Biology Reference
In-Depth Information
5.3.4.3.
How to find the global minima
Now, for a given constant
C
0
such that the level set
H
0
=
{
x
|
f
(
x
)
<
C
0
,
x
is the Lebesque measure of
H
0
,
then
C
0
is the minimum of
f
(
x
) and
H
0
is the global minimum set.
Otherwise, assume that
∈
G
} is nonempty, if
µ
(
H
0
)
=
0, where
µ
µ
(
H
0
)
>
0 and
C
1
is the mean value of
f
(
x
) on
H
0
. Then,
1
Ú
CHf x
H
=
() ( d
m
(2)
1
0
m
0
and
CC f x
≥≥(*).
(3)
0
One then gradually constructs the level set
H
k
and mean value
C
k
+1
of
f
(
x
)
on
H
k
as follows:
{
}
(4)
Hf x
=
()
<
Cx
,
Œ
G
k
k
and
1
m
Ú
C
+
=
()
H
f x
() .
d
m
(5)
k
1
k
H
k
With the assistance of OA sampling, a decreasing sequence of mean
values {
C
k
} and a sequence of level sets {
H
k
} are obtained.
Let
Lim
k
k
CC
=
*
(6)
Æ•
and
Lim
k
k
HH
=
*.
(7)
Æ•
It can thus be proven that
C
* is the minimum of
f
(
x
) on
G
, and
H
* is the
global minimum set.
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