Biology Reference
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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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