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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
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
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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