A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 69

Table 13 Multi-objective test functions
Name Variable bounds (dimension n) Formula Optimal solutions Comments

[ - 10 3 ,10 3 ]n (n =1)

f 1 ð x Þ ¼ x 2
f 2 ð

x 2 ½ 0 ; 2 SCH Convex

2

x Þ ¼ ð x 2 Þ

X 3
i¼1 ð

4, 4] n (n =3)

1
p Þ

FON [ Nonconvex

- x 1 ¼ x 2 ¼ x 3

2

f 1 ð x Þ ¼1 exp( x i Þ

1
p ;

1

p

2 ½

X 3
i¼1 ð x i þ

2

1
p Þ

f 2 ð x Þ ¼ 1 exp ð

Þ

p

x 1 = g ð x Þ

[0, 1] n (n = 30) ZDT1 x 1 2 ½0 ; 1 ; x i ¼ 0

i ¼ 2

Convex

f 1 ð x Þ ¼ x 1 ; f 2 ð x Þ ¼ g ð x Þ ½ 1

.

X n
i¼2 x i Þ

;...; n

g ð x Þ ¼ 1 þ 9 ð ð n 1 Þ

f 1 ð x Þ ¼ x 1 ; f 2 ð x Þ ¼ g ð x Þ ½ 1 ð x 1 . g ð x ÞÞ

[0, 1] n (n = 30) ZDT2 Nonconvex

x 1 2 ½ 0 x i ¼ 0

i ¼ 2 ;...; n

; 1 ;

2

.

X n
i

g ð x Þ ¼ 1 þ 9 ð

2 x i Þ

ð n 1 Þ

¼

p

x 1

x 1
g ð x Þ

[0, 1] n (n = 30) ZDT3 x 1 2 ½0 ; 1 ; x i ¼ 0 Convex, disconnected

f 1 ð x Þ ¼ x 1 ; f 2 ð x Þ ¼ g ð x Þ ½ 1 = g ð x Þ sin ð 10 p x 1 Þ

i ¼ 2 ;...; n

.

X n
i

g ð x Þ ¼ 1 þ 9 ð

2 x i Þ

ð n 1 Þ

¼