C++ Neural Networks and Fuzzy Logic: Preface

•  Linear objective function: Maximize Z = 3X

1

 + 4X

2

 + 5.7X

3

•  Linear equality constraint: 13X

1

 − 4.5X

2

 + 7X

3

 = 22

•  Linear inequality constraint: 3.6X

1

 + 8.4X

2

 − 1.7X

3

 d 10.9

•  Nonlinear objective function: Minimize Z = 5X

2

 + 7XY + Y

2

•  Nonlinear equality constraint: 4X + 3XY + 7Y + 2Y

2

 = 37.6

•  Nonlinear inequality constraint: 4.8X + 5.3XY + 6.2Y

2

 e 34.56

An example of a linear programming problem is the blending problem. An example of a blending problem is

that of making different flavors of ice cream blending different ingredients, such as sugar, a variety of nuts,

and so on, to produce different amounts of ice cream of many flavors. The objective in the problem is to find

C++ Neural Networks and Fuzzy Logic:Preface

Chapter 15 Application to Nonlinear Optimization

332

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the amounts of individual flavors of ice cream to produce with given supplies of all the ingredients, so the

total profit is maximized.

A nonlinear optimization problem example is the quadratic programming problem. The constraints are all

linear but the objective function is a quadratic form. A quadratic form is an expression of two variables with 2

for the sum of the exponents of the two variables in each term.

An example of a quadratic programming problem, is a simple investment strategy problem that can be stated

as follows. You want to invest a certain amount in a growth stock and in a speculative stock, achieving at least

25% return. You want to limit your investment in the speculative stock to no more than 40% of the total

investment. You figure that the expected return on the growth stock is 18%, while that on the speculative

stock is 38%. Suppose G and S represent the proportion of your investment in the growth stock and the

speculative stock, respectively. So far you have specified the following constraints. These are linear

constraints:

G + S = 1

This says the proportions add up to 1.

S d 0.4

This says the proportion invested in speculative stock is no more than 40%.

1.18G + 1.38S e 1.25

This says the expected return from these investments should be at least 25%.

Now the objective function needs to be specified. You have specified already the expected return you want to

achieve. Suppose that you are a conservative investor and want to minimize the variance of the return. The

variance works out as a quadratic form. Suppose it is determined to be:

    2G

2

  + 3S

2

 − GS

This quadratic form, which is a function of G and S, is your objective function that you want to minimize

subject to the (linear) constraints previously stated.