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the global best position identified from the entire population (or within a
neighborhood).
44
The rate of position change of the
i
th particle is given by its velocity
v
i
=(
v
i,
1
,v
i,
2
,...,v
i,d
). Equation (9.6) updates the velocity for each
particle in the next iteration step, whereas Equation (9.7) updates each
particle's position in the search space:
89
v
i,d
(
t
)=
τ
(
v
i,d
(
t
−
1) +
φ
1
(
p
i,d
−
x
i,d
(
t
−
1)) +
φ
2
(
p
gd
−
x
i,d
(
t
−
1)))
(9.6)
x
i,d
(
t
)=
x
i,d
(
t
−
1) +
v
i,d
(
t
)
(9.7)
where:
2
τ
=
φ
2
−
)
,
φ
=
φ
1
+
φ
2
,φ>
4
.
0
(9.8)
(
|
2
−
φ
−
4
φ
|
τ
is referred to as the constriction coecient.
Two common approaches of choosing
p
g
are known as gbest and
lbest methods. In the gbest approach, the position of each particle in the
search space is influenced by the best-fit particle in the entire population;
whereas the lbest approach only allows each particle to be influenced
by a fitter particle chosen from its neighborhood. Kennedy and Mendes
studied PSOs with various population topologies,
90
and have shown that
certain population structures could give superior performance over certain
optimization functions.
Further, the role of the inertia weight
φ
, in Equation (9.8), is considered
critical for the PSO's convergence behaviour. Improved performance can be
achieved through the application of an inertia weight applied to the previous
velocity:
v
i,d
(
t
)=
φv
i,d
(
t
−
1) +
φ
1
(
p
i,d
−
x
i,d
(
t
−
1)) +
φ
2
(
p
gd
−
x
i,d
(
t
−
1))
(9.9)
The inertia weight is employed to control the impact of the previous
history of velocities on the current one. Accordingly, the parameter
φ
regulates the trade-off between the global (wide-ranging) and local (nearby)
exploration abilities of the swarm. A large inertia weight facilitates global
exploration (searching new areas), while a small one tends to facilitate
local exploration, i.e., fine-tuning the current search area. A suitable value
for the inertia weight
φ
usually provides balance between global and local
exploration abilities and consequently results in a reduction of the number
of iterations required to locate the optimum solution. Initially, the inertia
