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In-Depth Information
8.1.2 Optimization of a Simple Function
From the presentation above and from what you know about the GEP-RNC
algorithm and the GA, you can see why the HZero algorithm is more flexible
than the GA. Basically, when the arrays of the RNCs contain just one ran-
dom constant, we get the same functionality of a GA. But by increasing the
number of elements in the arrays of RNCs, we are improving the odds of
finding the right combination of parameters that optimize our function. For
most problems, I use 10 random constants per gene, not only because it is
very convenient in terms of representation but also because it produces good
results. Let's see then how the HZero algorithm discovers the parameters
that optimize a function.
In parameter optimization the goal is to find a set of parameter values that
maximize (or minimize) a complex multidimensional function. Sometimes
minimizing a function makes more sense, and when that is the case, one just
has to multiply the function by minus one and then maximize. Obviously, in
gene expression programming, it is more convenient to maximize a function
as, in that case, better solutions have corresponding higher fitnesses, making
the selection process much easier.
The simple function we are going to maximize in this section, is the same
chosen by Michalewicz to illustrate the workings of the GA (Michalewicz
1996) and, therefore, serves also to show the differences between the two
algorithms. This simple, one variable, function is defined as:
f
(
x
)
x
sin(
10
S
x
)
1
.
0
(8.4a)
and the goal consists of finding, in the domain [-1, 2], the value
x
0
that maxi-
mizes the function, that is:
> @
f
(
0
x
)
t
f
(
x
)
for
all
x
-
1,
2
(8.4b)
The global maximum of this function in the given domain is obviously known
and, for simplicity, will be considered 2.85.
For this problem, a set of just five random numerical constants will be
used so that we can fit it all together (chromosomes and random constants)
in a row. As usual, the RNCs will be represented by the numerals 0-4, thus
giving R = {0, 1, 2, 3, 4}. The ephemeral random constant “?” will range
obviously over the rational interval [-1, 2] as the random constants are used
in their natural state by the algorithm, that is, no new constants can be cre-
ated through simple mathematical operations such as the addition or multi-
