![](bg5.png)
G. Rastrigin function, defined as:
f (x) = −
∑
N
d
i=1
(x
2
i
−10cos(2πx
i
) + 10),
where x
∗
= 0 and f (x
∗
) = 0 for −5.12 ≤x
i
≤5.12
H. Ackley’s function, defined as
f (x) = −20exp
− 0.2
q
1
30
∑
N
d
i=1
x
2
i
−
exp
1
30
∑
N
d
i=1
cos(2πx
i
)
+ 20 + e,
where x
∗
= 0 and f (x
∗
) = 0 for −32 ≤x
i
≤ 32
I. Griewank function, defined as:
f (x) =
1
4000
∑
N
d
i=1
x
2
i
−
∏
N
d
i=1
cos
x
i
√
i
+ 1,
where x
∗
= 0 and f (x
∗
) = 0 for −600 ≤x
i
≤ 600
4 DISCUSSION OF RESULTS
This section compares the performance of the ba-
sic PSO, BBPSO and BBPSO (EXP) algorithms dis-
cussed in Section 2. We implemented: ring, full,
star, mesh, toroidal and tree neighborhood topologies
and; k-means and hierarchical clustering algorithms
on each PSO algorithm. It is important to note that
neighborhood topologies were determined using par-
ticle indexes and were not based on any spatial infor-
mation. For both clustering algorithms, the euclidean
distance (spatial information) has been used to deter-
mined the distance among particles.
For the basic PSO algorithm, we used w = 0.72
and c
1
= c
2
= 1.49. These values have been shown
to provide good results (Clerc and Kennedy, 2002;
van den Bergh, 2002; van den Bergh and Engelbrecht,
2006).
For all the algorithms used in this section, the
swarm size was s = 50. 200 iterations were perform
by each algorithm (24 algorithms, since there were
implemented 6 topologies + 2 clustering techniques
in 3 PSO variants). Every resulting approach were ex-
ecuted 30 independent runs. These values were used
as defaults for all experiments which use static con-
trol parameters. Also, the distribution of the particles
were 10 ×5 when the mesh and toroidal topologies
were used. For hierarchical and k-means clustering
algorithms, 4 groups were asked for.
Since there are 3 algorithms, 8 different neighbor-
hood topologies, and 9 test functions, it was difficult
to show numerical results. In order to present such
results in a friendly-comparison way, we selected to
show the numerical results as box-plot graphics. In
Figures 2 and 3 shows such results.
From Figures 2 and 3, it is easily to see that the
BBPSO (EXP) version is highly dependent of the in-
terconnection topology, since it only presented a good
behavior when the fully connected, star and hierarchi-
cal approaches were used. However, when the remain
topologies were used, it presented the worst results
among the three algorithms. Therefore, we can say
that PSO and BBPSO algorithms showed a more ro-
bust behavior.
When PSO was used, the best topologies which
showed better results were toroidal, ring and mesh
(see Figures 2(a), 2(b), 2(c), 2(e), 3(b), 3(c), 3(d),
which corresponds to the sphere, schwefel, step, ro-
tated hyper-ellipsoid, ratrigin, ackley and griewank
test functions, respectively). For the generalized
schwefel test function (see Figure 3(a)) the k-means
clustering algorithm was the approach which per-
form the best. For the Rosenbrock test function all
topologies and clustering algorithms performed well.
In summary, when toroidal, mesh and ring topolo-
gies were used, PSO presented an good performance.
When BBPSO was used, the topologies which per-
formed better were full, star and hierarchical cluster-
ing algorithms in all test functions (see Figures 2 and
3).
In our opinion, BBPSO presented the best perfor-
mance, since it outperformed the other two PSO ap-
proaches in six out of nine test functions (see results
shown in Figures 2(a), 2(b), 2(c), 3(b), 3(c) and 3(d)
which corresponds to sphere, schwefel, step, ratrigin,
ackley and griewank test functions). BBPSO obtained
similar results with respect to the results obtained by
PSO in Rosenbrock test function (see results shown in
Figure 2(d)). The original PSO algorithm outperform
the others two PSO approaches in two out of nine test
functions (see results shown in Figures 2(e) and 3(a)
which corresponds to the rotated hyper-ellipsoid and
the generalized schwefel test functions).
From our results, we can conclude that, the topo-
logy plays a key role in PSO. The original PSO ap-
proach should be used with the toroidal, mesh and
ring topologies, whilst BBPSO should be used with
the fully connected, star or hierarchical clustering
methods.
5 CONCLUSIONS AND FUTURE
WORK
Our main conclusions are the following:
• We found that the use of mesh, toroidal and ring
topologies promote better convergence rates in the
PSO algorithm.
• The use of the fully connected, star and hierarchi-
cal clustering approaches promote better conver-
gence rates in the BBPSO algorithm.
• The topology most widely used (fully connected
topology) did not perform well in PSO algorithm
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