IEEE Computational Intelligence Magazine - August 2022 - 39

level since all its solutions are nondominated,
and solution set B is also
assigned to the first level since fewer
than half the solutions in B are dominated
by those in A. In fact, all the nondominated
solutions in A and B
constitute the Pareto front and it is reasonable
to assign both solution sets to
the first level. By contrast, solution set C
contains four solutions dominated by
those in A and solution set D contains
four solutions dominated by those in B,
hence both C and D are assigned to the
second level. In addition, solution set E
has four solutions dominated by those
in C and is assigned to the third level.
In terms of sparsity, it originally indicates
the ratio of zero variables in a
decision vector. Thus, the sparsity of a
solution set A can be defined as
sparsity A =- 11 / x ,
A D x A
()
1
!
where D denotes the number of variables
and x 0
(9)
denotes the number of
nonzero variables in x. While the sparsity
of solution sets is continuous, it
should be discretized to allow the existence
of multiple solution sets in the
same level. For this aim, solution set A is
superior over solution set B in terms of
sparsity if A is sparser than B and the
difference between their sparsity is not
less than
D
1
sparsity AsparsityB +
, i.e.,
()
The value
1
$
D
1
()
D
1
.
(10)
indicates the minimum
difference between the sparsity of
two solution sets. Since sparsity(A) =
sparsity(B) + D
means that each solution
in A has one less nonzero variable
than each solution in B on average, a
difference less than
D
1
is meaningless
and the two solution sets are regarded
as having the same sparsity in this case.
Then, the solution sets can be assigned
to several levels according to their sparsity
via the same procedure for assigning
the solutions sets according to their
dominance relations.
Since the diversity is used to distinguish
between the solution sets in the
same level, any criterion for diversity
z*
First Objective
(a)
Solutions
z*
First Objective
(b)
Reference Vectors
FIGURE 2 Difference between the diversity assessment based on two different sets of reference
vectors. (a)
c ( ,,,,,,).1021201=
(b) c ( ,,,,,,).1111111=
assessment can be adopted. While diversity
assessment is a challenging issue and
existing diversity indicators have more
or fewer limitations [26], this work does
not aim to suggest a powerful diversity
criterion to address the shortcomings of
existing indicators. By contrast, the proposed
CSD uses a reference vector based
diversity criterion for simplicity, where
the uniformly distributed reference vectors
have been widely used in the environmental
selection and diversity
assessment of MOEAs [46], [47]. To be
specific, each objective vector ()
fx in all
the solution sets is first normalized
according to the ideal point z)
nadir point z :
()
nad
f ()=
i x
zz
fz
i x
where z)
nad iM (11)
ii
-
-
)
i
) ,, ,, ,
=12 f
consists of the minimum
objective values in all the solution sets
and znad
consists of the maximum objective
values in all the solution sets.
Afterwards, a set of uniformly distributed
reference vectors R is generated, which
has the same size as the solution set having
the most solutions. The reference
vectors can be generated by the Das and
Dennis's method in general, and by the
mixture uniform design if the Das and
Dennis's method cannot generate the
required number of reference vectors
[48]. To assess the diversity of each soludiversity()
and
tion
set A, each reference vector is associated
with the solution closest to it.
Then, the number cx
of reference vectors
associated with each solution x
is counted:
cR argmin==! " |((),) ,xyAdis fy r ,
(12)
rx
where dis (( ), )fy r
!
denotes the distance
between each solution and reference
vector, and the standard deviation of all
cx
Ac c
A
=!
x
1
/^hr 2
x A
,
is regarded as the diversity of the
solution set:
(13)
where cr denotes the mean value of
all
c .x
It is worth noting that the proposed
CSD adopts a set of parallel reference
vectors rather than a set of reference
vectors starting from the origin. As illustrated
in Fig. 2, when the seven solutions
uniformly distribute on a convex
curve, they are associated with
,,,,,,
102120 1 reference vectors starting
from the origin, and are associated with
,,,,,,
111111 1 parallel reference vectors.
Obviously, the parallel reference vectors
can better assess the diversity of solution
sets lying on convex or concave surfaces.
While the generated reference vectors
always start from the origin, the intersection
point rl between each reference
AUGUST 2022 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 39
Second Objective
Second Objective
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