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4.1.2
Uncorrelated Gaussian Mutation with N Step Sizes
In the basic ( μ + )-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 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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