IEEE Computational Intelligence Magazine - August 2022 - 37

categories. The first category of indicators
assesses the convergence of a solution
set A, for instance, RNI (i.e., ratio
of non-dominated individuals) [37] calculates
the ratio of non-dominated solutions
in A, Purity [38] counts the
solutions in A that are non-dominated
with those in other solution sets, and
GD (i.e., generational distance) [39]
measures the mean distance between
each solution in A to the reference
points on the true Pareto front. The
second category assesses the diversity of
a solution set A, for example, Spacing
[40] calculates the standard deviation of
the minimum distances of each solution
to the others in A, CLn
[41] counts the
number of hypercubes having at least
one solution, and CPF (i.e., coverage
over Pareto front) [26] calculates the
coverage of A over the reference points
on the true Pareto front. The third category
assesses both the convergence and
diversity of a solution set, such as IGD
(i.e., inverted generational distance) [23]
calculating the mean distance between
each reference point on the true Pareto
front and the solutions in A, and HV
(i.e., hypervolume) [24] calculating the
area covered by A with respect to a reference
point.
The third category of indicators is
the most widely used since both the
convergence and diversity should be
considered in multi-objective optimization.
However, IGD requires a set of
uniformly distributed reference points
on the true Pareto front, which is difficult
to be sampled for MOPs with
irregular Pareto front and impossible for
real-world MOPs whose Pareto fronts
are unknown [42]. HV requires a reference
point consisting of slightly larger
objective values than the solutions,
which is also difficult to be determined
for a relatively fair comparison, where
the variation of the reference point may
change the performance rankings of
multiple MOEAs [25]. Therefore, it is
not easy to use these indicators in practice,
especially for engineers not familiar
with MOEAs. Moreover, most indicators
assess the convergence and diversity
of solutions in the objective space,
while only few consider the decision
variables of solutions [43], and none of
them consider the sparsity of solutions.
Since sparse solutions contain a few elements
corresponding to easy implementation
and high efficiency, they are
likely to be preferred by decision makers.
Therefore, the sparsity of solutions
should also be considered as a criterion
when assessing the performance of
MOEAs on sparse MOPs.
In view of the limitations of existing
indicators, this work proposes a new performance
indicator for the performance
assessment on sparse MOPs in Section
III, where the convergence, diversity, and
sparsity of multiple solution sets can be
assessed without using any reference
point. Then, the proposed indicator is
used to assess the performance of 11
MOEAs on 60 large-scale sparse MOPs
in Section IV.
III. The Proposed
Performance Indicator
A. Main Idea of the
Proposed Indicator
Unlike the convergence and diversity
that can be simultaneously assessed in
the objective space (e.g., by using
IGD and HV), the sparsity of solutions
is solely assessed in the decision
space. Hence,
it has
to separately
assess the convergence, diversity, and
sparsity by different criteria and integrate
them into a scalar. Considering
that the primary goal is to minimize
the objectives and the sparsity is a
pivotal factor in decision making, the
convergence and sparsity are given
priority over the diversity. Besides, it
is unreasonable to assign different
weights to the three criteria and sum
TABLE I Applicability of some representative MOEAs to large-scale sparse MOPs.
SUITABLE FOR
ALGORITHM
NSGA-II [29]
MOEA/D-DE [30]
CCGDE3 [2]
CCLSM [28]
MOEA/DVA [5]
LMEA [6]
WOF [7]
ReMO [9]
PCA-MOEA [10]
LSMOF [8]
IM-MOEA [32]
LMOCSO [11]
DGEA [12]
S3-CMA-ES [14]
SparseEA [20]
MOEA/PSL [22]
PM-MOEA [27]
MAIN SEARCH STRATEGY
Crossover and
mutation
Differential evolution
Random grouping
Differential grouping
Variable analysis
Variable clustering
Problem
transformation
Random embedding
Principal component
analysis
Problem reformulation
Gaussian process
Competitive swarm
optimizer
Adaptive offspring
generation
Covariance matrix
adaptation evolution
strategy
Bi-level encoding
Unsupervised neural
networks
Pattern mining
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AUGUST 2022 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 37
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LARGE-SCALE
MOPs
BINARY
MOPs
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SPARSE
MOPs
IEEE Computational Intelligence Magazine - August 2022

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