Nonlinear programming 2013

INTRODUCTION TO NONLINEAR PROGRAMMING (NLP)
In LP, our goal was to maximize or minimize a linear
function subject to linear constraints:
Maximize profit P = 7X1 + 10X2

Subject to:

Fabrication Time: 3X1 + 2X2 <= 36
Assembly Time: 2X1 + 4X2 <= 40
IC Chips: 10X1 <= 100
Non-negativity: X1, X2 >= 0

Linear functions have the form of a “sumproduct”:
a1X1 + a2X2 + a3X3 + …
So linear functions do not involve exponents,
logarithms, square roots, products of variables, and 2
so on. Functions having these components are
nonlinear.

INTRODUCTION TO NONLINEAR
PROGRAMMING (NLP)
 If an LP problem is feasible then, at least in theory,
it can always be solved because:
 We know the solution is a “corner point”: a point where
lines or planes intersect. There are a finite number of
possible solution points.
 The simplex algorithm will find that point

 Also, a very informative sensitivity analysis is
relatively easy to obtain for LP problems
 But in many interesting, real-world problems, the
objective function may not be a linear function, or
some of the constraints may not be linear
constraints  3

INTRODUCTION TO NLP
 Optimization problems that involve nonlinearities
are called nonlinear programming (NLP) problems.
 Many NLPs do not have any constraints. They are
called unconstrained NLPs.
 Solutions to NLPs are found using search
procedures. Solutions are more difficult to
determine, compared to LPs. One problem is
difficulty in distinguishing between a local and
global minimum or maximum point.

4

Example problem: Maximize f(x) = -x2 + 9x + 4
(An unconstrained problem that can be solved without a search)
30

25

20
f(x)

15 Solution process is straightforward using calculus:

f'(x) = -2x + 9 Set this equal to zero and obtain x = 4.5
10
f''(x) = -2 which is negative at x = 4.5 (or at any
other x-value) so we have indeed found a maximum rather
5 than a minimum point

So the function is maximized when x = 4.5, with a
0
maximum value of -4.52 + 9(4.5) + 4 = 24.25.
0 1 2 3 4 5 6 7 8 5 9
x

Problem: Maximize f(x)
450
Global
400
maximum

Local maximum
350

300

250
f(x)

200

150
This is trickier: a value x whose first derivative is zero and
100
whose second derivative is negative is not necessarily the
solution point! It could be a local maximum point rather
50
than the desired global maximum point.

0
0 1 2 3 4 5 6 7 8 6 9
x

Constrained Problem: Maximize f(x) subject to: x ≥ 7
450

400

350

300

Solution point
250
f(x)

200

150

In the case of this constrained
100 optimization problem basic calculus is
of no value, as the derivative at the
Feasible
50 solution point is not equal to zero
region
0
0 1 2 3 4 5 6 7 8 7 9
x

NLP EXAMPLE: SEARCHES CAN FAIL!
Maximize f(x) = x3 - 30x2 + 225x + 50
3000

2500

2000
f(x)

1500

1000

500

0
0 5 10 15 20 25
x

The correct answer is that the problem is
unbounded. There is no solution point! 8

Let‟s try Solver……

NLP EXAMPLE: SEARCHES CAN FAIL!
Maximize f(x) = x3 - 30x2 + 225x + 50
Solver results:
Initial Guess for X Final Value for X Function Value (f(x))
1 5 550
12 5 550
18 Does not converge: Unbounded!

In the first two cases Solver converged to a local maximum.
So the answer is incorrect!
In the third case Solver found the correct answer.
In more complex examples we couldn‟t plot the function and
would likely accept Solver‟s initial incorrect answer as being
correct. 9


NLP EXAMPLE: PRICING CHAIRS
The Hickory Cabinet and Furniture Company has
decided to concentrate on the production of chairs.
The fixed cost per month of making chairs is $7,500,
and the variable cost per chair is $40. Demand is
related to price according to the following linear
equation:
d = 400 − 1.2p,
where d is the demand and p is the price. Develop
the nonlinear profit function for this company and
determine the price that will maximize profit, the
optimal volume, and the maximum profit per month.
10

NLP EXAMPLE: PRICING CHAIRS
The Hickory Cabinet and Furniture Company has decided to concentrate on the production of chairs. The fixed cost per month of
making chairs is $7,500, and the variable cost per chair is $40. Price is related to demand according to the following linear
equation:
d = 400 − 1.2p,
where d is the demand and p is the price. Develop the nonlinear profit function for this company and determine the price that will
maximize profit, the optimal volume, and the maximum profit per month.

Profit = Revenue – Cost
Revenue = Units Sold (Demand) x Price = dp
= (400 – 1.2p)p
= 400p - 1.2p2
Cost = 7500 + 40d
= 7500 + 40(400-1.2p)
= 23,500 – 48p 11

NLP Investment Portfolio Selection Example:
Problem Definition and Model Formulation

Objective of the portfolio selection model is to:
■ minimize some measure of portfolio risk (for example
variance in the return on investment)
■ while achieving some specified minimum expected return
on the total portfolio investment.
Or..
■ maximize expected return on the total portfolio investment
■ without exceeding a specified maximum value for the risk
measure
12

Investment Portfolio Selection Example:
Example: (text pg. 491) A portfolio will consist of shares
of four stocks. Data for expected returns, variances, and
correlations is available for the stocks.

Stock (xi) Annual Return (ri) Variance (si)
Altacam .08 .009
Bestco .09 .015
Com.com .16 .040
Delphi .12 .023
Correlation Matrix
Stock A‟s return
A B C D
could vary from -
A 1 0.4 0.3 0.6
11% to +27% (2
std. dev. Interval) B 0.4 1 0.2 0.7
13
C 0.3 0.2 1 0.4
D 0.6 0.7 0.4 1

The math ():
Let: xi = proportion of the portfolio to invest in Stock i
ri = the expected return for Stock i
si2 = the variance for the return on Stock i (so si is the standard
deviation for stock i)
rij= the correlation between returns on Stocks i and j
for i = 1, 2, 3, 4

Then (skipping a LOT of math and statistical theory):

Expected portfolio return = R = x1r1 + x2r2 + x3r3 + x4r4

Variance of return = Z = x12s12 + x22s22 + x32s32 + x42s42 +
2x1x2Cov(x1,x2) + 2x1x3Cov(x1,x3) + 2x1x4Cov(x1,x4) +
2x2x3Cov(x2,x3) + 2x2x4Cov((x2,x4) + 2x3x4Cov(x3,x4)
= our measure of portfolio risk

Where Cov(xi,xj) = sisjrij (This is the „covariance‟ of Stocks i and j)
14

Investment Portfolio Selection Example:
Suppose that we require a return of at least 11% and wish
to minimize risk. Then the problem formulation, where all
variables are defined on the previous page is:

Decision variables: X1, X2, X3, X4

Objective: Minimize Z = risk

Subject to: X1 + X2 + X3 + x4 = 1
R >= .11
Non-negative

15

NLP EXAMPLE: FACILITY LOCATION (TEXT PG.
490)
Truckco is trying to determine where they should locate
a single warehouse. The positions in the x-y plane (in
miles) of their four customers and the number of
shipments made annually to each customer are as
follows:
Customer x-Coordinate y-Coordinate Number of Shipments
1 5 10 200
2 10 5 150
3 0 12 200
4 12 0 300

Truckco wants to locate the warehouse to minimize the
total distance trucks must travel annually from the 16
warehouse to the four customers.

NLP EXAMPLE: FACILITY LOCATION
Make the unreasonable assumption that you can go in a
straight line from the warehouse.

Recall that the distance from point (x1, y1) to point (x2,
y2) is:

( x1 x2 ) 2 ( y1 y2 ) 2

17

Nonlinear programming 2013

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