Information Technology Reference
In-Depth Information
•
the multivariate normal distribution
N
(
m
,
C
)withmean
m
and covariance
matrix
C
,
the
i
-th best point
x
(
t
+1)
i
:
λ
out of
x
(
t
+1)
1
,...,
x
(
t
+1)
λ
•
from equation 4.20,
μ
eff
=(
i
=1
w
i
)
−
1
is the variance effective selection mass, 1
•
≤
μ
eff
≤
μ
,
OP:
IR
N
IR
N×N
,
x
xx
T
.
•
→
→
The CMA-ES works in the following steps.
•
Parameter setting.
Set
λ, μ, w
i
=1
...μ
,c
σ
,d
σ
,c
c
,μ
cov
,c
cov
to default values.
Initialization.
Set evolution path
p
(0)
=
0
,
p
(0)
•
=
0
. Set covariance matrix
σ
c
IR
n
C
(0)
=
I
. Set initial step size
σ
(0)
∈
IR
+
, distribution mean
m
(0)
∈
problem dependent.
•
Iteration loop.
t
=0
,
1
...
until termination criterion fulfilled.
-
sample
new population of search points
x
(
t
+1)
k
(
m
(
t
)
,
(
σ
(
t
)
)
2
C
(
t
)
)for
k
=1
,...,λ
N
(4.20)
-
selection and recombination
μ
μ
w
i
x
(
t
+1)
i
:
λ
m
(
t
+1)
=
,where
w
i
=1
,w
i
>
0
(4.21)
i
=1
i
=1
-
step size control
c
σ
)
p
(
t
σ
+
c
σ
(2
c
σ
)
μ
eff
C
(
t
)
−
2
m
(
t
+1)
m
(
t
)
−
p
(
t
+1)
σ
=(1
−
−
(4.22)
σ
(
t
)
σ
(
t
+1)
=
σ
(
t
)
exp
c
σ
d
σ
1
p
(
t
+1)
||
||
σ
||
−
(4.23)
E
||N
(
0
,
I
)
-
covariance matrix adaptation
c
c
(2
c
c
)
μ
eff
m
(
t
+1)
m
(
t
)
−
p
(
t
+1)
c
c
c
)
p
(
t
)
+
h
(
t
+1)
σ
=(1
−
−
(4.24)
c
σ
(
t
)
p
(
t
+1)
c
)
C
(
t
)
)
(4.25)
c
cov
μ
cov
p
(
t
+1)
T
c
C
(
t
+1)
=(1
c
cov
)
C
(
t
)
+
+
δ
(
h
(
t
+1)
σ
−
w
i
OP
x
(
t
+1)
+
c
cov
1
μ
−
m
(
t
)
σ
(
t
)
1
μ
cov
i
:
λ
−
(4.26)
i
=1
The CMA-ES works with a
global
step, the scaling factor and an
individual
step, the covariance matrix. The global step is computed considering the mean
distance between the successful mutated offspring and the current population
center, see equation 4.23. A further concept that is used is the cumulative path-
length control by Ostermeier, Hansen and Gawelczyk [101]. The information of
previous generations is used. Cumulative step size adaptation makes use of an
evolution path with
cumulative path length
, relative to the mean distance of
selected offspring around the center, see equation 4.22.
p
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