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1
Introduction
Most Real world optimization problems require simultaneous treatment of multiple
objectives [1], and involve a number of inequality and/or equality constraints which
the optimal solutions must satisfy. A generic constrained multiobjective optimization
problem can be formulated as follows:
G
Min
.
f
(
x
),
i
=
1
2
"
m
,
i
G
S
.
t
.
g
(
x
)
0
j
=
1
2
"
,
p
(1)
j
G
h
(
x
)
=
0
j
=
p
+
1
"
,
m
j
x G
G
(
x
=
(
x
,
x
,...,
x
))
where
is the vector of the solutions
and
1
2
n
is the set of feasible solutions that satisfy p inequality con-
x
ʩ
,
ʩ
(
m
p
)
n
straints and
is a n-dimension rectangular
space confined by the low boundary and upper boundary of
equality constraints and
x G as follows.
l
x
u
,
l
k
n
(2)
k
k
k
where l and u are the lower boundary and upper boundary for a decision vari-
able x respectively. Usually, equality constraints are transformed into inequality
form as follows.
|
h j
|
−ʵ
0
j
=
p
+
1
,...,
m
(3)
ʵ is an allowed positive tolerance value.
Over the recent years, constraint handling has become an active area of research
for which numerous approaches have been proposed. Some of the commonly used
constraint-handling techniques are listed below.
a. Penalty functions methods: Penalty functions methods are one of the most com-
monly adopted forms of constraint handling[2] [15]. This method uses the constraint
violation to punish infeasible solutions. In this approach, the fitness of infeasible solu-
tions is degraded using a sum of constraint violations. The penalty functions methods
may work quite well for some constraint handling problem; however, some additional
parameters are required in implementations of most penalty functions schemes. The
result of the optimization process is known to be highly sensitive to these parameters.
As a result, the choice of these parameters is very critical to the success of penalty
functions methods for constrained optimization problems.
b. Ranking approaches: In order to eliminate the need for a penalty parameter, Ru-
narsson and Yao [2] introduced a stochastic ranking method based on the objective
function and constraint violation values, where a probability parameter is used to
determine if the comparison is to be based on objective or constraint violation values.
Besides, methods based on the preference of feasible solutions over infeasible solu-
tions have been proposed. For example, Deb [3] [18] proposed a constrained-
domination principle that is a feasibility-driven rule to compare individuals.
where
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