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(HMCR). The HMCR is defined as the probability of selecting a component from the
HM members, and 1-HMCR is, therefore, the probability of generating it randomly. If
comes from the HM, it can be further mutated according to the Pitching Adjust
Rate (PAR). The PAR determines the probability of a candidate from the HM to be
mutated. Obviously, the improvisation in HS is rather similar to the production of off-
spring in genetic algorithm (GA) [6] with the mutation and crossover operations.
However, GA creates new chromosomes using only one (mutation) or two (simple
crossover) existing ones, while the generation of new solutions in the HS method
makes full use of all the HM members.
Step 3.
Update the HM. First, the new solution from Step 2 is evaluated. If it yields a
better fitness than that of the worst member in the HM, it will replace that one. Oth-
erwise, it is eliminated.
Step 4.
Repeat Step 2 to Step 3 until a termination criterion (e.g., maximal number of
iterations) is met.
Similar to the GA and particle swarm algorithms [7], the HS method is a random search
technique. It does not require any prior domain knowledge, such as gradient information
of the objective function. However, different from those population-based approaches, it
only utilizes a single search memory to evolve. Therefore, the HS method has the distin-
guished feature of algorithm simplicity. Note that the HS memory stores the past search
experiences, and plays an important role in its optimization performance. In the next sec-
tion, we employ a fish swarm-based strategy to improve the management of the HM
members so that the HS method can cope with the multi-modal optimization problems.
x
′
j
3 Modified HS Method for Multi-modal Optimization
Multi-modal optimization is an important but challenging topic in the field of optimi-
zation [8]. Unfortunately, it is difficult for the HS method to locate all the global op-
tima of the multi-modal problems, because the HM members can be easily stagnated
into one or several of them during iteration. Thus, the key issue of applying the HS
method to the multi-modal optimization is to effectively maintain the diversity of the
HM members. The regular HM management policy has been explained in Section 2.
However, some additional approaches are needed to determine whether a solution
from Step 2 can replace the worst member in the HM. Indeed, the qualification of a
candidate solution as a new HM member should be based on not only its fitness but
also its similarity to the existing members.
Inspired by the artificial fish swarm algorithm [9], we propose a new control
mechanism, as shown in Fig. 2, for updating the HM in our modified HS method so as
to attack the multi-modal problems. Suppose the fitness of the current HM members
is denoted as
i
f
(
(
)
i
=
1
"
2
,
HMS
). After a new candidate solution
x
′
,
x
′
,
"
,
,
x
′
1
2
n
with the fitness
f
′
, is obtained, we first measure its distances
d
(
i
=
1
"
2
,
HMS
)
to all HM members:
[
]
[
]
i
i
i
n
′
′
"
′
"
d
=
x
,
x
,
,
x
−
x
,
x
,
,
x
,
(2)
i
1
2
n
1
2
where
is an appropriately selected distance metric.
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