IEEE Computational Intelligence Magazine - November 2023 - 36

TABLE I Mean IGD values obtained by LGSEA with different selections and generation operators on the GLT test suite.
training steps on the use ofGTM, which can be summarized as
follows.
❏ On Offspring Generation: In terms of the offspring generation,
MEA/GTM samples offspring solutions from the probability
model obtained by GTM, while LGSEA seeks to map some
points from the latent space to the decision space for guiding
the search. In contrast to MEA/GTM, which generates solutions
entirely based on the accuracy of the GTM model,
LGSEA somewhat relaxes the demand for GTMmodel quality,
as these mapped points are incorporated in the three alternative
operators as guiding vectors.
❏ On Training GTMModel: Unlike MEA/GTM, which initializes
the GTM model and performs a complete training step at
each iteration, the proposed LGSEA reuses the GTM model
with incremental training proceduresthatconduct thetraining
and evolutionary steps in an alternate manner. Through this
approach, the overhead ofthe GTM training can be reduced.
It should note that various RM-MEDA variants have well
investigated the modeling and sampling procedures of RMMEDA.For
instance,inarecent study [47],somenew components,
including the population partition, nondominated solutions
modeling, and hybrid offspring generation procedures, are
designed and embedded in RM-MEDA to improve its performance,
specifically using local PCA. However, a research gap
exists in the efficient application of the generative model GTM,
which has been initially implemented in MEA/GTM for addressingMOPs.
To address this gap, the proposed LGSEA proves to be
more practical than MEA/GTM, in terms of guiding the search
for offspring generation and incremental trainingGTMmodel.
IV. Experimental Study
To investigate the performance of the proposed LGSEA in
dealing with complicated MOPs, four parts ofexperiments are
conducted in this section.
1) General Performance and Analysis: In the first part, three
selection methods covered the main categories ofMOEAs,
i.e., NSGA-II [4], SMS-EMOA [7], and MOEA/D [9],
are embedded in LGSEA, and comparisons are conducted
within different selections in LGSEA with Operator 1, 2,
and 3. The parameters sensitivity ofK (the size ofthe mating
pool) in LGSEA is investigated, and the performance of
LGSEA with the parents selected from the neighbors and
global population are analyzed;
2) Comparisons With State-of-the-Arts: The optimal combinations
in LGSEA, e.g., the S-metric selection in SMS-EMOA and
the Operator 2,are comparedwith five newly developed and
36 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2023
FIGURE 5 The mean and standard deviations of the IGD
values obtained by LGSEA with different settings on
GLT1-GLT6 test instances over 30 independent runs.
classical learning-based MOEAs on 39 complicated test
instances;
3) Scalability ofAnalysis: LGSEA and five representative learning-based
MOEAs are examined on the WFG test suite
with a large number ofdecision variables;
4) Performance on Real-world Problems: Since LGSEA has shown
good performance on benchmark problems with complicated
PS and PF shapes, the proposed LGSEA is adopted to
deal with several real-world problems in this part.
A. Experimental Settings
This paper focuses on MOPs with both complicated PF and
PS shapes. In our experimental studies, we use several benchmark
test suites, including the GLT [48] test suite with complex
PF shapes, the LZ [10] test suite with intricate PS shapes,
the IMF [43] test suite with variable lineages, and the triobjective
WFG [49] and ZDT [50] test suites. For fair comparisons,
we adopt the recommended parameter settings for the
compared algorithms that have achieved the best performance
reported in the literature. All compared algorithms are implemented
in PlatEMO [51], and each algorithm is run independently
31 times for each instance in this study.
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