Digital Signal Processing Reference
In-Depth Information
Finally, we would like to point out the analogy between the tradi-
tional optimization approach and GAs. The binary strings correspond
to an orthogonal direction system, crossovers to moving randomly at the
same time in multiple directions from one point to another of the surface,
and mutation to searching along a single, randomly chosen direction.
Optimization of a simple function
A GA represents a general-purpose optimization method that searches
irregular, poorly characterized function spaces and is easily implemented
on parallel computers. The performance of the solutions is continuously
tested based on a fitness function. It's not always guaranteed that an
optimal candidate is found, but in most cases GAs do find a candidate
with high fitness. An important application area for GAs is pattern
recognition: the highly nonlinear problem of estimating the weights in a
neural network.
This section will apply the most important basic operations of a GA
to an example of function optimization [50].
The following function is considered:
g
(
x
)=
x
2
−
42
x
+ 152
where
x
is an integer. The goal is to find, based on a GA, the minimum
of this function in the interval [0
···
63]:
g
(
x
0
)
≤
g
(
x
)
,
x
∈
···
63]
.
for all
[0
To solve this optimization problem, some typical GA operators are
employed.
Number representation
The integer-valued
x
have to be transformed into a binary vector (chro-
mosome). Since 2
6
= 64, we will use six-bit binary numbers to represent
the solutions. This means that six bits are needed to represent a binary
vector (chromosome).
The transformation of a binary number
<b
5
···
b
0
>
into an integer
number
x
is done by the following rule:
Transform the binary number
<b
5
···
b
0
>
from basis 2 into basis
10:
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