1 Linear Programming Jeff Edmonds York University COSC 3101 Lecture 5 Def and Hot Dog Example Network Flow Def nNetwork Flow Def n Matrix View of Linear.

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Linear Programming

Jeff Edmonds York UniversityCOSC 3101Lecture 5

• Def and Hot Dog Example• Network Flow Defn

• Matrix View of Linear Programming• Hill Climbing Simplex Method• Dual Solution Witness to Optimality• Define Dual Problem• Buy Fruit or Sell Vitamines Duality• Primal-Dual Hill Climbing

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Linear Programming• Linear Program: An

optimization problem whose constraints and cost function are linear functions

• Goal: Find a solution which optimizes the cost.

E.g.Maximize Cost Function : 21x1 - 6x2 – 100x3 - 100x4

Constraint Functions:5x1 + 2x2 +31x3 - 20x4 211x1 - 4x2 +3x3 + 10x1 ³ 566x1 + 60x2 - 31x3 - 15x4 200…..

Applied in various industrial fields: Manufacturing, Supply-Chain, Logistics, Marketing…

To save money and increase profits !

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A Hotdog

A combination of pork, grain, and sawdust, …

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Constraints: • Amount of moisture• Amount of protein,• …

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The Hotdog Problem

Given today’s prices,what is a fast algorithm to find the cheapest hotdog?

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Abstract Out Essential Details

Cost: 29, 8, 1, 2

Amount to add: x1, x2, x3, x4

pork

grai

nw

ater

saw

dust

3x1 + 4x2 – 7x3 + 8x4 ³ 122x1 - 8x2 + 4x3 - 3x4 ³ 24

-8x1 + 2x2 – 3x3 - 9x4 ³ 8x1 + 2x2 + 9x3 - 3x4 ³ 31

Constraints: • moisture• protein,• …

29x1 + 8x2 + 1x3 + 2x4Cost of Hotdog:

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3x1 + 4x2 – 7x3 + 8x4 ³ 122x1 - 8x2 + 4x3 - 3x4 ³ 24

-8x1 + 2x2 – 3x3 - 9x4 ³ 8x1 + 2x2 + 9x3 - 3x4 ³ 31

29x1 + 8x2 + 1x3 + 2x4

Subject to:

Minimize:

Abstract Out Essential Details

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Network Flow as a Linear Program

• Given an instance of Network Flow: <G,c<u,v>> express it as a Linear Program: • The variables: • Maximize:• Subject to:

Flows f<u,v> for each edge.

<u,v>: F<u,v> c<u,v>. (Flow can't exceed capacity)v: u F<u,v> = w F<v,w> (flow in = flow out)

rate(F) = u F<u,t> - v F<t,v>

Mi,j XjNi³

subject to

Linear Programming Linear Program

Minimize: CTXSubject to: MX ³ N

• n variable xj that we are looking for values of.• An optimization function

– Each has a variable has a coefficient cj.– The dot product CTX gives one value to minimize.

• m constraints:– Some linear combination of the variables

must be at least some set value.– i MiX ³ Ni

• Generally implied that variables are positive.

Xj

Cj

minimize

Xj ³ 0

Mi,j XjNi³

subject to

Linear Programming Linear Program

Minimize: CTXSubject to: MX ³ N

• These are the linear programs in “standard” form– Minimize CTX hence X subject to X ³ – Maximize CTX hence X subject to X

• But you could mix and match.

Xj

Cj

minimize

Mi,j XjNi

subject to Maximize: CTX

Subject to: MX ³ N Xj

Cj

maximize

³=

=

Simplex Algorithm• Invented by George Dantzig in 1947• A hill climbing algorithm

Local Max

Global Max

Simplex Algorithm• Invented by George Dantzig in 1947• A hill climbing algorithm • Guaranteed to find an global optimal solution for Linear

Programs

Global Max

Simplex Algorithm• Computes solution to a Linear Program by evaluating

vertices where constraints intersect each other.• Worst case exponential time.• Practically very fast.• Ellisoid algorithm (1979)

First poly time O(n4L) algorithm.

C1C2(0,0)

x2

x1

Minimize

Cost = 5x1 + 7x2

Constraint Functions

C1: 2x1 + 4x2 ³ 100

C2: 3x1 + 3x2 ³ 90

x1, x2 ³ 0

• With n variables, x1, x2, … , xn, there are n dimensions. (Here n=2)

• Each constraint is an n-1 dimensional plain. (Here the 1-dim line)

• Each point in this space, is a solution.

• It is a valid solution if it is on the correct side of each constraint plain

Simplex Algorithm

C1C2(0,0)

x2

x1

Cost Function

Cost = 5x1 + 7x2


C1: 2x1 + 4x2 ³ 100

C2: 3x1 + 3x2 ³ 90

x1, x2 ³ 0

• Solutions on this line have one value of the objective function

• This has another

• These are not valid

• This is the optimal value

Simplex Algorithm

C1C2(0,0)

x2

x1

Cost Function

Cost = 5x1 + 7x2


C1: 2x1 + 4x2 ³ 100

C2: 3x1 + 3x2 ³ 90

x1, x2 ³ 0

• The arrow tells the direction that the optimal function increases.

• Note that the solution is a vertex (simplex).

• Each simplex is the intersection of n constraints. (Here n = #of variables = 2)

Simplex Algorithm

• The arrow tells the direction that the optimal function increases.

• Note that the solution is a vertex (simplex).

• Each simplex is the intersection of n constraints. (Here n = #of variables = 2)

• The simplex method takes hill climbing steps, from one simplex (valid solution) to another.

Simplex Algorithm

Maximize Cost : 21x1 - 6x2 – 100x3

Constraint Functions:5x1 + 2x2 +31x3 211x1 - 4x2 +3x3 566x1 + 60x2 - 31x3 200 ⁞-5x1 + 3x2 +4x3 8 ⁞x1, x2, x3 ³ 0

• With n variables, x1, x2, x3 … , xn, there are n dimensions. (Here n=3)

• Each constraint is an n-1 dimensional plain. (Here the 2-dim triangles)

• Each simplex (vertex) is the intersection of n such constraints. (Here looks like 6 but generally only n=3)

Simplex Algorithm

A simplex is specifiedby a subset of n of the m tight (=) constraints.

All other constraints must be satisfied.



• If we slacken one of our n tight constrains,

our solution slides along a 1-dim edge.

• Head in the direction that increases the potential function.

Simplex Algorithm







• Keep sliding until we tighten some constraint.

• This is one hill climbing step.

Simplex Algorithm







• Keep sliding until we tighten some constraint.

• This is one hill climbing step.

Simplex Algorithm


• Do all Linear Programs have optimal solutions ? No !

• Three types of Linear Programs:

1. Has an optimal solution with a finite cost value: e.g. nutrition problem

2. Unbounded: e.g maximize x, x 5, ³ x 0³

3. Infeasible: e.g maximize x, x 3, x 5 , ³ x 0³

Simplex Algorithm

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Primal

Dual

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Hill ClimbingWe have a valid solution.(not necessarily optimal)

Take a step that goes up.

measure

progress

Value of our solution.

Problems:

Exit Can't take a step that goes up.

Running time?

Initially have the “zero

Local Max

Global Max

Can our Network Flow Algorithm get stuck in a local maximum?

Make small local changes to your solution toconstruct a slightly better solution.

If you take small step,could be exponential time.

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Hill ClimbingAvoiding getting stuck

in a local maximum

Good ExecutionBad Execution

• Made better choices of direction• Hard

• Back up an retry• Exponential time

• Define a bigger step

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Network Flow

Can our Simplex Algorithm get stuck in local max?

Need to prove for every linear programfor every choice of steps an optimal solution is found!

No!

How?

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Primal-Dual Hill Climbing

Mars settlement has hilly landscapeand many layers of roofs.

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Primal-Dual Hill ClimbingPrimal Problem: • Exponential # of locations to stand.• Find a highest one.Dual problem:• Exponential # of roofs.• Find a lowest one.

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Primal-Dual Hill ClimbingProve:• Every roof is above every location to stand. R L height(R) height(L) height(Rmin) height(Lmax) Is there a gap?

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Prove:• For every location to stand either:• the alg takes a step up or• the alg gives a reason that explains why not

by giving a ceiling of equal height. i.e. L [ L’ height(L’) height(L) or R height(R) = height(L)]

or

But R L height(R) height(L)

No Gap

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or

Can't go up from this location and no matching ceiling.

Can't happen!

?

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or

No local maximum!

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Claim: Primal and dual have the same optimal value. height(Rmin) = height(Lmax)Proved: R L, height(R) height(L) Proved: Alg runs until it provides Lalg and Ralg

height(Ralg) = height(Lalg)

No Gap

height(Rmin) height(Ralg) = height(Lalg) height(Lmax)height(Rmin) height(Lmax)

Lalg witness that height(Lmax) is no smaller. Ralg witness that height(Lmax) is no bigger.

Exit

Mi,j XjNi³

subject to Linear Program Minimize: CTX

Subject to: MX ³ N

• n variable xj that we are looking for values of.• An optimization function

– Each has a variable has a coefficient cj.– The dot product CTX gives one value to minimize.

• m constraints:– Some linear combination of the variables

must be at least some set value.– i MiX ³ Ni

• Generally implied that variables are positive.

Xj

Cj

minimize

Xj ³ 0

Duality

Mi,j XjNi³

subject to Linear Program Minimize: CTX

Subject to: MX ³ N

• These are the linear programs in “standard” form– Minimize CTX hence X subject to X ³ – Maximize CTX hence X subject to X

• But you could mix and match.

Xj

Cj

minimize

Mi,j XjNi

subject to Maximize: CTX


Cj

maximize

³=

=

Duality

?

Duality

Dual Linear Program Ni

For every primal linear program, we define its dual linear program.

Everything is turned upside down.• The matrix of coefficients is transposed• For each constraint, a variable.

– Form the objective function vector from the constraint vector.– Generally, the constraint is ‘’ and then the variable is Yi 0

MTj,i

subject to

Yi

Mi,j XjNi³

subject to Primal Linear Program Minimize: CTX


Cj

minimize

Yi

Yi ³ 0

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Duality



Everything is turned upside down.• The matrix of coefficients is transposed• For each constraint, a variable.

– Form constraint vector from the objective function vector.– Generally, the constraint is ‘’ and then the variable is Yi 0– But if it is ‘=’, then the variable Yi is unconstrained.

MTj,i

subject to

Yi

Mi,j XjNi³



Cj

minimize

Yi

Yi

>=<

0

=

?

Duality


Cj


Everything is turned upside down.• The matrix of coefficients is transposed• For each constraint, a variable.• For each variable, a constraint.

– Form constraint vector from the objective function vector.• Max Min and

maximize NT.yMT.y CT MT

j,i

subject tomaximize

Yi

Mi,j XjNi³



Cj

minimize

Yi

Duality


Cj



j,i

subject tomaximize

Yi

Mi,j XjNi³



Cj

minimize

Yi

Everything is turned upside down.• Max Flow Min Cut• Buyer of nutrients Seller of nutrients in fruit in vitamins

Duality


Cj



j,i

subject tomaximize

Yi

Mi,j XjNi³



Cj

minimize

Yi

Every solution X of the primal is above every solution Y of the primal. X Y CTX NTY CXmin NYmax We will prove equality.

Dual of the dual is itself!

The Nutrition Problem• Each fruit

contains different nutrients

• Each fruit has different cost

An apple a day keeps the doctor away – but apples are costly!

A customer’s goal is to fulfill daily nutrition requirements at lowest cost.

The Nutrition Problem (cont’d)

• Let’s take a simpler case of just apples and bananas.

• Must take at least 100 units of Calories & 90 units of Vitamins for good nutrition.

• A customer’s goal is to buy fruits in such a quantity that it minimizes cost but fulfills nutrition.

Calories Vitamins Cost($)

2 3 5

4 3 7

Cost Function

Cost = 5x1 + 7x2


C1: 2x1 + 4x2 ³ 100

C2: 3x1 + 3x2 ³ 90

x1, x2 ³ 0

The Nutrition Problem (cont’d)

• Matrix Representation

Constraints:

2x1 + 4x2 ³ 100

3x1 + 3x2 ³ 90

Non-negativity: x1, x2 ³ 0

Cost function = 5x1 + 7x2

Real life problems may have many variables and constraints !

Cj

XjNiMi,j ³Xj

Semantics of Duality

A customer’s goal is to buy fruits in such a quantity that it minimizes cost but fulfills nutrition.

Cj

XjNiMi,j ³Xj

Primal LP: minimize C.x

Q.x ³ N

Coefficients in each column represent the amount of nutrients in a particular food

Cost of each fruitDaily nutrition

Quantity of each fruit

Semantics of Duality

Dual LP:maximize NT.y

QT.y CT

Coefficients in each row represent the amount of nutrients in a particular fruit

NiYi CjMj,i

Yi

But what are Yis in the dual ? Price of

each nutrient!

Daily nutritionCost of each fruit

Imagine a salesman trying to sell supplements for each fruit.

Semantics of Duality• Primal Problem: A customer’s goal is to buy fruits in such

a quantity that it minimizes cost but fulfills nutrition.

• Dual Problem: A salesman goal is to set a price on each nutrient, so that it maximizes profit but his supplements are cheaper than fruits. (Otherwise who will buy them?!)

Primal (Customer)

Dual (Salesman)

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Prove:• For every solution x of the primal either:• the alg takes a step up to a better primal• the alg gives a reason that explains why not

by giving a solution y of the dual of equal value. i.e. x [ x’ CTx’ > CTx or y CTx = NTy]

or

But x y Cx Ny

No Gap

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or

Can't go up from this location and no matching ceiling.

Can't happen!

?



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or

No local maximum!



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Claim: Primal and dual have the same optimal value. CTxmin = NTymax

No Gap

xalg witnesses that CTxmin is no smaller. yalg witnesses that CTxmin is no bigger.

Exit



• Given any solution x of the primal.Simplex

Algorithm


All other constraints must be satisfied.



• Given any solution x of the primal.

• If we slacken one of our n tight constrains, our solution slides along a 1-dim edge.


Simplex Algorithm







• Keep sliding until we tighten some constraint. Giving new solution x’

Simplex Algorithm




Simplex Algorithm





• Keep sliding until we tighten some constraint. Giving new solution x’



Simplex Algorithm



• Step to another x OR

• None of these “steps” increases the potential function.• the alg gives a reason that explains why not

by giving a solution y of the dual of equal value.



Simplex Algorithm


But practically it tends to be fast.

Bread and butter of optimization in industry.

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Thank You!

Questions ?

End

• Do all Linear Programs have optimal solutions ? No !

• Three types of Linear Programs:

1. Has an optimal solution with a finite cost value: e.g. nutrition problem

2. Unbounded: e.g maximize x, x 5, ³ x 0³

3. Infeasible: e.g maximize x, x 3, x 5 , ³ x 0³

Simplex Algorithm

Simplex

C1C2(0,0)

y

x

Cost Function

P = 5x + 7y


C1: 2x + 4y 100

C2: 3x + 3y 90

Non-Negativity: x,y ³ 0

Recall we need to evaluate our cost

function at the vertices where the constraint

functions intersect each other

Simplex (cont’d)

Our Equations

P = 5x + 7y

C1: 2x + 4y 100

C2: 3x + 3y 90

x,y ³ 0

Slack Form

Can be re-written as:

P = 5x + 7y

s1 = 100 - 2x - 4y

s2 = 90 - 3x - 3y

x, y ³ 0

s1,, s2 0³We introduce 2 new

variables called slack variables

We don’t want to deal with complex inequalities

Simplex (cont’d)Cost Function

P = 5x + 7y

s1 = 100 - 2x - 4y

s2 = 90 - 3x - 3y

s1, , s2 , x , y ³ 0

STEP 1:

• We want an initial point

• Let’s put x=0, y=0

Feasible solutionx=0, y=0

P = 0


P = 5x + 7y

s1 = 100 - 2x - 4y

s2 = 90 - 3x - 3y

s1, , s2 , x , y ³ 0

STEP 2:

• We want next point

• Let's try to increase x.

• x can be increased maximum to 30 (s2 becomes zero)

• Rewrite equations (Pivoting)

• Now put y, s2 = 0

Feasible solution

x = 30 – y – s2/3

s1 = 40 + 2/3s2 – 2y

P = 150 – 5/3s2 + 2y

x=30, y=0

P = 150


P = 150 – 5/3s2 + 2y

x = 30 – y – s2/3

s1 = 40 + 2/3s2 – 2y

s1, , s2 , x , y ³ 0

STEP 3:

• We want next point

• Let's try to increase y.

• y can be increased maximum to 20 (s1 becomes zero)

• Rewrite equations (Pivoting)

• Now put s1, s2 = 0

y = 20 + 1/3s2 – 1/2s1

x = 10 – 1/2s1 - 2/3s2

P = 190 - s1 – s2

x=10, y=20

P = 190

(We don’t increase s2 because it will decrease

P)

Note that we cannot increase s1 & s2 without

decreasing P. So we stop !

Feasible solution

Is this solution optimal? Or have

we run into a local minimum?

?

1 Linear Programming Jeff Edmonds York University COSC 3101 Lecture 5 Def and Hot Dog Example Network Flow Def nNetwork Flow Def n Matrix View of Linear.

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