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Probabilistic convergence analysis of the stochastic particle swarm optimization model without the stagnation assumption. (English) Zbl 1475.68478

Summary: As a stochastic optimization algorithm, it is more reasonable for particle swarm optimization (PSO) to study the probabilistic convergence. In this study, we analyze its convergence with probability 1 using the theory of probabilistic metric space. Firstly, we assume that the personal best of the particle and global best of the particle swarm are updated during the run; however, we do not assume that the personal best of the particle and global best of the particle swarm must be independent of the position of the particle. Such an assumption is more pragmatic and could be implemented in all PSO variants. Then, we develop a stochastic recurrence relation of the state of a particle under this assumption. Finally, we derive a sufficient condition that ensures the stochastic PSO algorithm is \(\tau\)-convergent with probability 1. In addition, we analyze the impact of the parameters in the unstable range on the individual iterations. Although these parameters could not guarantee the long-term convergence, their impact significantly influences the exploration ability of PSO; thus, it is crucial to understand them. Based on this analysis, we propose a novel strategy for balancing the exploitation and exploration abilities of the PSO algorithm.

MSC:

68W50 Evolutionary algorithms, genetic algorithms (computational aspects)
68W40 Analysis of algorithms
90C59 Approximation methods and heuristics in mathematical programming
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[1] E. Ozcan, C.K. Mohan, Particle swarm optimization: surfing the waves, In: Proceedings of the IEEE 1999 Congress on Evolutionary Computation-CEC99, Washington, USA, 1999, pp. 1939-1944. DOI: 10.1109/CEC.1999.785510
[2] Del Valle, Y.; Venayagamoorthy, G. K.; Mohagheghi, S.; Hernandez, J. C.; Harley, R. G., Particle swarm optimization: basic concepts, variants and applications in power systems, IEEE Trans. Evol. Comput., 12, 2, 171-195 (2008)
[3] AlRashidi, M. R.; El-Hawary, M. E., A survey of particle swarm optimization applications in electric power systems, IEEE Trans. Evol. Comput., 13, 4, 913-918 (2009)
[4] Riemann, R. C.; Sanchez, E. N.; Fernando, O. T.; Loukianov, A. G.; Harley, R. G., Particle swarm optimization for discrete-time inverse optimal control of a doubly fed induction generator, IEEE Trans. Cybern., 43, 6, 1698-1709 (2013)
[5] Kulkarni, R. V.; Venayagamoorthy, G. K., Particle swarm optimization in wireless-sensor networks: a brief survey, IEEE Trans. Syst. Man Cybern., 41, 2, 262-267 (2011)
[6] Xue, B.; Zhang, M.; Browne, W. N., Particle swarm optimization for feature selection in classification: a multi-objective approach, IEEE Trans. Cybern., 43, 6, 1656-1671 (2013)
[7] Gong, Y. J.; Li, J. J.; Zhou, Y., Genetic learning particle swarm optimization, IEEE Trans. Cybern., 46, 10, 2277-2290 (2017)
[8] Ishaque, K.; Salam, Z.; Amjad, M.; Mekhilef, S., An improved particle swarm optimization (pso)-based mppt for pv with reduced steady-state oscillation, IEEE Trans. Power Electron., 27, 8, 3627-3638 (2012)
[9] Chen, W. N.; Zhang, J.; Lin, Y.; Chen, N.; Zhan, Z. H.; Chung, S. H.; Li, Y.; Shi, Y. H., Particle swarm optimization with an aging leader and challengers, IEEE Trans. Evol. Comput., 17, 2, 241-258 (2013)
[10] Wang, H.; Sun, H.; Li, C. H.; Rahnamayan, S.; Pan, J. S., Diversity enhanced particle swarm optimization with neighborhood search, Inf. Sci., 223, 2, 119-135 (2013)
[11] Bonyadi, M. R.; Michalewicz, Z., Particle swarm optimization for single objective continuous space problems: a review, Evol. Comput., 25, 1, 1-54 (2017)
[12] Emara, H. M.; Abdel Fattah, H. A., Continuous swarm optimization technique with stability analysis, (Proceedings of the 2004 American Control Conference, Boston, USA (2004))
[13] Zhang, Y. D.; Wang, S. H.; Ji, G. L., A comprehensive survey on particle swarm optimization algorithm and its applications, Math. Problems Eng., 1, 1-38 (2015) · Zbl 1394.90588
[14] Bonyadi, M. R.; Michalewicz, Z., Analysis of stability, local convergence, and transformation sensitivity of a variant of particle swarm optimization algorithm, IEEE Trans. Evol. Comput., 20, 3, 370-385 (2016)
[15] E. Ozcan, C.K. Mohan, Analysis of a simple particle swarm optimization system, in: Proceedings of Intelligent Engineering Systems through Artificial Neural Networks, 1998, pp. 253-258.
[16] Clerc, M.; Kennedy, J., The particle swarm-explosion, stability, and convergence in a multidimensional complex space, IEEE Trans. Evol. Comput., 6, 1, 58-73 (2002)
[17] J. Liu, H. Liu, W. Shen, Stability analysis of particle swarm optimization, in: International Conference on Intelligent Computing, Berlin, Germany, 2007. DOI: https://doi.org/10.1007/978-3-540-74205-0_82
[18] Bergh, F. V.D.; Engelbrecht, A. P., A study of particle swarm optimization particle trajectories, Inf. Sci., 176, 8, 937-971 (2006) · Zbl 1093.68105
[19] Poli, R., Mean and variance of the sampling distribution of particle swarm optimizers during stagnation, IEEE Trans. Evol. Comput., 13, 4, 712-721 (2009)
[20] Trelea, I. C., The particle swarm optimization algorithm: convergence analysis and parameter selection, Inform. Proc. Lett., 85, 6, 317-325 (2003) · Zbl 1156.90463
[21] Jiang, M.; Luo, Y. P.; Yang, S. Y., Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm, Inform. Process. Lett., 102, 1, 8-16 (2007) · Zbl 1184.68621
[22] Esperanza, G. G.; Juan, L. F.M., Convergence and stochastic stability analysis of particle swarm optimization variants with generic parameter distributions, Appl. Math. Comput., 249, 286-302 (2014) · Zbl 1338.90471
[23] Cleghorn, C. W.; Engelbrecht, A. P., A generalized theoretical deterministic particle swarm model, Swarm Intell., 8, 1, 35-59 (2014)
[24] Liu, Q. F., Order-2 stability analysis of particle swarm optimization, Evol. Comput., 23, 2, 187-216 (2015)
[25] Qian, W.; Li, M., Convergence analysis of standard particle swarm optimization algorithm and its improvement, Soft. Comput., 22, 12, 4047-4070 (2017) · Zbl 1398.90213
[26] Bonyadi, M. R.; Michalewicz, Z., Stability analysis of the particle swarm optimization without stagnation assumption, IEEE Trans. Evol. Comput., 20, 5, 814-819 (2016)
[27] Alimohammady, M.; Esmaeli, A.; Saadati, R., Completeness results in probabilistic metric spaces, Chaos, Solitons Fractals, 39, 2, 765-769 (2009) · Zbl 1197.54043
[28] Razani, A.; Fouladgar, K., Extension of contractive maps in the menger probabilistic metric space, Chaos, Solitons Fractals, 34, 5, 1724-1731 (2007) · Zbl 1152.54367
[29] Mihet, D., Fixed point theorems in probabilistic metric spaces, Chaos, Solitons Fractals, 41, 2, 1014-1019 (2009) · Zbl 1198.54082
[30] Schweizer, B.; Sklar, A., Statistical metric spaces, Pac. J. Math., 10, 10, 313-334 (1960) · Zbl 0091.29801
[31] Sun, J.; Wu, X.; Palade, V.; Fang, W.; Lai, C. H.; Xu, W., Convergence analysis and improvements of quantum-behaved particle swarm optimization, Inf. Sci., 193, 81-103 (2012)
[32] Gunther, J., A convergence theory for probabilistic metric spaces, Quaestiones Mathematicae, 38, 4, 587-599 (2015) · Zbl 1397.54042
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