Particle Swarm Optimization and Quantum Particle Swarm Optimization to Multidimensional Function Approximation
Abstract:This work compares the results of multidimensional
function approximation using two algorithms: the classical Particle
Swarm Optimization (PSO) and the Quantum Particle Swarm
Optimization (QPSO). These algorithms were both tested on three
functions - The Rosenbrock, the Rastrigin, and the sphere functions
- with different characteristics by increasing their number of
dimensions. As a result, this study shows that the higher the function
space, i.e. the larger the function dimension, the more evident the
advantages of using the QPSO method compared to the PSO method
in terms of performance and number of necessary iterations to reach
the stop criterion.
 H. Robbins and S. Monro, “A stochastic approximation method,”
Ann. Math. Statist., vol. 22, no. 3, pp. 400–407, 09 1951. (Online).
 N. Barricelli, “Esempi numerici di processi di evoluzione,” Methodos,
no. 21-22, pp. 45–68, 1954, cited By 55.
 I. Rechenberg, “Cybernetic solution path of an experimental problem,”
Evolutionary Computation: The Fossil Record, pp. 301–310, 1968, cited
 L. J. Fogel, Intelligence Through Simulated Evolution: Forty Years of
Evolutionary Programming. New York, NY, USA: John Wiley & Sons,
 S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi,
“Optimization by simulated annealing,” Science, vol. 220,
no. 4598, pp. 671–680, 1983. (Online). Available:
 F. Glover, “Future paths for integer programming and
links to artificial intelligence,” Computers and Operations
Research, vol. 13, no. 5, pp. 533 – 549, 1986,
applications of Integer Programming. (Online). Available:
 X.-S. Yang, Nature-Inspired Metaheuristic Algorithms. Luniver Press,
 J. H. Holland, Adaptation in Natural and Artificial Systems: An
Introductory Analysis with Applications to Biology, Control and
Artificial Intelligence. Cambridge, MA, USA: MIT Press, 1992.
 M. Dorigo and T. St¨utzle, Ant Colony Optimization. Scituate, MA,
USA: Bradford Company, 2004.
 J. Kennedy and R. Eberhart, “Particle swarm optimization,” 1995.
 L. M., “Improving particle swarm optimization by hybridization of
stochastic search heuristics and self-organized criticality,” 2002.
 P. S. Arrhenius, “Xxxi. on the influence of carbonic acid in
the air upon the temperature of the ground,” Philosophical
Magazine, vol. 41, no. 251, pp. 237–276, 1896. (Online). Available:
 A. Taghvaei, P. G. Mehta, and S. P. Meyn, “Error estimates for the
kernel gain function approximation in the feedback particle filter,” in
2017 American Control Conference (ACC), May 2017, pp. 4576–4582.
 A. Taghvaei and P. G. Mehta, “Gain function approximation in the
feedback particle filter,” in 2016 IEEE 55th Conference on Decision
and Control (CDC), Dec 2016, pp. 5446–5452.
 G. A. Hoffmann, “Function approximation with learning networks in
the financial field and its application to the interest rate sector,” in
Proceedings of 1995 Conference on Computational Intelligence for
Financial Engineering (CIFEr), Apr 1995, pp. 178–182.
 I. Schalk-Schupp, F. Faubel, M. Buck, and A. Wendemuth,
“Approximation of a nonlinear distortion function for combined linear
and nonlinear residual echo suppression,” in 2016 IEEE International
Workshop on Acoustic Signal Enhancement (IWAENC), Sept 2016, pp.
 J. E. A., T. P. I. Ahamed, and R. T., “A function approximation approach
to reinforcement learning for solving unit commitment problem with
photo voltaic sources,” in 2016 IEEE International Conference on Power
Electronics, Drives and Energy Systems (PEDES), Dec 2016, pp. 1–6.
 A. B. de Souza, “Fundamentos de otimizacao por inteligencia de
enxames: uma visao geral,” Sba: Controle e Automacao Sociedade
Brasileira de Automatica, vol. 20, pp. 271 – 304, 09 2009.
 Z. Meng, P. Feng, P. Chao, L. Weixing, and G. Qi, “Trajectory
optimization using time-separating strategy with improved pso on
mechanical arms,” in 2017 36th Chinese Control Conference (CCC),
July 2017, pp. 2669–2674.
 C. H. R. Jethmalani, S. P. Simon, K. Sundareswaran, P. S. R. Nayak, and
N. P. Padhy, “Auxiliary hybrid pso-bpnn-based transmission system loss
estimation in generation scheduling,” IEEE Transactions on Industrial
Informatics, vol. 13, no. 4, pp. 1692–1703, Aug 2017.
 L. Gong, W. Cao, and J. Zhao, “An improved pso algorithm for
high accurate parameter identification of pv model,” in 2017 IEEE
International Conference on Environment and Electrical Engineering
and 2017 IEEE Industrial and Commercial Power Systems Europe
(EEEIC / I CPS Europe), June 2017, pp. 1–5.
 T. Guan and F. Zhuo, “An improved sa-pso global maximum power
point tracking method of photovoltaic system under partial shading
conditions,” in 2017 IEEE International Conference on Environment
and Electrical Engineering and 2017 IEEE Industrial and Commercial
Power Systems Europe (EEEIC / I CPS Europe), June 2017, pp. 1–5.
 L. Benchikhi, M. Sadgal, and A. El-Fazziki, “An optimization approach
of parameters in image processing based on pso: Case of quality
control,” in Proceedings of 2013 International Conference on Industrial
Engineering and Systems Management (IESM), Oct 2013, pp. 1–6.
 K. Wang, X. Yan, Y. Yuan, X. Jiang, G. Lodewijks, and R. R.
Negenborn, “Pso-based method for safe sailing route and efficient
speeds decision-support for sea-going ships encountering accidents,” in
2017 IEEE 14th International Conference on Networking, Sensing and
Control (ICNSC), May 2017, pp. 413–418.
 M. K. Pamba R.V., Sherly E., “Evaluation of frequent pattern growth
based fuzzy particle swarm optimization approach for web document
clustering.” in Computational Science and Its Applications ICCSA 2017.
ICCSA 2017. Lecture Notes in Computer Science, vol 10404., 2017.
 M. S. K. P. Jatana N., Suri B. and C. A. R., “Particle swarm
based evolution and generation of test data using mutation testing.” in
Computational Science and Its Applications ICCSA 2016. ICCSA 2016.
Lecture Notes in Computer Science, vol 9790, 2016.
 J. Sun, B. Feng, and W. Xu, “Particle swarm optimization with particles
having quantum behavior,” in Proceedings of the 2004 Congress on
Evolutionary Computation (IEEE Cat. No.04TH8753), vol. 1, June 2004,
pp. 325–331 Vol.1.
 C. Zhang, Y. Xie, D. Liu, and L. Wang, “Fast threshold image
segmentation based on 2d fuzzy fisher and random local optimized
qpso,” IEEE Transactions on Image Processing, vol. 26, no. 3, pp.
1355–1362, March 2017.
 R. Faia, T. Pinto, and Z. Vale, “Optimization of electricity markets
participation with qpso,” in 2016 13th International Conference on the
European Energy Market (EEM), June 2016, pp. 1–5.
 X. Xie, H. Wang, S. Tian, and Y. Liu, “Optimal capacity configuration
of hybrid energy storage for an isolated microgrid based on qpso
algorithm,” in 2015 5th International Conference on Electric Utility
Deregulation and Restructuring and Power Technologies (DRPT), Nov
2015, pp. 2094–2099.
 L. A. Rastrigin, “Systems of extremal control,” 1974.
 H. M¨uhlenbein, M. Schomisch, and J. Born, “Paper: The
parallel genetic algorithm as function optimizer,” Parallel Comput.,
vol. 17, no. 6-7, pp. 619–632, Sep. 1991. (Online). Available: