IEEE Computational Intelligence Magazine - November 2023 - 40
The ensemble of the proposed
generation operators may be an
interesting future work.
AMEA and SMEA can generate promising solutions on various
MOPs with complex PF and PS shapes, but their mating
restriction strategies, based on probability values, might
fail to balance local exploitation and global exploration
since their effectiveness is still problem-dependent. In contrast,
LGSEA employs GTM to learn the manifold structure
for guiding the search with some alternative operators,
which is helpful in generating promising offspring solutions.
The designed incremental training procedures also reduce
the overhead of the GTM training.
In summary, LGSEA outperforms the compared
algorithms on the five test suites in terms of both IGD and
HV. These results demonstrate that LGSEA is highly
competitive with several state-of-the-art learning-based
MOEAs.
D. Scalability Studies
To further demonstrate the superiority ofLGSEA, the scalability
ofLGSEA to the number of decision variables is examined
by varying the number ofdecision variables on WFG1-WFG9
test instances with 50, 80, 100, and 200 dimension variables
(600 iterations for 50, 80, and 100-D instances and 800 iterations
for 200-D instances). The IGD results achieved by six
compared MOEAs are given in Table IV. In these comparisons,
LGSEA obtains the highest number ofthe optimal results
(27 out of 36), followed by IM- MOEA, AMEA, and SMEA
with 6, 2, and 1 best results, respectively.
To visualize the results, Figure 7 plots the mean and standard
deviations of the IGD values obtained by LGSEA with
the different number of decision variables. LGSEA can produce
relatively small IGD values for WFG2-WFG9 and n
does not significantly influence the obtained IGD values,
while LGSEA is still not well for WFG1. Considering that
the PF of WFG1 is quite complicated, LGSEA may require
more evolutionary generations. In a word, the developed
LGSEA demonstrates its capability to scale up to large-scale
decision variables effectively.
TABLE IV Statistical results (Mean(Std. Dev.)[rank]) obtained by AMEA, IM-MOEA, SMEA, RM-MEDA, MEA/GTM, and LGSEA over 30
independent runs on the WFG test suites with different variables dimensional in terms of the IGD metric.
40 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2023
IEEE Computational Intelligence Magazine - November 2023
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