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In-Depth Information
4.1.2
Uncorrelated Gaussian Mutation with
N
Step Sizes
In the basic (
μ
+
,λ
)-ES usually a vector of
n
σ
=
N
step sizes is used, which
results in mutation ellipsoids:
z
:= (
σ
1
N
1
(0
,
1)
,...,σ
N
N
N
(0
,
1))
(4.5)
The corresponding strategy parameter vector is mutated with the extended log-
normal rule:
σ
1
e
(
τ
1
N
1
(0
,
1)
,...,σ
N
e
(
τ
1
N
N
(0
,
1)
σ
:=
e
(
τ
0
N
0
(0
,
1))
·
(4.6)
c
√
2
N
The parameters
τ
0
and
τ
1
have to be tuned. Recommended settings are
τ
0
=
√
2
√
N
and
c
= 1 as a reasonable choice for a (10,100)-ES [18]. Compri-
sing, an individual
a
consists of the object parameter set
x
i
with 1
c
and
τ
1
=
N
,the
mutation strength vector and the assigned fitness F(x). So it is specified by
≤
i
≤
a
=(
x
1
,...,x
N
,σ
1
,...,σ
N
,F
(
x
))
.
(4.7)
4.1.3
Correlated Mutation
For some fitness landscapes it is more beneficial to use a rotated mutation el-
lipsoid for the purpose of an improvement of the success rate. Rotation of the
mutation ellipsoid is achieved by the correlated mutation proposed by Schwefel
[132]. For an
N
-dimensional problem
k
=
N
(
N
1)
/
2 additional strategy para-
meters, the angles for the rotation of the mutation ellipsoid, are introduced. Let
σ
again be the vector of step sizes and
M
be the orthogonal rotation matrix.
The mutations are produced in the following way:
−
z
:=
M
(
σ
1
N
1
(0
,
1)
,...,σ
N
N
N
(0
,
1))
(4.8)
The probability density function for
z
becomes
1
2
z
T
·C
−
1
e
−
·
z
p
(
z
)=
(4.9)
(2
π
)
n
)
2
(det
C
·
with the rotation matrix
1
C
and entries
c
ii
=
σ
i
,
(4.10)
⎧
⎨
0
no correlations
,
c
ij,i
=
j
=
(4.11)
⎩
1
2
(
σ
i
σ
j
)tan(2
α
ij
) correlations.
−
1
The rotation matrix
C
is also called
covariance matrix
.
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