Particle Swarm Optimization and Quantum Particle Swarm Optimization to Multidimensional Function Approximation

Authors: Diogo Silva, Fadul Rodor, Carlos Moraes

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.

Keywords: PSO, QPSO, function approximation, AI, optimization, multidimensional functions.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1316690

References:

[1] H. Robbins and S. Monro, “A stochastic approximation method,” Ann. Math. Statist., vol. 22, no. 3, pp. 400–407, 09 1951. (Online). Available: http://dx.doi.org/10.1214/aoms/1177729586.
[2] N. Barricelli, “Esempi numerici di processi di evoluzione,” Methodos, no. 21-22, pp. 45–68, 1954, cited By 55.
[3] I. Rechenberg, “Cybernetic solution path of an experimental problem,” Evolutionary Computation: The Fossil Record, pp. 301–310, 1968, cited By 2.
[4] L. J. Fogel, Intelligence Through Simulated Evolution: Forty Years of Evolutionary Programming. New York, NY, USA: John Wiley & Sons, Inc., 1999.
[5] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, no. 4598, pp. 671–680, 1983. (Online). Available: http://science.sciencemag.org/content/220/4598/671.
[6] 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: http://www.sciencedirect.com/science/article/pii/0305054886900481.
[7] X.-S. Yang, Nature-Inspired Metaheuristic Algorithms. Luniver Press, 2008.
[8] 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.
[9] M. Dorigo and T. St¨utzle, Ant Colony Optimization. Scituate, MA, USA: Bradford Company, 2004.
[10] J. Kennedy and R. Eberhart, “Particle swarm optimization,” 1995.
[11] L. M., “Improving particle swarm optimization by hybridization of stochastic search heuristics and self-organized criticality,” 2002.
[12] 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: http://dx.doi.org/10.1080/14786449608620846.
[13] 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.
[14] 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.
[15] 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.
[16] 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. 1–5.
[17] 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.
[18] 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.
[19] 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.
[20] 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.
[21] 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.
[22] 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.
[23] 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.
[24] 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.
[25] 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.
[26] 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.
[27] 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.
[28] 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.
[29] 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.
[30] 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.
[31] L. A. Rastrigin, “Systems of extremal control,” 1974.
[32] 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: http://dx.doi.org/10.1016/S0167-8191(05)80052-3.