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x
2
★
NP Parameter vectors from generation G
Newly generated parameter vector
v
Fx
r
2
,
G
x
r
3
,
G
(
)
Minimum
★
★
★
★
x
i
,
G
★
★
x
r
3
,
G
★
x
r
1
,
G
x
r
2
,
G
★
★
★
★
(
)
v
=
x
r
1
,
G
+
Fx
r
2
,
G
x
r
3
,
G
x
1
Fig. 1.3.
Contour lines and the process for generating v in scheme DE1
where
i
∈
[1
,
NP
];
j
∈
[1
,
D
] ,
F
>
0, and the integers
r
1
,
r
2
,
r
3
∈
[1
,
NP
] are generated
randomly selected, except:
r
1
=
i
.
Three randomly chosen indexes,
r1
,
r2
,and
r3
refer to three randomly chosen vectors
of population. They are mutually different from each other and also different from the
running index
i
. New random values for
r1
,
r2
,and
r3
are assigned for each value of
index i (for each vector). A new value for the random number
rand
[0
,
1] is assigned for
each value of index
j
(for each vector parameter).
F
is a real and constant factor, which
controls the amplification of the differential variation. A two dimensional example that
illustrates the different vectors which play a part in DE2 are shown in Fig 1.3.
=
r
2
=
r
3
Crossover
In order to increase the diversity of the parameter vectors, the vector
u
=(
u
1
,
u
2
,...,
u
D
)
T
(1.10)
⎧
⎨
v
(
G
)
j
for j
=
n
D
,
n
+ 1
D
,...,
n
+
L
−
1
D
x
(
G
)
i
u
(
G
)
j
=
(1.11)
⎩
ot herwise
j
is formed where the acute brackets
D
denote the modulo function with modulus
D
.This
means that a certain sequence of the vector elements of
u
are identical to the elements
of
v
, the other elements of u acquire the original values of
x
(
G
i
. Choosing a subgroup of
parameters for mutation is similar to a process known as crossover in genetic algorithm.
This idea is illustrated in Fig 1.4 for D = 7, n = 2 and L = 3. The starting index
n
in (12)
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