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x
#
x
*
x
#
is
feasible?
x
*
is
feasible?
x
*
dominates
x
#
?
x
*
Eliminate
x
x
*
is
feasible?
*
f(
x
*
)
f(
x
#
)
?
Repl
ace
x
#
with
x
*
x
G
and
x
G
and replacement of
x
G
with
x
G
#
*
#
*
Fig. 4.
Comparison between
It has been observed that the infeasible HM members were able to be evolved in
the modified HS method. In other words, we do not have to search for new feasible
HM members by repeatedly examining them with the constraint functions, as in
Fig. 3. Compared with the original HS method, our approach needs only a considera-
bly smaller number of constraint functions evaluation, and thus, can provide more ef-
ficient solutions. This advantage will be demonstrated using computer simulations in
Section 5.
5 Simulations
In this section, we investigate the effectiveness of the two modified HS methods with
simulation examples of the multi-modal and constrained optimization problems.
5.1 Multi-modal Optimization Problems
In this example, the multi-modal optimization capability of our first modified HS
method is examined using the following two functions [11]:
f
(
x
,
y
)
=
200
−
(
x
2
+
y
−
11
)
2
+
(
x
+
y
2
−
7
)
2
,
−
5
≤
x
,
y
≤
5
(7)
1
f
2
(
x
,
y
)
=
x
sin(
4
π
x
)
−
y
sin(
4
π
y
+
π
)
+
1
,
−
1
≤
x
,
y
≤
1
(8)
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