Introduction to Network Mathematics (1)

- Optimization techniques

Yuedong Xu10/08/2012

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

reverse-engineer 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

10

1)log(

i ii ax

10

11,0

i ixx

Optimal solution ?

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 fi(x) ≤ 0; i=1,…,m hi(x) = 0; i=1,…,p

Objective function

x: Decision variables

Inequality constraint

Equality constraint

Overview

• Optimization tree

Overview

• Our coverage

NonlinearPrograms

ConvexPrograms

LinearPrograms

(Polynomial) IntegerPrograms(NP-Hard)

Flowand

Matching

Outline

• Toy Examples• Overview• Convex Optimization• Linear Programming• Linear Integer Programming

Convex Optimization

• Concepts– Convex combination:

x, y Rn

0 1 z = x +(1)yA convex combination of x, y.

A strict convex combination of x, y if 0, 1.

Convex Optimization

• Concepts– Convex set:

S Rn is convex if it contains all convex combinations of pairs x, y S.

convex 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！ )

c

x yx +(1)y

c(x)

c(y)c(x) + (1)c(y)

c(x +(1)y)

S Rn a convex set

c:S R a convex function if

c(x +(1)y) c(x) + (1)c(y), 0 1

Convex Optimization

• Concepts – convex functions

We are lucky to find that many networking problems have convex obj.s

• 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

Convex Optimization

• Concepts– Convex Optimization:

– A fundamental property: local optimum = global optimum

Convex

Convex

Linear/Affine

Minimize f0(x)

subject to fi(x) ≤ 0; i=1,…,m hi(x) = 0; i=1,…,p

Convex Optimization

• Method– You may have used• Gradient method• Newton methodto 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 OptimizationExample: Minimize x2 + y2

subject to 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 = 1Lagrangian: 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 algorithmsAn 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 SlacknessTo 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

NetworkLinks l of capacities cl

Sources iL(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

c1 c2

x1

x2

x3

121 cxx 231 cxx

Convex Optimization

• Application: TCP Flow Control– Primal problem

Llcy

xU

ll

iii

xi

, subject to

)( max0

Convex Optimization

• Application: TCP Flow Control– Lagrangian dual problem

)( )( max )( min

)( )( ),(

00: problem Dual

:function Dual

ll

ll

sss

xp

ll

ll

sss

xcpxUpD

xcpxUpxD

s

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

Link updates congestion signal

))(( )1( : source 1' tqUtx iii

)])(()([ )1( :link lllll ctytptp

Price of congestion at a link: packet drop prob., etc.

Total prices of a flowalong its route

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

– Esp. max-min fairness as alpha infinity

otherwise log

,1 if 1

1

xwxU

xwxU

ss

ss

Convex Optimization

• Application: TCP Flow ControlTake 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 1 1 2 2 n nc x c x c x

Subject to

11 1 12 2 1 1

1 1 2 2

n n

m m mn n m

a x a x a x b

a x a x a x b

11 1 12 2 1 1

1 1 2 2

n n

m m mn n m

a x a x a x b

a x a x a x b

……

Linear Programming

• Formulation– General form

Max or min 1 1 2 2 n nc x c x c x

Subject to

11 1 12 2 1 1

1 1 2 2

n n

m m mn n m

a x a x a x b

a x a x a x b

11 1 12 2 1 1

1 1 2 2

n n

m m mn n m

a x a x a x b

a x a x a x b

……

A linear objective functionA linear objective function

A set of linear constraints

• linear inequalities or

• linear equations

Linear Programming

• Formulation– Canonical form

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

Min 1 1 2 2 n nc x c x c x

1 2, , , 0nx x x

subject to

“greater than” only

“greater than” only

Non-negativeNon-negative

Linear Programming

• Example

Maximize 3 2x y

Subject to 3 12

8

2 10

0

0

x y

x y

x y

x

y

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

312

xy

8

xy

2

10

xy

x

y

Linear Programming

• Example

Maximize 3 2x y

Subject to 3 12

8

2 10

0

0

x y

x y

x y

x

y

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

312

xy

8

xy

2

10

xy

x

y

32

0

xy

32

6

xy

32

8

xy

32

15

xy

32

19

xy

32

22

xy

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.

– Local optimum = global optimum (convexity)

basicfeasiblesolution

at a corner

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.

Guaranteed “feasible” solution Complexity grows exponentially

x1=0

X2=0 X2=2X2=1

x1=1 x1=2

X2=0 X2=2X2=1X2=0 X2=2X2=1

Linear Integer Programming

• Method– Linear relaxation plus rounding

-cT

x1

x2

LP Solution

Integer Solution

a). Variables being continuousb). Solving LP efficientlyc). Rounding the solution

No guarantee of gap to optimality!

Linear Integer Programming

• Combined Method– Branch and bound algorithm

Step 1: Solve LP relaxation problemStep 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

1 {0,1}x

2 {0,1}x

3 {0,1}x

5 {0,1}x

4 {0,1}x

Objective Function

Unknowns or Variables

Constraints

's, 1, ,5ix i

, ,0,1 1, 5ix i

1 2 3 4 54 2 10 2 1

Maximize

x x x x x

1 2 3 4 512 1

Subject t

154 2

o

1x x x x x

Linear Integer Programming

• Example – Knapsack problem

1 2 3 4 54 2 10 2Maxim 1ize x x x x x

1 2 3 4 512 1 4 2Subject to 15,

0,1 , 1, ,5

1

i

x x x x x

x i

Is the problem difficult?Is the problem difficult?

What is the complexity in your approach?What is the complexity in your approach?

NP hard!NP hard!

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 writeThe Lagrangian dual functionOn 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

Min f0(x)

s.t. fi(x) ≤ 0; i=1,…,m hi(x) = 0; i=1,…,p

Primal: Dual:

),,( xLMin

If λ ≥ 0, dual minimum is lower bound of primal minimum

What shall we do next?

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!