IEEE Computational Intelligence Magazine - November 2021 - 44

probability density (. ,. )== of the second gene in
i.e., xx 15== Similarly, the sec(or
x3) is more similar to the first gene of x2
is generxx
,23,1
is set to x ,,22
ond gene of x1
therefore, == The interim solution x3
ated as [. ,. ]15 25 T
xx 25,, ..
32 21
greater similarity to x1
knowledge transfer.
B. Offspring Generation Based on
Adaptive Mirror Transformation
To avoid premature convergence and to explore more
promising areas, this work also proposes a mirror transformation
to generate offspring. OBL has been proven to be
effective in single-task and multi-task optimization [24],
[45]. However, the existing OBL strategies based on the
opposite point and the generalized opposite point can
search only one point or a 1-D line in the space (as shown
in Fig. 1), which suffers from limited exploration ability in
large-scale problems and sometimes might lead to premature
convergence [41]. To address this issue, the proposed mirror
transformation is designed to search larger regions defined
by the opposite point ,xr
the center point c, and the boundaries
of the search space.
As shown in Fig. 3, the proposed mirror transformation
searches two candidate areas A and B adaptively to improve the
exploration. Area A is confined by the opposite point xr
1
x
0.5
c
A
B
1
B
A
0.5
x
0.2
0.4
(c)
A Parent Solution
Center
FIGURE 3 The proposed mirror transformation on a 2-D case.
Opposite Point
Area A
0.6
0.8 10 0.2
0.4
(d)
Area B
0.6
0.8
1
c
0.5
c
x
0.2
0.4
(a)
1
B
A
0.6
0.8
1
0.2
0.4
(b)
0.6
0.8
1
B
0.5
A
c
and
1
x
. As shown in Fig. 2, the generated x3
than x2
task gene similarity. The mating of x1
and x2
has a
thanks to the use of interand
x3
, instead of x1
, is expected to result in more positive and inter-task
N 15 032
31 22
nv
,, ..
;
the center point c, while area B is bounded by the opposite
point xr
and the boundary of the search space. Since areas A
and B cover larger exploration regions, mirror transformation
can achieve better solution diversity than OBL based on the
opposite point or the generalized opposite point.
The probabilities of selecting regions A and B are initialized
%, respectively. Then, these two
as P 50A
=
% and P 50B
=
probabilities are updated by
P
PP1
A
=
A
BA
=where
PA
e
and PB
e
PP PP
PP
))
)
A
A
e
A
e
+ B
B
e
(6)
are the proportions of elite offspring individuals
in areas A and B being selected into the next generation,
respectively. The definitions of PA
and PB
x (,...,),xxn
=
1
facilitate the
search toward a more promising area based on the recent performance
of the algorithm.
Given a solution
if the selection of area A
or B is determined, a new offspring can be obtained by mirror
transformation as follows. Let
l (,..., )lln
= 1
and u (,..., )uun
=
1
1
be the lower and upper bounds of the search space, respectively.
The center of the search space and the opposite point of x
based on Eq. (2) can be calculated with
lu )/ ,...,(lu )/ )
c (,..., )ccn
(( ++ and xc2
= (,...,)
of x is then generated with
ol (),,..., ,
nn
1
i ii i
Area
=+ ) -=rand ul in1
Area Area
11 22 r =-)
The offspring o oon
x ,
(7)
==
respectively.
44 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2021
IEEE Computational Intelligence Magazine - November 2021

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