or 1?
– Selecting a route, 0 or 1?
– Assigning servers in a location
– Caching a movie, 0 or 1?
– Assigning a wireless channel, 0 or 1?
– and so on……
No fractional solution !
Linear Integer Programming
• Graphical interpretation
Only
discrete
values
allowed
Linear Integer Programming
• Method
– Enumeration – Tree Search, Dynamic
Programming etc.
x1=0
X2=0
X2=1
X2=2
x1=2
x1=1
X2=0
X2=1
X2=2
X2=0
Guaranteed “feasible” solution
Complexity grows exponentially
X2=1
X2=2
Linear Integer Programming
• Method
– Linear relaxation plus rounding
Integer Solution
-cT
x2
a). Variables being continuous
b). Solving LP efficiently
LP Solution
c). Rounding the solution
x1
No guarantee of gap to optimality!
Linear Integer Programming
• Combined Method
– Branch and bound algorithm
Step 1: Solve LP relaxation problem
Step 2:
Linear Integer Programming
• Example – Knapsack problem
Which boxes should be chosen
to maximize the amount of
money while still keeping the
overall weight under 15 kg ?
Linear Integer Programming
• Example – Knapsack problem
x1  {0,1}

x 4  {0,1}
Objective Function
M axim ize
4 x1  2 x 2  10 x 3  2 x 4  1 x 5

x 2  {0,1}
Unknowns or Variables
x i 's, i  1,

x 5  {0,1}
x 3  {0,1}
,5
Constraints
S ubject t o
12 x1  1 x 2  4 x 3  2 x 4  1 x 5  15
x i   0,1 , i  1,
,5
Linear Integer Programming
• Example – Knapsack problem
M axim ize 4 x1  2 x 2  10 x 3  2 x 4  1 x 5
S ubject to 12 x1  1 x 2  4 x 3  2 x 4  1 x 5  15,
x i   0,1 , i  1,
,5
Outline
•
•
•
•
•
•
Toy Examples
Overview
Convex Optimization
Linear Programming
Linear Integer Programming
Summary
Summary
Our course covers
• a macroscopic view of optimizations in
networking
• ways of solving optimizations
• applications to networking research
Summary
Keywords
• Convex optimization
– Lagrangian dual problem, KKT
• Linear programming
– Simplex, Interior-point
• Integer linear programming
– Dynamic programming, Rounding, or Branch
and bound
Summary
Complexity (“solve” means solving optimum)
• Linear programming
– [P], fast to solve
• Nonlinear programming
– Convex: [P], easy to solve
– Non-convex: usually [NP], difficult to solve
• (Mixed) Integer linear programming
– Usually [NP], difficult to solve
Summary
Thanks!
Backup slides
Convex Optimization
• Concepts
– Convex function: more examples (see Byod’s
book)
We are lucky to find that many networking problems have convex obj.s
Convex Optimization
• Example – waterfilling problem
To Tom: you can write
The Lagrangian dual function
On the white board.
Convex Optimization
• Method
– Roadmap
• Treat the original optimization problem as the
Primal problem
• Transform primal problem into dual problem
• Solve dual problem and validate the optimality
Convex Optimization
• Method
– Lagrangian dual function
Convex Optimization
• Method
– For given (  , ) , solving dual problem
Convex Optimization
• Method
– Comparison
What shall we do next?
Primal:
Dual:
Min f0(x)
s.t.
fi(x) ≤ 0; i=1,…,m
hi(x) = 0; i=1,…,p
Min L ( x ,  , )
If λ ≥ 0, dual
minimum is lower
bound of primal
minimum
Convex Optimization
• Method
– The Lagrangian dual problem
Much easier after getting rid of ugly constraints!
Convex Optimization
• Method
– Question: are these two problems the
same?
Convex Optimization
• Method
– Karush-Kuhn-Tucker (KKT) conditions
For the convex problem, satisfying KKT means strong duality!