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particles with
rcloud
=0.5 and
rexcl
=
rconv
=31.5, as suggested in [1]. For FMSO, there
are at most 10 child swarms each has a radius of 25.0. The size of the parent and the
child swarms are set to 100 and 10 particles, respectively [16]. For cellular PSO, a
5-Dimensional cellular automaton with 105 cells and Moore neighborhood with
radius of two cells is embedded into the search space. The maximum velocity of
particles is set to the neighborhood radius of the cellular automaton and the radius for
the random local search (
r
) is set to 0.5 for all experiments. The cell capacity θ is set
to 10 particles for every cell [6].
As depicted in the Table 2 and Table 3, the proposed algorithm outperforms other
tested PSO algorithms when the number of peaks increases.
Table 3.
Offline error ±Standard Error for f =5000 and f =10000
Proposed
algorithm
f=5000
MultiSw
armPSO
f=5000
CellularP
SO
f=5000
FMSO
f=5000
mQSO10
f=5000
Proposed
algorithm
f=10000
MultiSw
armPSO
f=10000
CellularP
SO
f=10000
FMSO
f=10000
mQSO10
f=10000
1
2.54±0.19
0.56
±0.04
2.55±0.12
3.44±0.11
3.82±0.35
1.52±0.17
0.27
±0.02
1.53±0.12
1.90±0.06
1.90±0.18
5
1.49±0.11
1.06
±0.06
1.68±0.11
2.94±0.07
1.90±0.08
0.88±0.11
0.70
±0.10
0.92±0.10
1.75±0.06
1.03±0.06
10
1.44
±0.10
1.51±0.04
1.78±0.05
3.11±0.06
1.91±0.08
0.91
±0.06
0.97±0.04
1.19±0.07
1.91±0.04
1.10±0.07
20
1.85
±0.11
1.89±0.04
2.61±0.07
3.36±0.06
2.56±0.10
1.32
±0.07
1.34±0.08
2.20±0.10
2.16±0.04
1.84±0.08
30
2.00
±0.09
2.03±0.06
2.93±0.08
3.28±0.05
2.68±0.10
1.35
±0.05
1.43±0.05
2.60±0.13
2.18±0.04
2.00±0.09
2.02
±0.08
2.04±0.06
3.14±0.08
3.26±0.04
2.65±0.08
1.27
±0.04
1.47±0.06
2.73±0.11
2.21±0.03
1.99±0.07
40
50
2.03
±0.08
2.08±0.02
3.26±0.08
3.22±0.05
2.63±0.08
1.30
±0.03
1.47±0.04
2.84±0.12
2.60±0.08
1.99±0.07
100
2.23±0.04
2.14
±0.02
3.41±0.07
3.06±0.04
2.52±0.06
1.32
±0.03
1.50±0.03
2.93±0.09
2.20±0.03
1.85±0.05
200
2.54±0.19
0.56
±0.04
2.55±0.12
3.44±0.11
3.82±0.35
1.51±0.02
1.48
±0.02
2.88±0.07
2.00±0.02
1.71±0.04
For all algorithms we reported the average offline error and 95% confidence
interval for 100 runs. Offline error of the proposed algorithm, mQSO10(5+5q) [1],
FMSO [16], cellular PSO [6], and Multi-Swarm PSO [9] for different dynamic
environment is presented in Table 2 and Table 3. For each environment, result of the
best performing algorithm(s) with 95% confidence is printed in bold.
5 Conclusions
In this paper, a new PSO algorithm is proposed to deal with the dynamic
environments. In the proposed PSO there is one Learning Automaton per each particle
which it is to learn for its corresponding particle how to act during the evolution. To
prevent redundant search in the same area, the LA belonging to particle
P
i
, which is
denoted by
L
i
, learns the relationship between the variance of the solutions,
normalized distance between the position
x
i
and position of its local optima and
normalized distance between the position of its local optima and global optima, and
the behavior of the particles
i
. Indeed the proposed PSO is a kind of indirect niching
method. In addition, the deluge water level is employed during the evolution.
Results of the experiments show that for many tested dynamic environments the
proposed algorithm outperforms all competent tested PSO algorithms.
References
1.
Blackwell, T., Branke, J.: Multi-Swarms, Exclusion, and Anti-Convergence in Dynamic
Environments. IEEE Transactions on Evolutionary Computation 10, 459-472 (2006)
2.
Blackwell, T., Branke, J.: Multi-Swarm Optimization in Dynamic Environments.
Applications of Evolutionary Computing, 489-500 (2004)
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