IEEE Computational Intelligence Magazine - August 2022 - 35
network science [17], signal processing
[18], and economics [19], their Pareto
optimal solutions are generally very
sparse in the decision space, i.e., most
decision variables of the Pareto optimal
solutions are zero. Since it is difficult for
existing large-scale MOEAs to generate
many decision variables of zeros, several
MOEAs have been tailored for solving
large-scale sparse MOPs in recent years
[20]-[22]. By suggesting novel variation
operators and dimensionality reduction
strategies, these MOEAs can not only
highly reduce the decision space of
large-scale MOPs, but also maintain the
sparsity of solutions.
In comparison to other large-scale
MOEAs, the above MOEAs can obtain
sparser and better-converged solutions
for large-scale sparse MOPs. However, a
deep comparative analysis of them and
other large-scale MOEAs has not been
performed, and the performance of
state-of-the-art MOEAs on popular realworld
applications is still unclear. More
seriously, most existing performance
indicators are unsuitable for assessing the
performance of MOEAs on sparse
MOPs. On the one hand, a set of reference
points on the true Pareto front are
required by some indicators (e.g., IGD
[23]), while the true Pareto front is
unknown for real-world MOPs;
although some other indicators (e.g., HV
[24]) require a single reference point not
on the true Pareto front, the reference
point is still difficult to be set for fairness
[25]. On the other hand, existing performance
indicators assess only the convergence
and diversity of solutions in the
objective space, without considering the
sparsity of solutions in the decision space
for a more accurate performance assessment
on sparse MOPs.
To address the above issues, this work
proposes a comprehensive performance
indicator for studying the performance of
state-of-the-art MOEAs on large-scale
sparse MOPs. The proposed indicator can
assess the convergence, diversity, and sparsity
of multiple solution sets simultaneously,
which first sorts the solutions sets
into several levels according to their
dominance relations and sparsity, then
tunes the rankings of the solution sets
according to their diversity. More importantly,
the proposed indicator does not
require the assistance of any reference
point or parameter, which facilitates the
performance comparisons on real-world
applications. In the experiments, we study
the performance of two classical MOEAs,
six large-scale MOEAs, and three sparse
MOEAs on 60 test instances taken from
eight benchmark large-scale sparse
MOPs and seven real-world applications.
The experimental results indicate that the
large-scale MOEAs do not outperform
classical MOEAs, while the sparse
MOEAs are obviously better than classical
MOEAs and large-scale MOEAs for
solving large-scale sparse MOPs. Moreover,
the experiments also verify that the
proposed indicator is more effective than
existing indicators in assessing the solution
sets for sparse MOPs.
The rest of this paper is organized as
follows. Section II introduces the largescale
sparse MOPs in practical applications,
and reviews the state-of-the-art
MOEAs for solving large-scale sparse
MOPs and popular performance indicators
for multi-objective optimization.
Section III is devoted to the description
of the proposed indicator. Section IV
presents the experimental results and
gives some analysis. Finally, conclusions
are drawn in Section V.
II. Related Work
A. Large-Scale Sparse Multi-Objective
Optimization Problems
An unconstrained MOP can be mathematically
defined as:
min ff f
xx x
s. t.
=
=
12
12
f
D
fx xx x
x
M
() (),( ), ,( )
(, ,, ),
^h
f !X
(1)
where solution x denotes a decision vector
containing D variables, ()
fx denotes
its objective vector containing M conflicting
objectives, and Ω is the decision
space. A solution x is said to dominate
and be better than another solution y
(denoted as xy' ), if and only if
()
()
'
ff
ff
ii
jj
xy
xy
# (),
(),
6f
1 7f
iM
jM
=
=
12
12
,, ,
,, ,
.
(2)
The solutions that are not dominated
by any other solutions in Ω are the
Pareto optimal solutions for the MOP,
and the objective vectors of all the
Pareto optimal solutions constitute the
Pareto front of the MOP. Since the
number of Pareto optimal solutions
may be infinite for continuous MOPs,
the goal of solving an MOP is to obtain
a solution set A as an approximation of
all the Pareto optimal solutions, where
the distance between A and the Pareto
front is denoted as its convergence, and
the spread and evenness of A are denoted
as its diversity [26]. For multi-objective
optimization, both the convergence
and diversity of A should be as good
as possible.
An MOP with a large number of
decision variables (i.e., D 100$ in general)
is called a large-scale MOP, and an
MOP whose Pareto optimal solutions
contain many variables of zeros is
known as a sparse MOP. For the MOPs
in many scientific and engineering
fields, they are characterized by both a
large number of decision variables and
sparse Pareto optimal solutions. Firstly,
the objective functions of many realworld
MOPs are calculated based on
large datasets (e.g., training sets in neural
network training [15], transaction datasets
in pattern mining [16], graphs in
critical node detection [17], and expected
returns
in portfolio optimization
[19]), which introduce many decision
variables to be optimized (e.g., weights
in neural network training, items in frequent
pattern mining, nodes in critical
node detection, and ratios of instruments
in portfolio optimization). Secondly,
most decision variables of the
Pareto optimal solutions are zero, which
is determined by the objective functions
of many real-world MOPs. For some
MOPs, the sparsity of solutions is
regarded as an objective to be optimized
(e.g., network complexity in neural network
training and the number of selected
nodes in critical node detection),
which leads to lightweight solutions to
be efficiently implemented in practice.
Although some other MOPs do not
optimize the sparsity of solutions explicitly,
the Pareto optimal solutions are still
very sparse due to the restriction of
objectives, since these MOPs are subset
AUGUST 2022 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 35
IEEE Computational Intelligence Magazine - August 2022
Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - August 2022
Contents
IEEE Computational Intelligence Magazine - August 2022 - Cover1
IEEE Computational Intelligence Magazine - August 2022 - Cover2
IEEE Computational Intelligence Magazine - August 2022 - Contents
IEEE Computational Intelligence Magazine - August 2022 - 2
IEEE Computational Intelligence Magazine - August 2022 - 3
IEEE Computational Intelligence Magazine - August 2022 - 4
IEEE Computational Intelligence Magazine - August 2022 - 5
IEEE Computational Intelligence Magazine - August 2022 - 6
IEEE Computational Intelligence Magazine - August 2022 - 7
IEEE Computational Intelligence Magazine - August 2022 - 8
IEEE Computational Intelligence Magazine - August 2022 - 9
IEEE Computational Intelligence Magazine - August 2022 - 10
IEEE Computational Intelligence Magazine - August 2022 - 11
IEEE Computational Intelligence Magazine - August 2022 - 12
IEEE Computational Intelligence Magazine - August 2022 - 13
IEEE Computational Intelligence Magazine - August 2022 - 14
IEEE Computational Intelligence Magazine - August 2022 - 15
IEEE Computational Intelligence Magazine - August 2022 - 16
IEEE Computational Intelligence Magazine - August 2022 - 17
IEEE Computational Intelligence Magazine - August 2022 - 18
IEEE Computational Intelligence Magazine - August 2022 - 19
IEEE Computational Intelligence Magazine - August 2022 - 20
IEEE Computational Intelligence Magazine - August 2022 - 21
IEEE Computational Intelligence Magazine - August 2022 - 22
IEEE Computational Intelligence Magazine - August 2022 - 23
IEEE Computational Intelligence Magazine - August 2022 - 24
IEEE Computational Intelligence Magazine - August 2022 - 25
IEEE Computational Intelligence Magazine - August 2022 - 26
IEEE Computational Intelligence Magazine - August 2022 - 27
IEEE Computational Intelligence Magazine - August 2022 - 28
IEEE Computational Intelligence Magazine - August 2022 - 29
IEEE Computational Intelligence Magazine - August 2022 - 30
IEEE Computational Intelligence Magazine - August 2022 - 31
IEEE Computational Intelligence Magazine - August 2022 - 32
IEEE Computational Intelligence Magazine - August 2022 - 33
IEEE Computational Intelligence Magazine - August 2022 - 34
IEEE Computational Intelligence Magazine - August 2022 - 35
IEEE Computational Intelligence Magazine - August 2022 - 36
IEEE Computational Intelligence Magazine - August 2022 - 37
IEEE Computational Intelligence Magazine - August 2022 - 38
IEEE Computational Intelligence Magazine - August 2022 - 39
IEEE Computational Intelligence Magazine - August 2022 - 40
IEEE Computational Intelligence Magazine - August 2022 - 41
IEEE Computational Intelligence Magazine - August 2022 - 42
IEEE Computational Intelligence Magazine - August 2022 - 43
IEEE Computational Intelligence Magazine - August 2022 - 44
IEEE Computational Intelligence Magazine - August 2022 - 45
IEEE Computational Intelligence Magazine - August 2022 - 46
IEEE Computational Intelligence Magazine - August 2022 - 47
IEEE Computational Intelligence Magazine - August 2022 - 48
IEEE Computational Intelligence Magazine - August 2022 - 49
IEEE Computational Intelligence Magazine - August 2022 - 50
IEEE Computational Intelligence Magazine - August 2022 - 51
IEEE Computational Intelligence Magazine - August 2022 - 52
IEEE Computational Intelligence Magazine - August 2022 - 53
IEEE Computational Intelligence Magazine - August 2022 - 54
IEEE Computational Intelligence Magazine - August 2022 - 55
IEEE Computational Intelligence Magazine - August 2022 - 56
IEEE Computational Intelligence Magazine - August 2022 - 57
IEEE Computational Intelligence Magazine - August 2022 - 58
IEEE Computational Intelligence Magazine - August 2022 - 59
IEEE Computational Intelligence Magazine - August 2022 - 60
IEEE Computational Intelligence Magazine - August 2022 - 61
IEEE Computational Intelligence Magazine - August 2022 - 62
IEEE Computational Intelligence Magazine - August 2022 - 63
IEEE Computational Intelligence Magazine - August 2022 - 64
IEEE Computational Intelligence Magazine - August 2022 - 65
IEEE Computational Intelligence Magazine - August 2022 - 66
IEEE Computational Intelligence Magazine - August 2022 - 67
IEEE Computational Intelligence Magazine - August 2022 - 68
IEEE Computational Intelligence Magazine - August 2022 - Cover3
IEEE Computational Intelligence Magazine - August 2022 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202311
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202308
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202305
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202302
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202211
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202208
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202205
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202202
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202111
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202108
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202105
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202102
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202011
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202008
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202005
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_202002
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_201911
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_201908
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_201905
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_201902
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_201811
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_201808
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_201805
https://www.nxtbook.com/nxtbooks/ieee/computationalintelligence_201802
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_winter17
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_fall17
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_summer17
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_spring17
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_winter16
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_fall16
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_summer16
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_spring16
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_winter15
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_fall15
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_summer15
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_spring15
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_winter14
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_fall14
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_summer14
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_spring14
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_winter13
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_fall13
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_summer13
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_spring13
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_winter12
https://www.nxtbook.com/nxtbooks/ieee/computational_intelligence_fall12
https://www.nxtbookmedia.com