-Chapter-11-Non-Linear-Programming ppt.ppt

Copyright © 2004 Brooks/Cole, a division of Thomson Learning, Inc.
1
CHAPTER 11:
NONLINEAR PROGRAMMING
to accompany
Operations Research: Applications & Algorithms,
4th edition, by Wayne L. Winston

2
Chapter 12 - Learning Objectives
1. Learn the differences between the LP and
the nonlinear program (NLP).
2. Study solution schemes or approaches for
NLP’s.
3. Understand the wide range of real
applications for which NLP’s are used.
4. Learn about the available software to
solve NLP’s.

3
What is a NLP?
NLP’s are closer to general and realistic
(and possibly unsolvable) models than
the LP’s. Some LPs are the linearized
versions of NLPs out of necessity.
NLP’s have non-proportional and non-
additive relationships.

4
The most general mathematical model is
likely to have nonlinear terms with
random (and possibly dependent)
coefficients. These difficulties are some of
the reasons why a deterministic LP is,
often, used as an approximation to a
stochastic NLP.

5
Why the term “Non-linear”?
• Up until this chapter, decision variables,
anywhere in model, were always in
additive (hence linear) form: 3x1+4x2, etc.
• There never was a case when other
algebraic operators were ever seen.
• In NLP, no such limitations exist. The LP
is actually a subset of NLP.

6
What causes nonlinearity?
Common operations such as multiplication
(x1x2), power (x2), and the others in Table 2
make a model nonlinear even if only one of
them appears just once anywhere in the
model.

7
A Real and a Simple NLP:
There used to be a time in some foreign
students’ life when he/she needed a
wooden crate to ship the books,
belongings, and the stereo (no PC then)
home.
Shipping companies (sea) had all sorts of
limits on the dimensions of the crate for
various price categories.

8
The problem often came to this: what
should the dimensions of the crate (a box,
often a prism) be so that the volume is
maximum or adequate. Weight did not
matter much. A cube will maximize the
volume, but a cube may not always be
feasible.

9
Maximum Volume of a Box
Let the height, width, and length be a,
b, and, c.
Volume = a b c (product is not linear)
The model is:
Max Volume;
subject to something? (constraints)

10
Without the constraints, optimizers such as
LINGO (later in this chapter) or LINDO (in
linear case) would set each dimension to
infinity to get volume that is infinite.
Obviously, the dimensions are the decision
variables and they must be positive. The
shipper may require that height is no more than
Y feet and total surface area is limited to X
square feet.

11
The Model :
Maximize Volume
S.t. :
2( ab + ac + bc) <= X ;
a<= Y;
(a, b, c,) > 0.
This problem was a real one. A carpenter
often built a special crate.

12
A Real Class Exercise to Illustrate
NLP:
An instructor decides to illustrate the concept of
optimization using a NLP fun example rather than
an LP case first.
The instructor buys poster papers and cuts them
into 11”x 13.75” pieces and gives one piece to each
student. The challenge is to construct a cylinder
with no lid such that the volume is maximum.

13
A Real Class Exercise to Illustrate
NLP cont’d:
Many students do a good job via trial and
error and some use calculus too. The
problem is another case in NLP modeling.
Let the cylinder have a radius of r and a
height of h in inches.

14
 The volume (V) is (pi* r2 * h). The
constraint is that the area used can not
exceed the available area of 151.25 in2.
The decision variables are r and h. The
main constraint is :
(pi)*(r2) + 2*(pi)*(r) *(h) <= 151.25
Both r and h are positive.

15
THE ANSWER
The radius ( r ) should be 4.08 inches and the
height should be 3.86 inches to have an
open cylinder (no lid) with a maximum
Volume using a sheet the available sheet of
11”x13.75”. Notice that the dimensions do
matter although their product was used in
the constraint. An odd shaped sheet may
not be feasible even if the model gives a
solution.

16
NLP vs. LP Applications
If possible, the analyst should strive to model
a decision process as a LP. Many
management and production type problems
have long been solved as LPs.
Different set of problems (engineering design
and stock selection, for example) must
contain non-linear terms that can not be
avoided.

17
Unlike the LPs, one is not always sure if a
given NLP solution is optimal or not. In
NLP, decision variables are not
automatically non-negative. This allows
certain physical values such as temperature
to assume negative values, if necessary.

18
Concepts of Limits & Derivatives
It appears that geometry and algebra were
sufficient until this chapter. This ends with
NLPs. Examples 1, 2, and 3 refresh our
memory on necessary calculus needed in
NLP.

19
In example 3, f(p) is similar to objective
functions seen in previous chapters, but it
is unconstrained. The derivative, f ‘(p) is
the rate of change of f(p) or the slope of
the revenue function, f(p).
This is a common application in
econometric analysis. If the price, p, is
more than $1 already, additional price
increases will result in a revenue loss.

20
Example 5 is the same as the others, but it has
two variables. Example 6 illustrates the role
of second and partial derivatives in NLPs.
This chapter has much calculus. Why? Calculus
is the backbone of NLP much like matrix
algebra was for LP earlier in the text.

21
It is possible to use software (LINGO and
EXCEL) to solve NLP’s just like using
LINDO for LP’s without worrying
much about the underlying
mathematics.

22
LINGO is a great tool for NLP!
• Previously, LINGO was used to solve
some special LP’s (e.g. TSP, assignment,
etc.) with unique formulations.
• LINGO is not limited to special models.
•LINGO allows the user to include
unusual operators such as absolute
value, logarithm, and exponentiation
in the modeling process also.

23
LINGO is a great tool for NLP!
LINGO is not limited to special models.
LINGO can be used to solve (or attempt
to solve) NLP’s of any form. It is also
possible to have negative and/or integer
(even binary) decision variables with
LINGO.

24
CONVEXITY and CONCAVITY in NLP
Classic calculus (definitions 3 and 4 and
Figures 9 and 10) facts are very
important in NLP. In general, the sign
of the second derivative of a function
(objective function in NLP) tells if the
function is convex or concave or
neither.

25
Concave vs. Convex NLP’s
Both have the so called convex constraint sets.
A concave NLP has a concave objective
function and it is a maximizing model.
A convex NLP has a convex objective function
and it is minimizing model.
To tell which set we have, apply the classic
second derivative test.

26
USING EXCEL TO SOLVE NLP
Figure 8 shows how to solve example 8
(previously solved using LINGO) with EXCEL.
It is also shown how the EXCEL SOLVER fails to
find the optimum in another problem shown
below:
Max Z = (x-1) (x-2) (x-3) (x-4) (x-5)
Where x ranges from 1 to 5

27
Example 9: The Oil Mix Problem, a Real
(and Useful) Case of NLP.
This is one of those problems that clearly explain
why NLP (and OR in general) is important.
Tables 3 and 4 and Figure 6 show the problem and
its solution using LINGO. This problem (saves $30
million/year) has to be nonlinear in part.

28
 The objective function (revenue-cost) is linear
along with most of the constraints (except
No.8, 9, 16-21) in Figure 6.
 The decision variables are R, U, and P.
Constraints 8 and 9 calculate chemical
contents, causing nonlinearity.

29
Example 9 Continued,
Ideally, we would have no or lesser amount
of nonlinearity, but there is no way to express
certain chemical ratios linearly. Notice the
rows 22-29 in Figure 6: decision variables
must be declared to be positive if they have
to be positive.

30
Example 10: Facility Location Problem of
Section
This example has a linear objective function and
mildly non-linear constraints. Figure 7 shows
how LINGO software is used in determining the
optimal location (in x, y coordinates) of a new
warehouse.

31
Example 11: Rubber Production Problem
Figure 8 shows how common formulas for
strength, elasticity, and hardness are used as
constraints in a physics like manner. This is quite
typical in NLP’s.

32
Multivariate Functions vs. Convexity and
Concavity Concepts.
Similar to second derivative tests for single variable functions,
the Hessian matrix and the ith principal minor tests are
performed. Definitions are on page 812. Theorems 3 and 3’,
and examples 17, 18, 19, and 20 illustrate these concepts.
NOTE: While important, these mathematical details are not
critical for most practioners.

33
Section 12.4: Solving One-variable NLP’s
Manually.
This section provides a detailed treatment of the
fundamentals involved in solving constrained NLP’s
that have just one decision variable.

34
Example 21: Production Level Selection
This example is all about supply and demand.
Notice the sales price is 10-x ; it goes down as we
are able to supply/sell more. Profit is found by
subtracting the cost from the revenue. The
problem becomes
Max P(x) = 5x-x2 where x ranges from 0 to 10.
Case 1 check tells us x=2.5 is a local optimal
solution.

35
Embellishing Example 2.1
The answer was x=2.5, a continuous value. This
means that the product is divisible type that can be
sold in fractional quantities. What if x has to be an
integer? You can have an integer NLP:
MODEL:
MAX=5*x-x^2;
x>0;
x<10;
@GIN( x);
END
Objective value: 6.000000
Variable value X 2.000000

36
Golden Section Search of Section 12.5
So far, we have dealt with “nice” objectives functions
that were differentiable. If this is not the case or the
roots of derivative can not be found easily, then the
general NLP schemes, described so far, do not work.
The Golden Section Method can be used if the
function is of unimodal kind.

37
Example 23 : On the Golden Section
Method
Max –X2-1
S.t. x ranging from –1 to 0.75
The first and second derivatives are –2X and –2.
X=0 is the answer using calculus. This problem does
not actually need the Golden Section Method, but it
is done for illustration.

38
The Golden Section Method is only able to
tell that the answer lies in the interval of
from -0.072 to 0.0815 when the correct
answer is known to be zero.

39
Unconstrained Optimization Section 12.6
Theorems 6, 7, 7’, 7” provide the basics of
unconstrained NLP’s that may have two or more
decision variables. Example 24 illustrates
these types of problems. If the problem is
constrained, it is important to know that the
solution (obtained from LINGO) may not always
be the true optimum. It might be a local
optimum. This is not the case if the problem
is unconstrained.

40
Section 12.7 : The Method of Steepest
Ascent :
This is the prime method used in realistic
NLP’s that are unconstrained. Example 27
shows the steps needed to implement this
method in a small problem. Note that this
example has no constraints. It simply says
The X’s must belong to real number set.

41
Section 12.8 : Lagrange Multipliers
We use this concept if the NLP comes with all
equality constraints. As shown in equation 12.
Example 28 shows how to perform the
mathematics of the Lagrange multipliers. This
may be a tough task at times.
Good News: LINGO will bypass all the math. You
just type the problem and run it.

42
Shadow Prices in Operations Research
Do you remember this concept from the LP’s? Lagrange
Multipliers are the equivalent of the shadow prices in NLP.
They are the rate of change of the optimal value as a fraction
of the changes in the RHS values of the NLP model.
Example 28 illustrates this concept. LINGO output in Figure
28 gives the so called Lagrange Multipliers under the price
Column.

43
QUADRATIC PROGRAMMING
(QP) of Section of Section 12.10
This is a very special and a highly realistic form of
NLP. The constraints are linear and the objective
function has a unique and a mildly non-linear
form. The terms of the objective function can be
either in square of one variable or the
multiplication of any of the two variables.

44
QP Cont’d.
QP’s application in portfolio optimization is so important
that LINGO has a special structure for it. The goal is to find
how to allocate our funds to several securities while
minimizing the portfolio variance and achieving a minimum
return of 12%. Example 33 shows how EXCEL and LINGO
can be used to solve QP’s.

45
The QP is in the same class of important problems as the
transportation, assignment, and ,of course, the traveling
salesperson problems presented earlier. LINGO has
special ready to use structures for all these problems.
This portfolio – QP application is used daily by many
investment firms.

46
Section 12.11 Separable Programming.
This concept is all about linearizing mildly nonlinear terms
encountered in objective functions and/or the
constraints. Figures 35, 36, and 37 illustrate the concept
using geometry.
There is no software for linearizing terms, but many
common software can be used in unique efforts.
Separable programming is an advanced topic in O.R.

47
Section 12.12 The Method of Feasible
Directions
This method takes the steepest ascent method of section
12.7 into a case where the NLP now has constraints.
Example 35 illustrates this advanced concept employed
by many optimizers.
This example is less nonlinear than even the quadratic
form. LINGO
can easily solve this problem.

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