IEEE Computational Intelligence Magazine - November 2021 - 45

where rand is a random real number in [, ]01 . li
lc ux Ax c
lx uc Ax c
lx uu Bx c
ll ux Bx c
Area
i
Area
i
Area
i
Area
i
==
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Area
i
r
r
i
,
i
i
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,
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i ii i
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i ii i
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if is selectedand
if is selectedand
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$
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1
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(8)
The four rules correspond to the four cases shown in
Fig. 3 respectively. With mirror transformation, the algorithm
can explore a more promising area A or B in an adaptive
manner. The mirror transformation also effectively
complements with the evolutionary operators used in the
original MFEA.
IV. Experimental Results and Analyses
To verify the performance of the proposed algorithm, we conducted
experimental studies on both benchmark and real-world
problems. Four types of problems were considered:
❏ single-objective multi-tasking test suite;
❏ multi-objective multi-tasking test suite;
❏ single-objective many-tasking test suite;
❏ real-world problems on neural network parameter optimization.
In the experiments, MFEA-GSMT is compared with other
state-of-the-art MFEAs to reveal the strengths and weaknesses
of the algorithm. The effects of the two proposed strategies
on the performance of the algorithm are also fully investigated.
To make a statistical significance comparison, all compared
algorithms are run 20 times independently on each test problem.
The Wilcoxon rank-sum test at a 5% significance level is
executed to compare the proposed MOMFEA-GSMT with
other algorithms. The significantly better, significantly worse,
and comparable results are symbolized with " + " , " - " , and
" . " , respectively.
A. Results on Single-Objective Multi-Tasking Test Suite
The single-objective multi-tasking test suite proposed in the
CEC 2017 evolutionary multi-task optimization competition
[46] is used to test the performance of the proposed algorithm.
The results on the other two benchmarks, i.e., multiobjective
multi-tasking test suite and single-objective
many-tasking test suite, are provided in the Supplementary
Materials due to the page limit. In this benchmark, each
MTO problem includes two single-objective optimization
tasks. The MTO problems are divided into nine categories
based on the intersection degree of their global optima, (i.e.,
complete intersection (CI), partial intersection (PI), and no
intersection (NI)), and the inter-task landscape similarity (i.e.,
high similarity (HS), medium similarity (MS), and low similarity
(LS)).
On this single-objective multi-task test suite, the performance
of the proposed MFEA-GSMT is validated via comparison
with SOEA [11], MFEA [11], and two latest MFEAs,
i.e., MFEA-II [28] and MFEA/GHS [24]. MFEA-II learns
Area and ui
Area
denote the lower and upper bounds of the search area respectively.
They are set as follows:
r
the inter-task correlation online and adjusts the random mating
probability (rmp) dynamically to reduce inter-task ineffective
knowledge transfer. MFEA-GHS is a recent MFEA that
integrates genetic transformation and hyper-rectangle search
strategies. The parameters of the compared algorithms are set
as follows.
❏ Simulated binary crossover (SBX) and polynomial mutation
(PM) [47] are used in all compared algorithms, where the
distribution indexes are set to be 2 and 5, respectively. The
crossover probability and mutation probability are set to be
1 and /n1
, respectively.
❏ The random mating probability rmp in MFEA, MFEA/
GHS, and MFEA-GSMT is set to 0.3. Following the suggestion
of [28], rmp is dynamically adjusted through online
learning in MFEA-II.
❏ A population size N = 50 and a maximum number of function
evaluations 50,000 are used in SOEA. For fair comparisons,
N = 100 and a maximum number of function
evaluations 100,000 are applied to MFEA, MFEA-II,
MFEA/GHS, and MFEA-GSMT.
❏ The generation interval of using the mirror transformation
in MFEA-GSMT is set to G = 5.
The mean and standard deviation of the best solutions
obtained by all algorithms are reported in Table I. The best
mean value of each task is highlighted in bold. The convergence
trend diagram of the best obtained objective value on
each task is plotted in Fig. 4. As can be seen in the results,
MFEA-GSMT obtains the best overall performance. The
observations are detailed as follows:
❏ All MFEAs outperform the single-tasking SOEA on the
majority of tasks, which implies that inter-task knowledge
transfer introduced by MFEAs can accelerate the convergence
speed and find a better solution set.
❏ The proposed MFEA-GSMT outperforms other MFEAs
on 15 out of 18 tasks, especially on approximating the
optimal value of each task. The success of MFEA-GSMT
lies in two factors. First, MFEA-GSMT has a higher probability
of jumping out of local optima via the mirror
transformation. Second, the knowledge transfer based on
the inter-task similar gene can achieve more positive
knowledge transfer and better trade-off of exploitation
and exploration.
❏ Negative transfer knowledge is an issue faced in MFEAs
when the global optimum of each task is far away from
each other in the unified search space. MFEA-II introduces
the online learning of rmp based on the population density
distribution, yet it is less precise especially in high-dimensional
space with a small number of samples.
❏ The superiority of MFEA-GSMT to MFEA/GHS in
the majority of tasks is attributed to the effectiveness
of the proposed mirror transformation. MFEA/GHS
introduces the opposite point and searches based on
the difference between two population centers, which
might not be able to maintain as good diversity as mirror
transformation.
NOVEMBER 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 45
IEEE Computational Intelligence Magazine - November 2021

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