IEEE Computational Intelligence Magazine - August 2022 - 48

search strategies for tackling highdimensional
decision spaces are inefficient
in finding sparse solutions.
Thirdly, the dimensionality reduction
based MOEAs (i.e., WOF-SMPSO and
LSMOF-NSGA-II) considerably outperform
the MOEAs based on variable
grouping and novel variation operators,
since the former can quickly converge
to quasi-optimal regions but the
latter cannot converge by using
100 D# function evaluations. In particular,
LMOCSO is verified to be
effective with
15000 D# function
evaluations [11], which is unaffordable
for real-world applications.
Fourthly, the sparse MOEAs (i.e.,
SparseEA, MOEA/PSL, and PMMOEA)
obtain the best CSD values and
have obviously better rankings than the
other MOEAs, where the effectiveness
of these MOEAs in searching for sparse
Pareto optimal solutions can be confirmed.
Fig. 3 depicts the objective values
of the solution sets with median
CSD obtained by the compared
MOEAs on SMOP1, SMOP6, and
SMOP7 with 5000 decision variables,
where the solution sets obtained by
SparseEA, MOEA/PSL, and PMMOEA
have better convergence than
those obtained by classical MOEAs and
large-scale MOEAs. Lastly, although all
of the three sparse MOEAs maintain the
sparsity of solutions by using the bi-level
encoding in (3), PM-MOEA outperforms
SparseEA while MOEA/PSL
outperforms both SparseEA and PMMOEA.
This is because SparseEA does
not adopt any dimensionality reduction
strategies, while PM-MOEA uses an
evolutionary pattern mining approach
to reduce the dimensionality of the
binary vector, and MOEA/PSL uses
two unsupervised neural networks to
reduce the dimensionality of both the
binary vector and real vector.
To verify the effectiveness of the proposed
indicator, Table IV presents the
IGD values obtained by the compared
MOEAs, where 10000 reference points
are sampled by the method suggested in
[42] for IGD calculation. It can be
observed from the table that the average
rankings of the compared MOEAs are
similar to those in Table III, where
MOEA/PSL obtains the best performance,
PM-MOEA obtains the second
best performance, and SparseEA obtains
the third best performance. Nonetheless,
the IGD values of some results are
inconsistent with the corresponding
CSD values in Table III. For example,
PM-MOEA obtains better CSD value
than MOEA/PSL on SMOP1 with
5000 decision variables, while MOEA/
PSL gains better IGD value than PMMOEA
on the same test instance.
According to the decision variables of
the solution sets obtained for SMOP1
with 5000 decision variables shown in
Fig. 4, the solution set obtained by PMMOEA
is sparser than that obtained by
MOEA/PSL. Hence, PM-MOEA
obtains the best CSD value since it can
find very sparse solutions with the assistance
of the evolutionary pattern mining
approach, and the proposed CSD
considers the sparsity of solution sets. By
contrast, MOEA/PSL gains the best
IGD value since it can better optimize
the nonzero variables by using the
restricted Boltzmann machine, and IGD
does not take the sparsity of solution sets
into consideration. As a consequence,
the proposed CSD is more effective
than IGD in assessing the solution sets
for sparse MOPs.
C. Comparisons on
Real-World SMOPs
Next, the performance of the 11
MOEAs is compared on seven realworld
applications with 100 to 6241
decision variables. These real-world
problems hold some characteristics
NN4
PO4
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
10-3
10-2
10-1
Complexity of Model
NSGA-II
100
0 2468
Risk
IM-MOEA LMOCSO
MOEA/D-DE MOEA/DVA LMEA
SparseEA
×10-6
1
0.965
0.97
0.975
0.98
0.985
0.99
0.995
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Inter-Link Density
WOF-SMPSO LSMOF-NSGA-II
MOEA/PSL PM-MOEA
CD4
FIGURE 5 Solution sets (in objective space) with median CSD obtained by 11 MOEAs on neural network training, portfolio optimization, and
community detection with approximately 5000 decision variables.
48 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2022
Error Rate

IEEE Computational Intelligence Magazine - August 2022

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