mit and james orlin © 2003 1 non linear programming 1 nonlinear programming (nlp) –modeling...
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MIT and James Orlin © 2003
1
Non Linear Programming 1
Nonlinear Programming (NLP)– Modeling Examples
MIT and James Orlin © 2003
2
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, , ..., 0nx x
ASSUMPTIONS:
Proportionality Assumption
– Objective function– Constraints
Additivity Assumption– Objective function– Constraints
MIT and James Orlin © 2003
3
What is a non-linear program?
maximize 3 sin x + xy + y3 - 3z + log zSubject to x2 + y2 = 1 x + 4z 2 z 0
A non-linear program is permitted to have non-linear constraints or objectives.
A linear program is a special case of non-linear programming!
MIT and James Orlin © 2003
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Nonlinear Programs (NLP)
Nonlinear objective function f(x) and/or Nonlinear constraints gi(x).
Today: we will present several types of non-linear programs.
1 2 , , ,
( )
( ) , 1, 2, ,
n
i i
Let x x x x
Max f x
g x b i m
MIT and James Orlin © 2003
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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. Dem.
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. Distances are “Euclidean.”
MIT and James Orlin © 2003
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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
MIT and James Orlin © 2003
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Here are the objective values for 55 different locations.
0
50
100
150
200
250
300
350
valuesfor y
Ob
ject
ive
valu
e
x = 0
x = 2
x = 4
x = 6
x = 8
x = 10
x = 12
MIT and James Orlin © 2003
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Facility Location. What happens if P must be within a specified region?
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
MIT and James Orlin © 2003
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The model
2 219 8 2( ) ( )x y
2 25 14 13( ) ( )x y
+ …+Minimize
Subject to x 7 5 y 11 x + y 24
MIT and James Orlin © 2003
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0-1 integer programs as NLPs
minimize j cj xj
subject to j aij xj = bi for all i
xj is 0 or 1 for all j
is “nearly” equivalent to
minimize j cj xj + 106 j xj (1- xj).
subject to j aij xj = bi for all i
0 xj 1 for all j
MIT and James Orlin © 2003
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Some comments on non-linear models
The fact that non-linear models can model so much 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 use
techniques 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.
MIT and James Orlin © 2003
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Variant of exercise from Bertsimas and Freund
Buy a machine and keep it for t years, and then 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)
MIT and James Orlin © 2003
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Machine values
00.5
11.5
22.5
33.5
44.5
0.2 1
1.8
2.6
3.4
4.2 5
5.8
6.6
7.4
8.2 9
9.8
Time
Mil
lio
ns
of
do
llar
s
revenue
salvage
total
MIT and James Orlin © 2003
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How long should we keep the machine?
Work with your partner on how long we should keep the machine, and why?
MIT and James Orlin © 2003
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Non-linearities Because of Time
Discount rates decreasing value of equipment over time
– wear and tear, improvements in technology Tax implications (Depreciation) Salvage value
Secondary focus of the previous model(s): Finding the right model can be subtle
MIT and James Orlin © 2003
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Non-linearities in Pricing
The price of an item may depend on the number sold – quantity discounts for a small seller– price elasticity for monopolist
Complex interactions because of substitutions: – Lowering the price of GM automobiles will
decrease the demand for the competitors
MIT and James Orlin © 2003
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Non-linearities because of congestion
The time it takes to go from MIT to Harvard by car depends non-linearly on the congestion.
As congestion increases just to its limit, the traffic sometimes comes to a near halt.
MIT and James Orlin © 2003
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Non-linearities because of “penalties”
Consider any linear equality constraint:
e.g., 3x1 + 5x2 + 4x3 = 17
Suppose it is a “soft” constraint and we permit solutions violating it. We can then write:
3x1 + 5x2 + 4x3 - y = 17
And we may include a term of –10y2 in the objective function.
– This adds flexibility to the solution by discourages violation of our “goals”
MIT and James Orlin © 2003
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Portfolio Optimization
In the following slides, we will show how to model portfolio optimization as NLPs
The key concept is that risk can be modeled using non-linear equations
Since this is one of the most famous applications of non-linear programming, we cover it in much more detail
MIT and James Orlin © 2003
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Risk vs. Return
In finance, one trades of 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 standard deviation.)
MIT and James Orlin © 2003
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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
for Stock 2 is 10%– If you put $100 in Stock 1, and $200 in Stock 2,
your expected return is $9 + $20 = $29.
MIT and James Orlin © 2003
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Variances do not add (at least not simply)
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 “umbrellas” and “sunglasses” is less than the risk of either investment by itself.
In general:
Var(X1 + X2 + …+ Xn) = 1( ) 2 ( , )
n
i i ji i jVar X Cov X X
MIT and James Orlin © 2003
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Reducing risk
Diversification is a method of reducing risk, even when investments are positively correlated (which they often are).
If only two investments are made, then the risk reduction depends on the covariance.
MIT and James Orlin © 2003
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Portfolio Selection (cont’d)
Two Methods are commonly used:
– Min Risk
s.t. Expected Return Bound
– Max Expected Return - (Risk)
where reflects the tradeoff between return and risk.
MIT and James Orlin © 2003
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Portfolio Selection Example
There are 3 candidate assets for out portfolio, X, Y and Z. The expected returns are 30%, 20% and 8% respectively (if possible we would like at least a 12% return). Suppose the covariance matrix is:
What are the variables?
3 1 0 5
1 2 0 4
0 5 0 4 1
.
.
. .
X Y Z
X
Y
Z
Let X,Y,Z be percentage of portfolio of each asset.
MIT and James Orlin © 2003
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Portfolio Selection Example
Min
st
Max
st
2 2 23 2 2 0.8X Y Z XY XZ YZ
1.3 1.2 1.08 1.12
1
0, 0, 0
X Y Z
X Y Z
X Y Z
2 2 2
1.3 1.2 1.08
(3 2 2 0.8 )
X Y Z
X Y Z XY XZ YZ
1
0, 0, 0
X Y Z
X Y Z
MIT and James Orlin © 2003
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More on Portfolio Selection
There can be institutional constraints as well, especially for mutual funds.
No more than 15% in the energy sector Between 20% to 25% high growth At most 3% in any one firm etc. We end up with a large non-linear program. The unconstrained version becomes the “CapM
model” in finance.
Portfolio Example
MIT and James Orlin © 2003
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RegressionEstimate for Midterm = x * HW3 + y
Midterm = x * HW3 + y + residual
x y
0.6 40
HW3 Estimate Midterm 1 Residual Residual squared91 94.6 89 -5.6 31.3680 88 97.5 9.5 90.2561 76.6 58.5 -18.1 327.6188 92.8 92 -0.8 0.6486 91.6 93.5 1.9 3.6156 73.6 87 13.4 179.5660 76 99 23 52987 92.2 85 -7.2 51.8450 70 67 -3 9
sum of squares 1222.87
Find the best linear fit for estimating the midterm grade from the homework grades
MIT and James Orlin © 2003
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Writing regression as an NLP
Minimize j (rj)2
subject to
r1 = (91x + y) – 89
r2 = (80x + y) – 97.5
r3 = (61x + y) – 58.5
…
r9 = (50x + y) – 67
Minimize j (rj)2
subject to
rj = Hj x + y – Mj for each j
In an optimization framework, one can constrain coefficients.
MIT and James Orlin © 2003
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Midterm 2 vs Homeworks (2002)
30
40
50
60
70
80
90
100
30 40 50 60 70 80 90 100
Avg of last 3 homeworks
Mid
term
Gra
de
r2 =.082
MIT and James Orlin © 2003
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Midterm 1 vs. homework 3 (2001)
40
50
60
70
80
90
100
40 50 60 70 80 90 100
homework 3 grades
mid
term
gra
de
s
r2 =.29
MIT and James Orlin © 2003
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An application of regression to finance
A famous application in Finance of determining the best linear fit is determining the of a stock.
CAPM assumes that the return of a stock s in a given time period is
rs = a + rm + ,
rs = return on stock s in the time period
rm = return on market in the time period
= a 1% increase in stock market will lead to a % increase in the return on s (on average)
MIT and James Orlin © 2003
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Regression, and estimating
Return on Stock A vs. Market Return
-60.00%
-40.00%
-20.00%
0.00%
20.00%
40.00%
60.00%
80.00%
-40.00% -20.00% 0.00% 20.00% 40.00% 60.00% 80.00%
Market
Sto
ck
What is the best linear fit for this data? What does one mean by best?
MIT and James Orlin © 2003
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Regression, and estimating
Return on Stock A vs. Market Return
-60.00%
-40.00%
-20.00%
0.00%
20.00%
40.00%
60.00%
80.00%
-40.00% -20.00% 0.00% 20.00% 40.00% 60.00% 80.00%
Market
Sto
ck
The value is the slope of the regression line. Here it is around .6 (lower expected gain than the market, and lower risk.)
MIT and James Orlin © 2003
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Solving NLP’s by Excel Solver
MIT and James Orlin © 2003
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Summary
Applications of NLP to location problems, portfolio management, regression
Non-linear programming is very general and very hard to solve
Special case of convex minimization NLP is easier, because a local minimum is a global minimum
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