IEEE Computational Intelligence Magazine - August 2022 - 51

different from benchmark problems,
including highly discrete decision spaces
and irregular Pareto fronts. In particular,
the objective functions of feature selection,
pattern mining, community detection,
critical node detection, and
knapsack problem are calculated based on
binary variables, which pose challenges to
many MOEAs
(i.e., MOEA/D-DE,
MOEA/DVA, LMEA, WOF-SMPSO,
LSMOF-NSGA-II, IM-MOEA, and
LMOCSO) that can only handle real
variables. Although these MOEAs can
optimize real variables and round them
before function evaluations, the introduction
of flat landscapes is likely to deteriorate
their search abilities. As evidenced by
the CSD values shown in Table V, these
MOEAs are underperformed by the
binary genetic operator based NSGA-II.
Although the uniform crossover and bitflip
mutation are the simplest genetic
operators, they are relatively effective for
solving binary MOPs and bring passable
performance to NSGA-II.
Nevertheless, the high-dimensional
decision spaces hinder NSGA-II from
gaining the best CSD values on most
test
instances. By contrast, the three
MOEAs tailored for sparse MOPs still
exhibit the best overall performance,
where SparseEA and MOEA/PSL
obtain the best average ranking and are
followed by PM-MOEA. Fig. 5 plots
the objective values of the solution sets
with median CSD obtained by the
compared MOEAs on neural network
training (NN4), portfolio optimization
(PO4), and community detection (CD4)
with approximately 5000 decision variables.
For NN4, it can be found that
MOEA/D-DE, MOEA/DVA, and
LMOCSO cannot obtain good solutions
in terms of the first objective (i.e.,
model complexity) and the second
objective (i.e., error rate), whereas
NSGA-II, LMEA, WOF-SMPSO,
LSMOF-NSGA-II, and IM-MOEA can
obtain a few solutions with extremely
low error rates, but these solutions (i.e.,
neural networks) probably overfit the
training set as analyzed in [20]. By contrast,
SparseEA, MOEA/PSL, and PMMOEA
can obtain many solutions with
both low complexities and error rates,
where the low complexities can effectively
alleviate overfitting. For PO4, only
the solutions obtained by SparseEA,
MOEA/PSL, and PM-MOEA hold
good spread along the Pareto front,
while the solutions obtained by all the
other MOEAs shrink to the upper left
corner. For CD4, the solution set
obtained by PM-MOEA is well converged
and diversified, the solution sets
obtained by NSGA-II, SparseEA, and
MOEA/PSL have good convergence
but bad spread, and the solution sets
obtained by all the other MOEAs are
badly converged. As a result, the superiority
of the sparse MOEAs is verified
on both benchmark problems and realworld
applications.
It is worth noting that SparseEA has
worse average ranking than MOEA/PSL
and PM-MOEA on benchmark problems
but competitive average ranking on realworld
applications. To illustrate this inconsistency,
Table VI shows the HV values
obtained by the compared MOEAs,
where the reference point (, ,, )
11 1f
is
used for HV calculation. It can be found
from the table that SparseEA obtains
worse HV values than MOEA/PSL and
PM-MOEA, which is inconsistent with
the results in Table V. In particular, Spar1
SparseEA
HV
0.8
Sparsity
CSD
0.6
9.9976e-1
9.9977e-1
3.8634e-1
PM-MOEA
9.9976e-1
9.9957e-1
7.7268e-1
D.
Further Verification of the
Proposed CSD
To further verify the effectiveness of the
proposed CSD in separately assessing
convergence, diversity, and sparsity, several
FS4
seEA has better CSD value but worse HV
value than PM-MOEA on NN4.
According to the solution sets obtained
for NN4 shown in Fig. 5, the solutions
obtained by SparseEA have slightly worse
convergence than those obtained by
PM-MOEA, leading to slightly worse
HV value of SparseEA. On the other
hand, Fig. 6 draws the decision variables
of the solution sets obtained for NN4,
where SparseEA, MOEA/PSL, and PMMOEA
obtain very sparse solution sets
and the solution set obtained by SparseEA
is the sparsest, hence SparseEA
obtains the best CSD value on NN4. To
summarize, the solution sets obtained by
SparseEA have slightly worse convergence
but better sparsity than those
obtained by PM-MOEA, hence SparseEA
gains better CSD values but worse
HV values than PM-MOEA. Moreover,
the effectiveness of the proposed indicator
in the performance assessment of
convergence, sparsity, and diversity can
be further demonstrated.
0.4
0.2
0 123 4
Ratio of Selected Features
×10-4
FIGURE 7 Two solution sets (in objective space) obtained by SparseEA and PM-MOEA on FS4,
which have the same convergence and diversity but different sparsity.
AUGUST 2022 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 51
Error Rate
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