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(a)
(b)
(c)
(d)
(e)
Fig. 4.4.
The mutation ellipsoids of the different mutation operators in a two-
dimensional search space: For the purpose of better understanding we assume all mu-
tations fall into the ellipsoids instead of taking the probability density functions into
account. From left to the right: (a) uncorrelated mutation with one step size, (b) uncor-
related mutation with
N
= 2 step sizes, (c) correlated mutation resulting in rotation,
(d) directed mutation with different skewness and (e) biased mutation making use of
the
bias coe
cient vector
.
step size, but again with a
N
-dimensional bias coecient vector
ξ
=(
b
1
,...,b
N
)
with
b
i
=
ξ
i
·
σ
i
. Hence, the mutation
z
becomes
z
:=
σ
(
N
1
(
ξ
1
,
1)
,...,
N
N
(
ξ
N
,
1))
.
(4.37)
Mutations of the sBMO lie within a hypersphere. The bias
ξ
is mutated analog
to the biased mutation of the standard BMO in order to enable self-adaptation,
see equation 4.36.
4.2.3
Cube Biased Mutation Operator (cBMO)
Another variant of the BMO is the use of only one bias value
b
,whichspansa
hypercube with the edge length
b
, but
N
step sizes. The mutation
z
becomes
z
:= (
σ
1
N
1
(
b,
1)
,...,σ
N
N
N
(
b,
1))
.
(4.38)
Mutations of the cBMO lie within the hyperellipsoid of the standard mutation,
but the center of this hyperellipsoid is shifted by the vector (
b,...,b
). Again,
ξ
is mutated analog to the biased mutation of the standard BMO.
4.2.4
Comparison of Computational Effort of the Randomized
Operators
Let
N
be the number of strategy variables. Uncorrelated Gaussian mutation
with one step size requires only
O
(1) operations, uncorrelated Gaussian muta-
tion with
N
steps sizes
O
(
N
) respectively. Correlated mutation needs
O
(
N
2
)
for the rotation of the mutation ellipsoid and is thus computationally expensive.
The directed mutation requires
O
(2
N
) operations, but the computation of the
asymmetric probability density function is more complex. Also the BMO needs
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