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Proceedings ArticleDOI

Performance of Multi-chaotic PSO on a shifted benchmark functions set

01 Apr 2015-Vol. 1648, Iss: 1, pp 550023
TL;DR: It is concluded that using the multi-chaotic approach can lead to better results in optimization of shifted functions.
Abstract: In this paper the performance of Multi-chaotic PSO algorithm is investigated using two shifted benchmark functions. The purpose of shifted benchmark functions is to simulate the time-variant real-world problems. The results of chaotic PSO are compared with canonical version of the algorithm. It is concluded that using the multi-chaotic approach can lead to better results in optimization of shifted functions.

Summary (1 min read)

INTRODUCTION

  • In recent years there has been a significant development in the area of evolutionary computational techniques (ECTs) such as the PSO algorithm [1] [2] [3] [4] .
  • In this research the performance of PSO algorithm with multi-chaotic PRNG [9] is investigated on two shifted benchmark functions.
  • The shifted benchmark functions are designed in order to better simulate the time-variant real-world problems.

PARTICLE SWARM OPTIMIZATION ALGORITHM

  • The PSO algorithm is inspired in the natural swarm behavior of birds and fish.
  • Each particle in the population represents a candidate solution for the optimization problem that is defined by the cost function (CF).
  • The maximum velocity was limited to 0.2 times the range as it is usual.
  • Finally the linear decreasing inertia weight [3, 4] is used in the typically referred GPSO design that was used in this study.
  • A new w for each iteration is given by (3), where t stands for current iteration number and n stands for the total number of iterations.

TEST FUNCTIONS

  • In order to investigate on the performance of multi-chaotic PSO algorithm on functions closer to real problem than static test function, two shifted function were chosen.
  • Shifted function global optimum moves with each start of the algorithm but keeps their basic characteristic thus simulates the time-variant real problems.
  • Following shifted test functions were used in this study.

EXPERIMENT

  • Two different instances of multi-chaotic PSO [9] are investigated here.
  • In the multi-chaotic approach two different CPRNGs are switched when the algorithm seems to stagnate (for details see [9] ).
  • In the first design in this study the optimization starts with Lozi map based CPRNG and it is switched to CPRNG based on Arnold´s Cat map.
  • In the second design the CPRNGs are used in opposite order.

CONCLUSION

  • In this study the performance of multi-chaotic PSO was investigated on two different shifted benchmark functions.
  • The aim was to investigate the performance of this design on closer to real-world problems.
  • Results presented in this work support claim that using two different CPRNGs within one run of the algorithm may improve the performance of PSO algorithm on various optimization tasks.
  • The second designed combination of Arnold´s Cat map based CPRNG and Lozi map based CPRNG outperformed the canonical version in both cases.

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Performance of Multi-chaotic PSO on a Shifted Benchmark
Functions Set
Michal Pluhacek, Roman Senkerik and Ivan Zelinka
Tomas Bata University in Zlín, Faculty of Applied Informatics
Department of Informatics and Artificial Intelligence
nám. T.G. Masaryka 5555, 760 01 Zlín, Czech Republic
Abstract. In this paper the performance of Multi-chaotic PSO algorithm is investigated using two shifted benchmark
functions. The purpose of shifted benchmark functions is to simulate the time-variant real-world problems. The results of
chaotic PSO are compared with canonical version of the algorithm. It is concluded that using the multi-chaotic approach
can lead to better results in optimization of shifted functions.
Keywords: Particle swarm optimization, PSO, Chaos, Shifted functions, Multi-chaotic, Pseudo-random numbers
INTRODUCTION
In recent years there has been a significant development in the area of evolutionary computational techniques
(ECTs) such as the PSO algorithm [1-4]. One of the promising approaches is the implementation of chaotic
sequences as Pseudo-random number generators (PRNGs) [5 - 11]. In this research the performance of PSO
algorithm with multi-chaotic PRNG [9] is investigated on two shifted benchmark functions. The shifted benchmark
functions are designed in order to better simulate the time-variant real-world problems.
PARTICLE SWARM OPTIMIZATION ALGORITHM
The PSO algorithm is inspired in the natural swarm behavior of birds and fish. It was introduced by Eberhart and
Kennedy in 1995 [1]. Each particle in the population represents a candidate solution for the optimization problem
that is defined by the cost function (CF). In each iteration of the algorithm, a new location (combination of CF
parameters) for the particle is calculated based on its previous location and velocity vector (velocity vector contains
particle velocity for each dimension of the problem). Within this research the PSO algorithm with global topology
(GPSO) [6] was utilized. The chaotic PRNG is used in the main GPSO formula (1), which determines a new
“velocity”, thus directly affects the position of each particle in the next iteration.
)()(
21
1 t
ijj
t
ijij
t
ij
t
ij
xgBestRandcxpBestRandcvwv
(1)
Where:
v
i
t+1
- New velocity of the ith particle in iteration t+1.
w – Inertia weight value; v
i
t
- Current velocity of the ith particle in iteration t.; c
1
, c
2
- Priority factors; pBest
i
Personal best solution found by the ith particle; gBest - Best solution found in a population; x
ij
t
- Current position of
the ith particle (component j of dimension D) in iteration t.; Rand – Pseudo random number, interval (0, 1). CPRNG
is applied only here.
The maximum velocity was limited to 0.2 times the range as it is usual. The new position of each particle is then
given by (2), where x
i
t+1
is the new particle position:
11
t
i
t
i
t
i
vxx
(2)
Finally the linear decreasing inertia weight [3, 4] is used in the typically referred GPSO design that was used in
this study. The inertia weight has two control parameters wstart and wend. A new w for each iteration is given by
(3), where t stands for current iteration number and n stands for the total number of iterations. The values used in
this study were wstart = 0.9 and wend = 0.4.

n
tww
ww
endstart
start
(3)
Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014 (ICNAAM-2014)
AIP Conf. Proc. 1648, 550023-1–550023-4; doi: 10.1063/1.4912778
© 2015 AIP Publishing LLC 978-0-7354-1287-3/$30.00
550023-1

CHAOTIC MAPS
In this section two discrete chaotic systems that were used as CPRNGs are presented.
Lozi Map
The Lozi map is a simple discrete two-dimensional chaotic map. The map equations are given in (4). The
parameters used in this work are: a = 1.7 and b = 0.5 with respect to [11]. For these values, the system exhibits
typical chaotic behavior and with this parameter setting it is used in the most research papers and other literature
sources.
nn
nnn
XY
bYXaX
1
1
1
(4)
Arnold’s Cat Map
The Arnold’s Cat map is a simple two dimensional discrete system that stretches and folds points (x, y) to (x+y ,
x+2y) mod 1 in phase space. The map equations are given in Eq. 5. This map was used with parameter k = 0.1.
)1(mod
)1(mod
1
1
nnn
nnn
kYXY
YXX
(5)
TEST FUNCTIONS
In order to investigate on the performance of multi-chaotic PSO algorithm on functions closer to real problem
than static test function, two shifted function were chosen. Shifted function global optimum moves with each start of
the algorithm but keeps their basic characteristic thus simulates the time-variant real problems. Following shifted
test functions were used in this study.
Shifted Sphere function is given by (6).
¦
dim
1
2
)()(
i
ii
shiftxxf
(6)җ
Function minimum: Position for E
n
: (x
1
,x
2
…x
n
) = shift; Value for E
n
: y = 0;
Shifted Rastrigin`s function is given by (7).
¦
dim
1
2
)2cos(10)(dim10)(
i
iiii
shiftxshiftxxf
S
(7)җ
Function minimum: Position for E
n
: (x
1
,x
2
…x
n
) = shift; Value for E
n
: y = 0;
Shift
i
is an random number from interval <-5.11,5.11>. Where <-5.11,5.11> are the low and high bounds for the
population individuals. Shift value is generated on the start of optimization process.
EXPERIMENT
The control parameters of PSO algorithm were set as follows:
Pop. size: 40; N. of iterations: 5000; w
start
: 0.9; w
end
: 0.4; Dimension: 10; Runs: 50. Two different instances of
multi-chaotic PSO [9] are investigated here. In the multi-chaotic approach two different CPRNGs are switched when
the algorithm seems to stagnate (for details see [9]). In the first design in this study the optimization starts with Lozi
map based CPRNG and it is switched to CPRNG based on Arnold´s Cat map. In the second design the CPRNGs are
used in opposite order. The results are summarized in Tables 1 and 2. Furthermore mean gBest history for each
function is depicted in Fig. 1 and Fig. 2.
TABLE 1. Mean results comparison Shifted Sphere function
Sphere GPSO Lozi - Arnold Arnold - Lozi
Mean CF Value:
0.000289213 0.00183071
0.000042082
Std. Dev.:
0.00128856 0.012187 0.000171946
CF Value Median:
0. 0. 0.
Max. CF Value:
0.00863989 0.0870667 0.00121531
Min. CF Value:
0. 0. 0.
550023-2

1000 2000 3000 4000 5000
Generatio
n
0.1
0.2
0.3
0.4
0.5
gBest Valu
e
Arnold
ҟ
Lozi
Loz
i
ҟ
Arnol
d
GPSO
FIGURE 1. Mean gBest history for the Sphere function
1000 2000 3000 4000 5000
Generatio
n
10
20
30
40
gBest Valu
e
Arnold
ҟ
Lozi
Loz
i
ҟ
Arnol
d
GPSO
FIGURE 2. Mean gBest history for the Rastrigin´s function
550023-3

TABLE 2. Mean results comparison Shifted Rastrigin´s function
Rastrigin GPSO Lozi - Arnold Arnold - Lozi
Mean CF Value:
2.49478 2.88599
2.22995
Std. Dev.:
1.53435 1.64208 1.52427
CF Value Median:
1.99001 2.98488 1.98992
Max. CF Value:
6.96471 6.96471 6.2411
Min. CF Value:
0. 0. 0.
CONCLUSION
In this study the performance of multi-chaotic PSO was investigated on two different shifted benchmark
functions. The aim was to investigate the performance of this design on closer to real-world problems. Results
presented in this work support claim that using two different CPRNGs within one run of the algorithm may improve
the performance of PSO algorithm on various optimization tasks. The second designed combination of Arnold´s Cat
map based CPRNG and Lozi map based CPRNG outperformed the canonical version in both cases. This promising
result should motivate future research of this approach.
ACKNOWLEDGMENTS
This work was supported by Grant Agency of the Czech Republic - GACR P103/13/08195S, Grant of SGS No.
SP2014/159, VŠB - Technical University of Ostrava, Czech Republic, by the Development of human resources in
research and development of latest soft computing methods and their application in practice project, reg. no.
CZ.1.07/2.3.00/20.0072 funded by Operational Programme Education for Competitiveness, co-financed by ESF and
state budget of the Czech Republic, by European Regional Development Fund under the project CEBIA-Tech No.
CZ.1.05/2.1.00/03.0089 and by Internal Grant Agency of Tomas Bata University under the project No.
IGA/FAI/2014/010.
REFERENCES
1. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: IEEE International Conference on Neural Networks, 1995, pp.
1942-1948.
2. Kennedy, J., Eberhart, R.C., Shi, Y.: Swarm Intelligence. Morgan Kaufmann Publishers, (2001).
3. Nickabadi, A., Ebadzadeh, M.M., Safabakhsh, R.: A novel particle swarm optimization algorithm with adaptive inertia
weight. Applied Soft Computing 11(4), 3658-3670 (2011).
4. Yuhui, S., Eberhart, R.: A modified particle swarm optimizer. In: IEEE World Congress on Computational Intelligence., 4-9
May 1998, pp. 69-73.
5. R. Caponetto, L. Fortuna, S. Fazzino, M.G. Xibilia, Chaotic sequences to improve the performance of evolutionary
algorithms, Evolutionary Computation, IEEE Transactions on , vol.7, no.3, pp. 289- 304, June 2003
6. Araujo, E., Coelho, L., Particle swarm approaches using Lozi map chaotic sequences to fuzzy modelling of an experimental
thermal-vacuum system, Applied Soft Computing, v.8 n.4, p.1354-1364, September, 2008
7. Alatas B., Akin E., Ozer B. A., Chaos embedded particle swarm optimization algorithms, Chaos, Solitons & Fractals, Volume
40, Issue 4, 30 May 2009, Pages 1715-1734, ISSN 0960-0779
8. Pluhacek M., Senkerik R, Davendra D., Kominkova Oplatkova Z., and Zelinka I., "On the behavior and performance of chaos
driven PSO algorithm with inertia weight," Computers & Mathematics with Applications, vol. 66, pp. 122-134, 2013.
9. Pluhacek, M., Senkerik, R., Zelinka, I.: Particle swarm optimization algorithm driven by multichaotic number generator. Soft
Comput 18(4), 631-639 (2014). doi:10.1007/s00500-014-1222-z
10. Pluhacek M., Senkerik R,, Davendra D., Zelinka I., Designing PID Controller For DC Motor System By Means Of Enhanced
PSO Algorithm With Discrete Chaotic Lozi Map, In: Proceedings of the 26th European Conference on Modelling and
Simulation, ECMS 2012, pp. 405 - 409, 2012, ISBN 978-0-9564944-4-3.
11. Sprott, J. C., “Chaos and Time-Series Analysis“, Oxford University Press, 2003
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References
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Q1. What are the contributions mentioned in the paper "Performance of multi-chaotic pso on a shifted benchmark functions set" ?

In this paper the performance of Multi-chaotic PSO algorithm is investigated using two shifted benchmark functions.