IEEE Computational Intelligence Magazine - August 2022 - 40

ALGORITHM 1 Procedure of the proposed CSD.
Input: ,,
12
levellevel
cc
12
Output: (),( ),12 f (indicator values of the solution sets)
AA f (solution sets to be compared)
CSDA CSDA
1 Remove dominated solutions from each solution set temporarily and determine
the superiority between each two solution sets in terms of convergence by (4);
2 [, ,] ! Obtain the level of each solution set in terms of convergence;
3 Determine the superiority between each two solution sets in terms of sparsity
by (10);
4 [, ,]
ss
12
levellevel ! Obtain the level of each solution set in terms of sparsity;
5 Normalize the objective vectors of all solutions by (11);
6 R ! Generate a set of uniformly distributed reference vectors;
7 Calculate the intersection points of all reference vectors by (14);
8 Calculate the foots of all solutions by (15);
9 [( ),
12
(),]
10 [( ), (),]
11 return (),( ),
12
diversityA diversityA f ! Calculate the diversity of each solution set
by (12);
CSDA CSDA f ! Calculate the CSD value of each solution set by (17);
CSDA CSDA f12 ;
vector r and the surface ff
+=f 1M
ril
== f
=
is calculated:
i
/ 1
j
and the foot () of each solution x
drawn to the surface ff
+= f 1M
fxl
12
is calculated:
ffl () ()
ii
xx
iM
1
=+
-/j=1 f xj()
M
M
,
= 12 f,, ,.
(15)
Hence, the distance between each
solution and reference vector can be
regarded as the Euclidean distance
between the foot of the solution and the
intersection point of the reference vector.
C. Procedure of the Proposed
Indicator
After assigning the solution sets according
to their convergence, assigning the solution
sets according to their sparsity, and
calculating the diversity of the solution
set, their CSD values can be obtained by
integrating the three criteria. Specifically,
the harmonic average of the levels of
each solution set in terms of convergence
and sparsity is first calculated by
ha A
()=
where levelA
c
11
2
levellevel
A
c
and levelA
s
+
A
s
denote the levels
of A in terms of convergence and
sparsity, respectively. Then, the ranking
rankA of each solution set A is obtained
by sorting all the solution sets according
to their harmonic averages of lev(16)
,
++g
M
,, ,, ,
j
r iM
r
12
12
++g
(14)
els. Lastly, the indicator value CSD(A)
of each solution set A is calculated by
A
CSDA diversity()
rankA-1
()=
+ / max diversity(),
i 1
B
rankB = i
=
(17)
where maxrankB i= diversit ()y B indicates
the maximum diversity value among the
solution sets with a ranking of i. This
way, the indicator values of the solution
sets are always larger (i.e., worse) than
those with lower rankings. In other
words, the indicator value is mainly
determined by the convergence and
sparsity of a solution set, and the solution
sets having the same ranking are
distinguished by their diversity. The procedure
of the proposed CSD is summarized
in Algorithm 1.
As for the computational complexity
of the proposed CSD, assuming that the
number of solution sets is L, the number
of solutions in each set is N, and each
solution has M objectives and D variables,
the time complexity of assessing
their convergence is (),OL NM
22
the
time complexity of assessing their sparsity
is O(LND), the time complexity of
assessing their diversity is (),OLNM2
and the time complexity of integrating
the three criteria is ().OL2
In short, the
total time complexity of the proposed
CSD is (),OL NM
22
tion DL .NM%
D. Discussions
As a matter of fact, most performance
indicators assess either convergence or
diversity, while only a few can assess both
40 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2022
with the assumpconvergence
and diversity [49]. Up to now,
IGD and HV are still the most practical
and popular indicators for the convergence
and diversity assessment in multi-objective
optimization, whereas no indicator takes
the sparsity of solutions into consideration.
Thus, the proposed CSD serves as
the first indicator for the convergence,
sparsity, and diversity assessment in sparse
multi-objective optimization. In contrast
to IGD and HV, the proposed CSD does
not need the true Pareto front or any reference
point, thus eliminating the influence
of bias introduced by reference
points. Moreover, it can be found that the
time complexity of the proposed CSD is
the same as some indicators (e.g., IGD)
while much faster than some others (e.g.,
HV). As a result, the proposed CSD is
practical in real-world scenarios.
It should be noted that the proposed
CSD is not Pareto compliant (i.e., nondominated
solutions are always better
than dominated ones), since it gives
equal priority to convergence and sparsity
that may prefer dominated but
sparse solutions. Besides, due to the
polynary nature of CSD, the indicator
values of all the solution sets should be
recalculated if a solution set is changed.
Nevertheless, it will not consume many
additional computational resources since
the time complexity of CSD is polynomial
rather than exponential.
IV. Experimental and Analysis
This section analyzes the performance
of some representative MOEAs on
benchmark and real-world large-scale
sparse MOPs. To illustrate the effectiveness
of the proposed indicator, it is used
to assess the quality of the obtained
solution sets together with some existing
indicators. All the statistical results are
obtained by using the open-source platform
PlatEMO1 [53], where the results
can be easily reproduced by readers.
A. Experimental settings
1) Algorithms
A total of 11 MOEAs are compared in
the experiments, including NSGA-II
1https://github.com/BIMK/PlatEMO
IEEE Computational Intelligence Magazine - August 2022

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