IEEE Computational Intelligence Magazine - August 2022 - 52

SMOP5
0.8
1
0.6
0.4
0.2
0.2
0.4
f1
FIGURE 8 Two solution sets (in objective space) obtained by SparseEA
and PM-MOEA on SMOP5, which have the same convergence and
sparsity but different diversity.
typical solution sets obtained by the
compared MOEAs are investigated. Firstly,
Fig. 7 plots two solution sets obtained
by SparseEA and PM-MOEA on FS4,
which are completely the same and thus
have the same HV value. However, the
two solution sets have different sparsity,
thus leading to distinct CSD values. Secondly,
Fig. 8 draws two solution sets
obtained by SparseEA and PM-MOEA
on SMOP5, which have very similar
convergence (i.e., GD values) and sparsity.
However, the solution set obtained by
PM-MOEA distributes more uniformly
than that obtained by SparseEA, thus
leading to different diversity as well as
CSD values. Lastly, Fig. 9 depicts two
solution sets obtained by MOEA/D-DE
and LMOCSO on SMOP3, which have
the same sparsity and diversity. However,
the solution set obtained by MOEA/DDE
converges better than that obtained
by LMOCSO, thus leading to different
GD values as well as CSD values. As a
consequence, the proposed indicator is
capable of distinguishing between the
solution sets with only different convergence,
diversity, or sparsity.
V. Conclusions
The sparsity of solutions is a pivotal factor
in decision making, but it is ignored
0.6
0.8
1
SMOP3
SparseEA
GD
Sparsity
CSD
Diversity
2.5095e-4
8.9987e-1
4.9237e-2
4.9237e-2
PM-MOEA
2.2862e-4
8.9999e-1
3.7605e-2
3.7605e-2
2.5
3
1.5
2
0.5
1
0 0.5 1 1.5 2 2.5 3
f1
FIGURE 9 Two solution sets (in objective space) obtained by MOEA/
D-DE and LMOCSO on SMOP3, which have the same diversity and
sparsity but different convergence.
by most MOEAs and performance indicators.
Therefore, this work has proposed
a comprehensive indicator to compare
the performance of MOEAs on largescale
sparse MOPs. While existing indicators
use predefined reference points to
assess only the convergence and diversity
of the solution sets obtained by MOEAs,
the proposed indicator can assess the convergence,
diversity, and sparsity without
the assistance of any reference point.
According to the experiments, the proposed
indicator is more suitable for
assessing the quality of the solution sets
for sparse MOPs than existing indicators.
The experimental results have indicated
that large-scale MOEAs exhibit
similar performance to classical MOEAs
on large-scale sparse MOPs, even
though the former suggested various
search strategies to tackle high-dimensional
decision spaces. By contrast, the
sparse MOEAs proposed in recent years
significantly outperform large-scale
MOEAs and classical MOEAs, which is
mainly due to their effectiveness in
maintaining the sparsity of solutions.
However, the performance of these
sparse MOEAs fluctuates on different
problems, which implies that their performance
can be further enhanced.
Since the proposed indicator is designed
52 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2022
for assessing the convergence, diversity,
and sparsity of solution sets, it is reasonable
to embed the proposed indicator in
the selection strategies of MOEAs to
improve their performance on largescale
sparse MOPs.
Acknowledgment
This work was supported in part by the
National Key R&D Program of China
under Grant 2018AAA0100100, in part
by the National Natural Science Foundation
of China under Grant 61822301,
Grant 61876123, Grant 61876162, Grant
61906001, Grant 62136008, Grant
62172002, and Grant U21A20512, in part
by the Collaborative Innovation Program
of Universities in Anhui Province under
Grant GXXT-2020-013 and Grant
GXXT-2020-051, and in part by the
Research Grants Council of the Hong
Kong Special Administrative Region,
China under Grant PolyU11202418,
Grant PolyU11209219, and Grant
PolyU11211521.
References
[1] A. Zhou, B. Y. Qu, H. Li, S. Z. Zhao, P. N. Suganthan,
and Q. Zhang, " Multiobjective evolutionary algorithms:
A survey of the state of the art, " Swarm Evolut.
Comput., vol. 1, no. 1, pp. 32-49, 2011, doi: 10.1016/j.
swevo.2011.03.001.
[2] L. M. Antonio and C. A. Coello Coello, " Use of cooperative
coevolution for solving large scale multiobjective
GD
Sparsity
Diversity
CSD
MOEA/D-DE LMOCSO
1.8558e-1
1.0000e-5
0.0000e+0
1.0303e-1
1.8811e-1
1.0000e-5
0.0000e+0
2.5635e-1
f2
f2
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