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2. Reflection: In each iteration step, determine P
fhigh
, P
fsec hi
, P
low
vertices, indicating
vertex points that have the highest, the second highest, and the lowest function
values, respectively. Let f
fhigh
, f
fsec hi,
represent the corresponding observed
function values. Find P
cent
, the center of the simplex excluding P
fhigh
in the
minimization case. Generate a new vertex P
refl
by re
fl
ecting the worst point
according to the following equation (see Fig.
9
a):
P
refl
¼ð
þaÞ
P
cent
a
ð
Þ
1
P
fhigh
21
where P
refl
is the re
ection coef
cient (
ʱ
> 0). Nelder and Mead suggested the
ʱ
=1.Iff
low
≤
f
refl
≤
ection by replacing P
fhigh
with
P
refl
, and step 2 is entered again for a new iteration.
3. Expansion: Should re
use of
f
fsec hi
, accept the re
fl
ection produce a function value smaller than f
low
(i.e.,
f
refl
< f
low
), the re
ection is expanded in order to extend the search space in the
same direction and the expansion point is calculated by the following equation
(see Fig.
9
b)
fl
P
exp
¼ c
P
refl
þ
ð Þ
1
P
cent
ð
22
Þ
=2.If
f
exp
< f
low
, the expansion is accepted by replacing P
fhigh
with P
exp
; otherwise, P
refl
replaces P
fhigh
. The algorithm continues with a new iteration in step 2.
4. Contraction: When f
refl
> f
fsec hi
and f
refl
≤
where
ʳ
is the expansion coef
cient (
ʳ
> 1). Nelder and Mead suggested
ʳ
f
fhigh
, then P
refl
replaces P
fhigh
and
contraction is tried (see Fig.
9
c). If f
refl
> f
fhigh
, then direct contraction without the
replacement of P
fhigh
by P
refl
is performed (see Fig.
9
d). The contraction vertex is
calculated by the following equation:
P
cont
¼ b
P
fhigh
þð
1
bÞ
P
cent
ð
23
Þ
where
ʲ
is the contraction coef
cient (0 <
ʲ
< 1). Nelder and Mead suggested
f
fhigh
, the contraction is accepted by replacing P
fhigh
with P
cont
and then a new iteration begins with step 2.
5. Shrink:Iff
cont
> f
fhigh
in step 4, contraction has failed and shrinkage will be the
next attempt. This is done by shrinking the entire simplex (except P
low
) by (see
Fig.
9
e, f):
ʲ
= 0.5. If f
cont
≤
P
i
d
P
i
þð
1
dÞ
P
low
ð
24
Þ
where
ʴ
is the shrinkage coef
cient (0 <
ʴ
< 1). Nelder and Mead suggested
ʴ
= 0.5. The algorithm then evaluates function values at each vertex (except
P
low
) and returns to step 2 to start a new iteration.
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