Finding Knees in Multi-objective Optimization

Fig. 3. Comparison of NSGA-II with (a) angle-measure, (b) 4-angle-measure and (c) utilitymeasure
on a simple test problem.
Then, the DO2DK test problem is defined as follows:
¡
x¤�¦ min f1 g ¡ x¤ r ¡ � πx1� x1¤�� sin 2 s¥ 1 ¨ ¨�� 1 2s �
2
1
s¥ 2 � π 1� � ¨
¡
x¤�¦ min f2 g ¡ x¤ r ¡ ¡ ¡
πx1� ¨ x1¤
π¤�¨ 1¤
cos 2
g ¡ ¨ x¤�¦
�
1
n 9
n 1 ∑ xi
i � 2
1
K cos¡ 2Kπx1¤�� 2 s 2
0 � xi � 1 i ¦ 1£ 2£ ����� £ n
r ¡ x1¤�¦ 5 ¨ 10 ¡ x1 � 0� 5¤ 2 ¨
The parameter s in that function skews the front.
Let us first look at an instance with a very simple front which is convex and has a
single knee, using the parameters K ¦ 1£ n ¦ 30, and s ¦ 0. Figure 3 compares the nondominated
front obtained after running NSGA-II with the three proposed methods, the
angle-measure, the 4-angle-measure, and the utility-measure for 10 generations with a
population size of 200. As can be seen, all three methods clearly focus on the knee. The
run based on the utility-measure has the best (most regular) distribution of individuals
on the front. As expected, the 4-angle-measure puts a stronger focus on the knee than
the standard angle-measure.
Now let us increase the number of knees ¡ K ¦ 4¤ and skew the front ¡ s ¦ 1� 0¤ .
The non-dominated front obtained after 10 generations with a population size of 100 is
depicted in Figure 4. As can be seen, both measures allow to discover all knees. The
utility-measure shows a wider distribution at the shallow knees, while the angle-based
measure emphasizes the stronger knees, and also has a few solutions reaching into the
concave regions.