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One coordinate is linearly added to the function. The function is converging
towards infinity.
x
0
100
,
100]
N
.
∈
[
−
A.4 Numerical Functions in Bit String Representations
Onemax
N
, maximize
Onemax is a binary function. For a bit string
x
∈{
0
,
1
}
N
N
f
(
x
)=
x
i
with
x
∈{
0
,
1
}
(A.9)
i
=1
optimum
x
∗
=(1
,...,
1)
T
with
f
(
x
∗
)=
N
.
Ackley
Minimize
⎛
⎞
exp
1
N
cos(2
πx
i
)
+20+
e
(A.10)
N
N
1
N
⎝
−
⎠
−
f
(
x
)=
−
20
·
exp
0
.
2
x
i
i
=1
i
=1
with properties
•
multi-modal
•
huge number of local optima
and optimum
x
∗
=(0
,...,
0)
T
with
f
(
x
∗
)=0.
A.5 Constrained Numerical Functions
TR - Tangent Problem
Minimize
N
x
i
f
(
x
)=
(n-dim. sphere model)
(A.11)
i
=1
constraints
N
g
(
x
)=
x
i
−
t>
0
,
t
∈
IR
(tangent)
(A.12)
i
=1
For n=k and t=k the minimum lies at:
x
∗
=(1
,...,
1)
T
,
with
f
(
x
∗
)=
k.
(A.13)
TR2 is TR with
N
=2and
t
=2.
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