IEEE Computational Intelligence Magazine - November 2021 - 40

solution quality and convergence performance. The contributions
of this work are summarized as follows.
❏ The limitations of existing MFEAs are reviewed and discussed.
To improve the knowledge transfer efficiency in
high-dimensional space with only a small number of samples,
a new knowledge transfer strategy based on inter-task
gene similarity instead of individual similarity is proposed.
The new strategy can perform more fine-grained and accurate
knowledge transfer.
❏ An adaptive mirror transformation based on oppositionbased
learning is introduced to maintain good population
diversity and explore more promising search areas during
the search. It complements the evolutionary operators of
MFEA well and helps alleviate premature convergence.
❏ The proposed algorithm and the effects of the two key
strategies are fully validated on both a benchmark and realworld
problems. The advantages and disadvantages of the
two proposed strategies are discussed to afford some new
insights into designing efficient MFEAs.
The remainder of this article is organized as follows.
Section II introduces the related background and related work.
Section III details the proposed algorithm. Section IV presents
the experimental results and discussion. Finally, Section V
concludes this article.
II. Background and Related Work
This section introduces some basic definitions of the optimization
problem and opposition-based learning followed by the
related work on evolutionary multi-tasking optimization.
A. Optimization Problem
Without loss of generality, an optimization problem can be
defined as follows:
)
where X 3 Rn
minFf f
subject to
() (),, ()
:
xx x
x
= 6 1
! X
vector of n variables, and ():FxRm
f
m
@
T
(1)
denotes the decision space, x ! X is a decision
X " indicates an m-dimensional
objective function vector. If m > 1, the problem is called a
multi-objective optimization problem (MOP); otherwise, the
problem is a single-objective optimization problem (SOP) with
m = 1. Given two decision vectors xu
dominate xv
6f
x*
and xx ,vu
im ff1 xx
!# and () ().FF A solution
! X is called a Pareto-optimal solution if it is not dominated
{, ,},( )( )
iu iv
(also written as xx ),uv
!
'
xx
uv
by any other feasible solution. The set of Pareto-optimal solutions
are known as the Pareto set (PS) and the mapping of PS in
the objective space is known as the Pareto front (PF).
B. Opposition-Based Learning
The concept of opposition-based learning (OBL) was first
proposed by Tizhoosh in [31], where the opposite point is
considered as a better candidate than a random guess for the
exploration of optimization search. Opposition-based sampling
can lead the search to the optimal solution more efficiently
than random re-sampling. OBL samples not only a
solution point but also its opposite point. To facilitate the
understanding of OBL, the preliminary definitions of opposite
number, opposite point, and generalized opposite point are
provided as follows:
Opposite number [31]: given a real number
xl u! [, ],
where l and u are the lower and upper bounds of x, respectively,
the opposite number xr of x is defined as
Opposite point [31]: given a point x (,..., )xxn
n-dimensional space with xl uii i
x luxr =+ - .
= 1
! [, ] and = 1
1
referred to as
xl ,,...,.
r =+ -=ii ux in1ii
(2)
An illustration of an opposite point in a two-dimensional
space is provided in Fig. 1, where the original solution point x
and its opposite point xr
are symmetrical with respect to the
center of the search space. The definition of opposite point can
be further extended to generalized opposite point [32] as follows:
Generalized opposite point [32]: given a point x =
in n-dimensional space with xl uii i
, where li and ui are the lower and upper bounds of ,xi
(,..., )xxn
in , ...,1=
1
1
x2
u2
X
X
l2
l1
u1 x1
FIGURE 1 An example of a solution x and its opposite point xr on the
2-D case.
t =+a ) -=() ,,..., ,
ii ii
a 1 ).=
xl ux in1
where a is a real number in [, ]01 (The generalized opposite
point is equivalent to the opposite point if
Based on the above definitions, the procedure of OBL can
be outlined as follows. Given a candidate solution point x in
the search space, OBL identifies the opposite point xr
generalized opposite point )xt
()
(or the
of x. If () ()FF (or
xx#
r
FF( )),xx#
t
(or ).xt
the search continues with x; otherwise, with xr
Note that with the opposite point, OBL can explore
only one point at a time, whereas with the generalized opposite
point, it is possible for OBL to explore on a line with different
settings of a as shown in Fig. 1.
40 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2021
! [, ] and
the generalized opposite point (,...,)tt=x xxn of x is defined as
t
x xxn
in
in , ..., , where li
and ui are the lower and upper bounds of x in the i-th dimension,
respectively, the opposite point
rr= (,...,) of x is
r
is said to
if and only if
x(c ∈[0.5, 1)
"
IEEE Computational Intelligence Magazine - November 2021

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