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Next, we calculate the number of the HM members,
M
, which are in the vicinity
V
of
(
)
′
′
′
,
1
"
. In other words, only the HM members, whose
d
are smaller
than
V
, are counted here. The average fitness of these HM members,
F
, is given as
follows:
x
x
,
,
x
n
M
∑
=
f
i
(3)
i
1
F
=
.
M
(
)
,
1
"
will replace the worst member of the HM only if it meets the
following three conditions:
Thus,
x
′
x
′
,
,
x
′
n
1)
f
is better than that of the worst HM member,
2)
M
is smaller than a preset threshold
′
M
,
V
3)
f
′
is better than
F
.
It is concluded from the above explanations that our approach can prevent the over-
similarity among the HM members so that the diversity of the HS solutions can be
maintained. That is to say, the modified HS method is well-suited for handling the
multi-modal problems. Nevertheless, the proposed technique has two drawbacks.
Firstly, parameters
V
and
M
are always applications dependent, and are usually
chosen based on
trial and error
. They can considerably affect the multi-modal opti-
mization performance of the modified HS method. Unfortunately, there is no analytic
way yet to guarantee their best values. Secondly, as in (2), the distances between
(
)
′
′
′
,
1
"
and all the present HM members have to be calculated. This require-
ment can certainly result in a time-consuming procedure, in case of a large
HMS
.
x
x
,
,
x
n
4 Modified HS Method for Constrained Optimization
4.1 Constrained Optimization Problems
Most of the practical optimization problems are actually constrained optimization
problems, whose goal is to find an optimal solution that satisfies certain given con-
straints [10]. In general, a constrained optimization problem is described as follows:
G
G
find
x
=
(
x
,
x
,
"
,
x
)
to minimize
f(
x
)
,
1
2
n
G
G
subject to
g
(
x
)
≤
0
,
i
=
1
"
2
,
I
and
h
(
x
)
=
0
,
j
=
1
"
2
,
J
i
j
G
G
G
where
are the inequality and
equality constraint functions, respectively. As a matter of fact, the equality constraint
functions can be transformed into inequality constraint functions:
f(
x
)
is the objective function, and
g
(
x
)
and
h
(
x
)
i
j
G
h
(
x
)
− ε
≤
0
,
(4)
j
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