IEEE Computational Intelligence Magazine - February 2022 - 94

tasks. When hundreds of thousands of
non-dominated solutions are obtained, it
is likely that the decision maker wants to
examine only its subset (e.g., 10, 100 or
1000 solutions). At the same time, he/she
may want to know the reason why the
presented subset is selected.
To summarize this
section, it
is
important to compare EMO algorithms
under multiple population size specifications
since different algorithms work
well under different specifications. When
different population size specifications
are used for different EMO algorithms,
fair comparison can be performed by
selecting a pre-specified number of solutions
from the final population or from
all the evaluated solutions in each run of
each algorithm.
IV. Choice of Performance
Indicators
For performance comparison,
results between two solution sets by a
Pareto compliant indicator are always
consistent with Pareto dominance-based
comparison results. Let S1
and S2
solution sets where S1
be two
is better than S2
based on the Pareto dominance relation
(see [16] for the " better " relation between
solution sets). In this case, S1
is always
evaluated as better than S2
by a Pareto
compliant indicator. The hypervolume
indicator is Pareto compliant whereas
IGD is not. That is, S1
hypervolume value than S2
mentioned case while S1
worse IGD value than S .2
can have a
This issue will
be further discussed later for IGD.
For a two-objective linear Pareto
the
hypervolume [24] and IGD [45] indicators
are often used. However, as shown
in this section, different results are
obtained depending on the reference
point specification for hypervolume calculation
and the reference point set
specification for IGD calculation. The
best solution set for each indicator is not
always a uniform solution set which fully
covers the entire Pareto front.
A. Difficulties in the Use of
Hypervolume
One reason for the frequent use of the
hypervolume indicator is that no other
indicator is known as Pareto compliant
[51]. Roughly speaking, comparison
1
front, it has been theoretically shown
that a uniform solution set is optimal for
hypervolume maximization [52]-[54].
Let us consider the optimal distribution
of n solutions on a linear Pareto front
for hypervolume maximization with the
reference point
r (, ).rr=
In Fig. 16, it is
assumed that the Pareto front is a line
between (0, 1) and (1, 0). When the
inequality relation
r 11 1)+-n
$
/(
holds, a uniform solution set including
(1, 0) and (0, 1) maximizes the hypervolume
indicator as shown in Fig. 16(a).
Intuitively, Fig. 16(a) shows the solution
set with the best diversity and the
best uniformity.
On the other hand, when r <
/(
r (, .)11=
11 ),1n+- the optimal distribution
of solutions does not include the two
extreme points (0, 1) and (1, 0), e.g., see
Fig. 16(b) with
Note in
Fig. 16(b) that a uniformly distributed
inside solution set is optimal. Figure 16
1
always has a better
in the aboveshows
that different solution sets are
optimal depending on the reference
point specification. However, its effect
is not large in the case of a two-objective
linear Pareto front since the same
solution set is always optimal when
/(
r 11 1). For the case of three
$ +-n
or more objectives, its effect is totally
different depending on the Pareto front
shape as shown later in this section.
For nonlinear Pareto fronts, uniformly
distributed solutions are not optimal
[54]. Figure 17 shows two examples
where SMS-EMOA [4] is used to find
near-optimal distributions of nine solutions
for DTLZ2 and Minus-DTLZ2
with
r (, ).22=
In Fig. 17(a), the two0
01
Minimize
f1
(a)
Minimize f1
(b)
FIGURE 16 Optimal solution distributions for hypervolume maximization with nine solutions
()9n = for different reference points r (, ).rr=
(b) r (, ).11=
$ +-n ==
/( )/ .;
(a) r 11 19 81125
1
objective DTLZ2 test problem [17] with
no distance variable is used. Since the
number of distance variables is zero, all
feasible solutions are Pareto optimal (i.e.,
they are always on the Pareto front).
Thus, the role of SMS-EMOA is to find
the best distribution of nine solutions.
The same setting as in the previous sections
is used in Fig. 17 except for the
population size ()9n = and the termination
condition (1,000,000 solution
evaluations). To find a near-optimal distribution,
the best solution set is selected
from ten runs of SMS-EMOA.
Figure 17(a) shows the obtained solution
set, which can be viewed as being
very close to the optimal distribution
since it is obtained using large computation
load (i.e., 10 independent runs
with 1,000,000 solution evaluations).
In the same manner, Fig. 17(b) is
obtained on the two-objective MinusDTLZ2
test problem [23] with no distance
variable. Figure 17 shows that the
optimal distribution of solutions is not
uniform on nonlinear Pareto fronts.
More solutions are located around the
center than the two extreme points
(1, 0) and (0, 1) in Fig. 17. For the convex
Pareto front in Fig. 17(b), the two
extreme points are not obtained. For
example, the right-bottom point in
Fig. 17(b) is not (1, 0). This is because
the two extreme points of the convex
Pareto front do not have large hypervolume
contributions [54], [55].
The non-uniformity of optimal distributions
is more clearly demonstrated
94 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2022
Minimize f2
Minimize f2
IEEE Computational Intelligence Magazine - February 2022

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - February 2022
