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x
10
6
8
GA with traditional crossover
GA with multiple crossover
Standard PSO
Hybrid algorithm
7
6
5
4
3
2
1
0
0
2000
4000
6000
8000
10000
Ite
r
ation
800
600
400
200
0
2000
3000
4000
5000
6000
7000
8000
9000
10000
Fig. 5 The evolutionary trajectory of
the single-objective optimization algorithm on the
Rosenbrock test function
De
nition 1 Pareto Dominance: It says that the vector
~
u
¼½
u
1
;
u
2
; ...;
u
n
domi-
~
~
\
~
nates the vector
v
¼½
v
1
;
v
2
; ...;
v
n
and it illustrates
u
v, if and only if:
8
i
2f
1
;
2
; ...;
n
g :
u
i
v
i
^9
j
2f
1
;
2
; ...
n
g :
u
j
\
v
j
R
n
De
nition 2 Non-dominated: A vector of decision variables
~
x
2
X
is non-
x
0
2
dominated, if there is not another
~
X which dominates
~
x. That is to say that
:
f
x
0
Þ
\
f
, where
f
x
0
2
x
0
8~
x
2
X
; 6 9~
X
;~
x
6
¼
~
ð~
ð~
x
Þ
¼
f
f
1
;
f
2
; ...;
f
m
g
denotes the
vector of objective functions.
R
n
,where X
is the design feasible region, is Pareto-optimal if this vector is non-dominated in X.
x
2
De
nition 3 Pareto-optimal: the vector of decision variables
~
X
De
nition 4 Pareto-optimal set: In multi-objective problems, a Pareto-optimal set
or in a more straightforward expression, a Pareto set denoted by P
consists of all
Pareto-optimal vectors, namely:
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