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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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