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For a mutation in two dimensions the vector of objective variables has to be
mutated with the following trigonometric matrix
⎛
⎞
cos(
α
ij
)
−
sin(
α
ij
)
⎝
⎠
.
(4.12)
sin(
α
ij
)
−
cos(
α
ij
)
For a self-adaptation process the
k
angles
α
1
,...α
k
have to be mutated. Schwefel
[132] proposed
α
=
α
+
γ
·N
(0
,
1)
(4.13)
with
γ
=0
.
0873 corresponding to 5
◦
.
4.1.4
Asymmetric Density Functions - Directed Mutation
Similar to the idea of correlated mutation, the assumption of directed mutation
is that in some parts of the landscape a success rate improvement can be achieved
by skewing the search into a certain direction. Hildebrand [55] proposes an asym-
metrical mutation operator. This approach demands an asymmetry parameter
set of
N
additional parameters
c
=(
c
1
,...,c
N
). These parameters determine
the mutation direction and therefore only cause linear growth of the strategy
parameters instead of the quadratic growth of correlated mutation. In order to
decouple asymmetry from the step sizes, a normalized directed mutation op-
erator has recently been proposed by Berlik [13]. The density function for the
normalized directed mutation is the following:
⎧
⎨
√
1
−a
x
2
2(
σ
(
a
)
σ
)
2
2
π
(1+
√
1
−a
)
σ
norm
(
a
)
σ
e
for
a
≤
0
,x
≤
0
√
1
−a
(1
−a
)
x
2
2(
σ
(
a
)
σ
)
2
2
π
a
)
σ
norm
(
a
)
σ
e
for
a
≤
0
,x>
0
(1+
√
1
−
f
σ,a
(
x
)=
(4.14)
√
1+
a
(1+
√
1+
a
)
σ
norm
(
a
)
σ
e
(1+
a
)
x
2
2(
σ
(
a
)
σ
)
2
⎩
2
π
for
a>
0
,x
≤
0
√
1+
a
x
2
2(
σ
(
a
)
σ
)
2
2
π
(1+
√
1+
a
)
σ
norm
(
a
)
σ
e
for
a>
0
,x>
0
and its normalization function
σ
norm
(
a
). The generation of corresponding ran-
dom numbers is a rather complicated undertaking. The idea of skewing the
mutations into a certain direction is similar to our approach. But our approach
is easier to implement. Figure 4.4 visualizes the effect of the directed mutation
operator when mutations are skewed into a certain direction. The highest prob-
ability to reproduce mutations is still in the neighborhood environment of the
child.
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