IEEE Computational Intelligence Magazine - August 2022 - 38

them up, as they have different magnitudes
and the predefined weights
will introduce bias.
On the other hand, the reference
points are generally inevitable in the
assessment of convergence and diversity.
Some indicators (e.g., GD and IGD)
measure the difference between a solution
set and a set of reference points,
where the reference points should be
uniformly sampled on the Pareto front.
Some others (e.g., CLn
and HV) calculate
the area covered by a solution set
with respect to a reference point,
which can be the nadir point (i.e., the
point consisting of the maximum
objective values of the Pareto front) of
the problem. While the reference points
required by these unary indicators are
difficult to be determined in practice, it
is desirable to design a polynary indicator
to compare multiple solution sets
with each other. That is, the reference
points for each solution set consist of
all
the other compared solution sets,
and any other reference point is no
longer required.
To meet the above considerations,
this work proposes a performance indicator
to assess the convergence, sparsity,
and diversity of the solution sets for
sparse MOPs, termed CSD. Given multiple
solution sets, the proposed CSD
first assigns them into several levels
according to their convergence and
sparsity, then sorts the solution sets in
the same level according to their diversity.
This way, the convergence and sparsity
are prioritized by the proposed
indicator and all the three criteria can
be easily integrated into a scalar. Moreover,
no additional weight or reference
point is required.
B. Criteria for Assessing Convergence,
Sparsity, and Diversity
The Pareto dominance introduces partial
orders between solutions, and solutions
can be sorted into several levels
according to their dominance relations
(i.e., using non-dominated sorting [44]).
Similarly, the proposed CSD sorts multiple
solution sets according to the
dominance relations between the solutions
in each two sets. More specifically,
a non-dominated solution set A is superior
over a non-dominated solution set
B in terms of convergence if
|: .
|: .
)
"
"
yx xy
xy xy
!! '
!! '
BA
AB
7
7
,
,
2
2
05
05
B
A
.
(4)
That is, A is superior over B if more
than half the solutions in B are dominated
by at least half the solutions in A.
Obviously, the above relation is irreflexive,
antisymmetric, and nontransitive.
which is contradictory with the precondition
that B is a non-dominated solution
set. Therefore, two non-dominated
solution sets cannot be superior over
each other.
■
Solution Set A (Level 1)
Solution Set B (Level 1)
Solution Set C (Level 2)
Solution Set D (Level 2)
Solution Set E (Level 3)
In [45], solution set A outperforms
solution set B only if all the solutions in B
are dominated by those in A. While such
a definition is so strict that all the solution
sets may not be superior over any other,
the ratio of dominated solutions in B
should be set to a small value. On the
other hand, the relation may be symmetric
(i.e., A and B are superior over each
other) if the ratio of dominated solutions
is smaller than 0.5. Hence, the ratio is set
to 0.5 here for the sake of distinguishability
and antisymmetry.
After determining the relations
First Objective
FIGURE 1 Levels of five non-dominated solution sets in terms of convergence.
between each two solution sets, the
solution sets not inferior to any others
are assigned to the first level and temporarily
ignored. Then, the remaining
solution sets not inferior to any others
are assigned to the second level, and the
operation repeats until all the solution
sets are assigned. As illustrated in Fig. 1,
solution set A is assigned to the first
38 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2022
The theoretical proof of the antisymmetric
nature is given as follows.
Proof: Assuming that two non-dominated
solution sets P and Q are superior
over each other, according to the second
condition in (4) when P is superior
over Q:
|| {| :}xy xySP Q!!7
205
1 =
.| |.P
'
(5)
Besides, according to the first condition
in (4) when Q is superior over P:
|| {| :}xy yxSP Q
2 = !!7
205
.| |.P
Since || .| |,|| .| |,SP SP05
SS ,P12 3 we have
+
,
SS .12! 4
! +
12
,
(7)
Let x SS according to the
definition of S1, there exists a y Q! satisfying
xy
.'
Besides, according to the
definition of S2, there exists a y Q!l
satisfying
yx .'l
That is,
yx ,y''l
(8)
'
(6)
12 0522 and
Second Objective
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