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In-Depth Information
Double Sum
Minimize
⎛
⎞
2
N
i
⎝
⎠
IR
N
f
(
x
)=
(
x
j
)
with
x
∈
(A.2)
i
=1
j
=1
with properties
•
unimodal, non-separable
•
scalable
100
,
100]
N
,
minimum
x
∗
=(0
,...,
0)
T
•
x
∈
[
−
with
f
(
x
∗
)=0.
A.2 Multimodal Numerical Functions
Rosenbrock
Minimize
n
−
1
(100(
x
i
−
1)
2
IR
N
x
i
+1
)
2
+(
x
i
−
f
(
x
)=
with
x
∈
(A.3)
i
=1
with properties:
•
multi-modal for
N>
4,
•
non-separable, scalable
•
very narrow valley from local optimum to global optimum
100
,
100]
N
,
•
x
∈
[
−
minimum
x
∗
=(1
,...,
1)
T
with
f
(
x
∗
) = 0. For higher dimensions the optimum
exhibits a local optimum at
x
=(
1
,...,
1)
T
.
−
5000
4000
5000
4000
3000
2000
3000
2000
1000
0
1000
0
-1000
-6
-6
6
6
-4
-4
4
4
-2
-2
2
2
0
0
0
0
2
-2
2
-2
4
4
-4
-4
Fig. A.2.
Plot of the rosenbrock function (left) and rosenbrock with noise in fitness
(right)
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