IEEE Computational Intelligence Magazine - November 2021 - 43

kk
12
(). nki ,1
and nkj ,2
tions, respectively. vki ,1
are the means of the two gene distribuand
vkj ,2
x [, ]13 T
1 =
denote standard deviations of
the two gene distributions, respectively. A smaller KLD value
indicates that the two gene distributions are more similar.
The procedure of calculating inter-task gene similarity is
summarized in Algorithm 2.
2) Selective Crossover
The proposed inter-task knowledge transfer occurs in the
selective crossover of inter-task similar genes. The original
MFEA may encounter negative inter-task transfer when the
optima of two tasks are far away from each other. To address
this issue, the proposed algorithm learns the online inter-task
gene similarity and uses the similarity in the proposed selective
crossover.
The detailed procedure of the selective crossover is provided
in Algorithm 3. Given two selected parent solutions x1
as well as their corresponding skill factors x1
is firstly generated as the copy of x1
and x2
solution x3
ward, for each gene i in x3
is identified as
i in x2
!
(line 3). Then the value of gene i in x3
x and xx ,2 an interim
(line 1). Afteri
argmin {,..., }jn1
=
, the corresponding most similar gene
^h
KLD N N,,
xx
i in x2 (line 4). Finally, two offspring solutions o1
generated by performing SBX crossover [44] on x1
(line 6). Compared with x2
, the generated x3
similar to x1
x1 and x3
x2
xx
12
ij
is replaced by that of gene
and o2
and x3
is much more
. Therefore, the knowledge transfer via crossover of
tends to be more positive than the crossover of x1
and
applied in most of the existing MFEAs.
A simple two-task example in a two-dimensional space is
provided in Fig. 2 to illustrate the generation of the interim
solution x3
N TT
R = diag([ ., .] )),
2
n11 diag 02 02==R
03 03 T
13
and ([ ., .] ,
n2
=
in the selective crossover. Suppose the two optimiN
25 15 T
respectively. Two parent solutions
zation tasks are modelled with Gaussian distributions
([ ,] ,([. ,. ]))
Algorithm 3 Procedure of selective crossover
Require: xx,, ,:
12 xx
12
12
,: two offspring solutions.
2: For in :1=
3:
5: End for
6: oo
{, }12
do
4: xx .,,i32
i ! argminjn ij
! i
! 1 ..., } ,,
{, KLD NN| |xx
(
! SBXCrossover xx
(, ).13
12
xx ).
1: xx .31
xx Two parent solutions and their skill factors. KLD:
Inter-task gene similarity matrix. n: the number of decision variables.
Ensure: oo
!
are
and x [. ,. ]25 15 T
2 =
tasks (i.e., they have different skill factors). To generate x3
of the first gene in x1
. Since the probability density (, .)102
(or x3) has smaller KLD value with the
N ==
2
nv
are associated with different
, the
inter-task gene similarity is applied to identify the most similar
genes in x2
Algorithm 2 Calculation of inter-task gene similarity
Require: P: the current population. n: the number of decision variables.
K: the number of optimization tasks.
Ensure: KLD: Inter-task gene similarity matrix.
1: PP
{,..., }
1
3:
4:
5:
2: For kK :1=
For in :1=
ki ki
End for
6: End for
7: For
8:
9:
10:
For kK :12
(
kK :11
=
=
End for
11: End for
do
do
KLD N N,,|12 |ki kj) ! Calculate the inter-task gene similarity
according to Eq. (5), where ,,..., .
ij =1 n
do
do
(, ),,
K ! Divide P into K subpopulations based on the skill
factors of the individuals.
//K: the number of optimization tasks
//n: the number of decision variables
nv ! Calculate the mean and standard deviation of
Pk
on the i-th gene according to Eq. (4)
4
N(µ1, Σ1)
2
4
3
2
x2
1
0 0
1
2
x1
FIGURE 2 An example of interim solution generation in selective crossover for two-dimensional two-task optimization.
3
x1: A Population Member of Task tx1
x2: A Population Member of Task tx2
x3: Genes From x2 to Near x1
Probability Density of Task tx1
Probability Density of Task tx1
Probability Density of Task tx2
Probability Density of Task tx2
N(µ2, Σ2)
on First Dimension
on Second Dimension
on First Dimension
on Second Dimension
x1 = (1, 3) x3 = (1.5, 2.5)
x2 = (2.5, 1.5)
4
NOVEMBER 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 43
IEEE Computational Intelligence Magazine - November 2021

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - November 2021
