lecture 17 - nlp - formulations

8/3/2019 Lecture 17 - NLP - Formulations

1/18

BUS321

Management Science

Lecture

NLP - Modeling Non-linear Progra

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Lecture 17:Non-Linear Programming

Modeling Examples

2

Objectives

Objectives

Know how to model non-linear programs (NLPs)(Ch. 11.2)

Applications of NLP to location problems, portfoliomanagement, and regression


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BUS321

Management Science

Lecture



Linear Programming Model

1 1 2 2

11 1 12 2 1n n 1

21 1 22 2 2n n 2

m1 1 2 2 mn n m

Maximize .....

subject to

a x + a x + ... +a x b

a x + a x + ... +a x b

a x + a x + ... +a x b

n n

m

c x c x c x

x

+ + +

1 2, , ..., 0

nx x

ASSUMPTIONS:

Proportionality Assumption

Objective function

Constraints

Additivity Assumption

Objective function

Constraints

Wha

tis

a

Non-Linear

Prog

ram?

maximize 3 sin x + xy + y3 - 3z + log zSubject to x2 + y2 = 1

x + 4z 2z 0

A non-linear program is permitted to have non-linear constraints or objectives.

A linear program is a special case of non-linearprogramming!


3/18

BUS321

Management Science

Lecture



Non-Linear Programs(NLP)

Nonlinear objective function f(x) and/orNonlinear constraints gi(x).

Today: we will present several types of non-linearprograms.

( )1 2, , ,

( )

( ) , 1, 2, ,

n

i i

Let x x x x

Max f x

g x b i m

=

=

Unconstrained Facility Location

0

2

4

6

8

10

12

14

16

y

0 2 4 6 8 10 12 14 16

C (2)

(7)

B

A (19)

P ?

D (5)

x

Loc. Demand

A: (8,2) 19

B: (3,10) 7

C: (8,15) 2

D: (14,13) 5

P: ?

This is the warehouse location problem with a single

warehouse that can be located anywhere in the plane.


4/18

BUS321

Management Science

Lecture



Costs proportional to distance;known daily demands.

An NLP

2 28 2( ) ( )x y + d(P,A) =

2 214 13( ) ( )x y + d(P,D) =

minimize 19 d(P,A) + + 5 d(P,D)

subject to: P is unconstrained

Herearethe objective valuesfor55

differentlocations.

0

50

100

150

200

250

300

350

values

for y

Objectivevalue

x = 0

x = 2

x = 4

x = 6

x = 8

x = 10

x = 12


5/18

BUS321

Management Science

Lecture



Facil ity Location. What happensif P must be

within aspecified region?

02

4

6

8

10

12

14

16

y

0 2 4 6 8 10 12 14 16

C (2)

(7)

B

A (19)

P ?

D (5)

x

Themodel

2 219 8 2( ) ( )x y +

2 25 14 13( ) ( )x y +

+ +Minimize

Subject to x 75 y 11

x + y 24


6/18

BUS321

Management Science

Lecture



0-1integer programsas NLPs

minimize Sj cj xj

subject to Sj aij xj = bi for all ixj is 0 or 1 for all j

is nearly equivalent to

minimize Sj cj xj + 106 Sj xj (1- xj).

subject to Sj aij xj = bi for all i

0 xj 1 for all j

Someco

mment

s

on non-linear

m

odels

The fact that non-linear models can model somuch is perhaps a bad sign How can we solve non-linear programs if we have

trouble with integer programs?

Recall, in solving integer programs we usetechniques that rely on the integrality.

Fact: some non-linear models can be solved,and some are WAY too difficult to solve. More

on this later.


7/18

BUS321

Management Science

Lecture



Another Example: Machine Values

Buy a machine and keep it for t years, andthen sell it. (0 t 10)

All values are measured in $ million

Cost of machine = 1.5

Revenue = 4(1 - .75t)

Salvage value = 1/(1 + t)

Machine values

00.5

1

1.5

2

2.5

3

3.5

4

4.5

0.

2 1

1.

8

2.

6

3.

4

4.

2 5

5.

8

6.

6

7.

4

8.

2 9

9.

8

Time

Millionsofdollars

revenue

salvage

total

How longs

hou

ld wekeep thema

chine?


8/18

BUS321

Management Science

Lecture



Non-linearities Because of T ime

Discount rates Decreasing value of equipment over time

wear and tear, improvements in technology

Tax implications

Salvage value

Non-linear

ities

in Pricing

The price of an item may depend on the numbersold

quantity discounts (small sellers)

price elasticity (monopoly)

Complex interactions because of substitutions: Lowering the price of GM automobiles will decrease the

demand for the competitors


9/18

BUS321

Management Science

Lecture



Non-linearities because ofcongestion

The time it takes to go from Lincoln Park to IITby car depends non-linearly on the congestion.

As congestion increases just to its limit, thetraffic sometimes comes to a near halt.

Non-linearities because ofpenalties

Consider any linear equality constraint:

e.g., 3x1 + 5x2 + 4x3 = 17

Suppose it is a soft constraint and we permit solutionsviolating it. We can then write:

3x1 + 5x2 + 4x3 - y = 17

And we may include a term of 10y2 in the objectivefunction.

This adds flexibility to the solution by discouragingviolation of our goals


10/18

BUS321

Management Science

Lecture



Portfolio Optimization

In the following slides, we will show how tomodel portfolio optimization as NLPs

The key concept is that risk can be modeledusing non-linear equations

Since this is one of the most famous applicationsof non-linear programming, we cover it in muchmore detail

Ris

k vs.

Retur

n In finance, one needs to trade-off risk and

return. For a given rate of return, one wants to minimize risk.

For a given rate of risk, one wants to maximize return.

Return is modeled as expected value.Risk is modeled as variance (or standarddeviation).


11/18

BUS321

Management Science

Lecture



Expectations Add

Suppose that X and Y are random variables

E(X + Y) = E(X) + E(Y)

Interpretation: Suppose that the expected return in one year for

Stock 1 is 9%.

Suppose that the expected return in one year forStock 2 is 10%

If you put $100 in Stock 1, and $200 in Stock 2,your expected return is $9 + $20 = $29.

Variances do notadd (atleast notsimply)

Suppose that X and Y are random variables

Var(aX + bY) =a2 Var(X) + b2 Var(Y) + 2ab Cov(X, Y)

Example. The risk of investing in umbrellasand sunglasses is less than the risk of eitherinvestment by itself.

In general:Var(X1 + X2 + + Xn) = 1 ( ) 2 ( , )

n

i i ji i jVar X Cov X X

=

lecture 17 - nlp - formulations

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