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8.3 Maximum Seeking with GEP
In this section we are going to seek the maximum of two different functions.
The first, is the following well-studied two-parameter function:
f
(
x, y
)
=
-
x
sin( 4
x
)
-
1.1
y
sin( 2
y
)
(8.7a)
subjected to the constraints:
0
d
x
d
10
and
0
d
y
d
10
(8.7b)
For this function the maximum value is known, which, for simplicity, will be
considered 18.5.
Although simple, this kind of functions with known global optima are
very useful as they can be used to measure accurately the performance of
different algorithms. For instance, the global minimum of the function (8.7)
above could not be found by traditional minimum seeking algorithms such
as the Nelder-Mead or the Broyden-Fletcher-Goldfarb-Shanno algorithms
(Haupt and Haupt 1998).
On the other hand, less conventional methods such as the GA or GEP have
no problems at all in finding the global maximum (or minimum) of this func-
tion (Table 8.3). Indeed, considering 18.5 the maximum output for function
(8.7), both the HZero and the GEP-PO algorithms could find the exact pa-
rameters for which the function evaluation returns output values equal to or
greater than 18.5 in virtually all runs.
As shown in Table 8.3, the HZero algorithm considerably outperforms the
GEP-PO system at this simple task (note that in the GEP-PO experiment
populations evolved 10 times longer). And taking into account the greater
complexity of the GEP-PO algorithm, it is obviously advisable to use the
simpler HZero system to solve function optimization problems with just one
or two dimensions. However, the GEP-PO algorithm can find much more
precise parameter values as it designs its solutions with as great a precision
as necessary and, therefore, for those cases where such a precision is needed,
the GEP-PO algorithm should be used instead. Furthermore, the HZero algo-
rithm might sometimes converge prematurely, becoming stuck in some local
maximum. In those cases, the higher flexibility of the GEP-PO algorithm
becomes indispensable for navigating such treacherous landscapes.
By analyzing the best-of-experiment solutions discovered with both algo-
rithms, the fine-tuning capabilities of the GEP-PO algorithm become clear.
