Progressively Interactive Evolutionary Multi-Objective Optimization ...

F2
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
y=1
Upper level
PO front
0.9
B
0.8 0.7071 0.6
A
Lower level
PO fronts
0.5
−1
−2
−1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6
F1
Figure 1: Pareto-optimal fronts of upper
level (complete problem) and some
representative lower level optimization
tasks areshown for problemTP1.
C
F2
2
1.5
1
0.5
Upper Level
front
0
0
x1=1
x1=y
C
0.5
A
y=0.25
y=0.5
1
F1
y=0.75
Lower level
fronts
x1=0
y=1
1.5
B
y=1.5
x1=1
Figure 2: Pareto-optimal front of the upper
level problem for problem TP2 with
xi = 0 for i = 2, . . . , K.
Forafixedvalueof y,thePareto-optimalsolutionsofthelowerleveloptimizationproblem
are given as follows: {xl ∈ R K x1 ∈ [0, y], xi = 0, for i = 2, . . . , K}. However, for
the upper level problem, Pareto-optimalsolutions correspond tofollowing conditions:
{x ∈ R K+1 x1 = y, xi = 0, for i = 2, . . . , K, y ∈ [0.5, 1.0]}. If an algorithm fails to
find the true Pareto-optimal solutions of the lower level problem and ends up finding
a solution below the ‘x1 = y’ curve in Figure 2 (such as solution C), it can potentially
dominateatruePareto-optimalpoint(suchaspointA)therebymakingthetaskoffinding
true Pareto-optimal solutions a difficult task. We use K = 14 here, so that the total
number of variablesis 15inthis problem.
4.3 TestProblemTP3
This problemis takenfrom(Eichfelder,2007):
2
x1 + x2 + y + sin
Minimize F(x) =
2 (x1 + y)
cos(x2)(0.1 + y)(exp(− x1
0.1+x2 )
subjectto
⎧
(x1−2)
argmin f(x) =
(x1,x2)
⎪⎨
(x1, x2) ∈
⎪⎩
2 +(x2−1) 2
+ 4
x2y+(5−y1) 2
g1(x) = x2 − x 2 1 ≥ 0
g2(x) = 10 − 5x 2 1 − x2 ≥ 0
g3(x) = 5 − y
− x2 ≥ 0
6
g4(x) = x1 ≥ 0
G1(x) ≡ 16 − (x1 − 0.5) 2 − (x2 − 5) 2 − (y − 5) 2 ≥ 0,
0 ≤ x1, x2, y ≤ 10.
,
16
x 2 1 +(x2−6) 4 −2x1y1−(5−y1) 2
80
+ sin( x2
10 )
For this problem, the exact Pareto-optimal front of the lower or the upper level optimization
problem are not derived mathematically. For this reason, we do not consider
thisproblemanyfurtherhere. SomeresultsusingourearlierBLEMOprocedurecanbe
found elsewhere(Deband Sinha,2009b).
81
⎫
⎪⎬
,
⎪⎭
2
(5)