Abstract
Optimization in dynamic optimization problems (DOPs) requires the optimization algorithms not only to locate, but also to continuously track the moving optima. Particle swarm optimization (PSO) is a population-based optimization algorithm, originally developed for static problems. Recently, several researchers have proposed variants of PSO for optimization in DOPs. This paper presents a novel multi-swarm PSO algorithm, namely competitive clustering PSO (CCPSO), designed specially for DOPs. Employing a multi-stage clustering procedure, CCPSO splits the particles of the main swarm over a number of sub-swarms based on the particles positions and on their objective function values. The algorithm automatically adjusts the number of sub-swarms and the corresponding region of each sub-swarm. In addition to the sub-swarms, there is also a group of free particles that explore the environment to locate new emerging optima or exploit the current optima which are not followed by any sub-swarm. The adaptive search strategy adopted by the sub-swarms improves both the exploitation and tracking characteristics of CCPSO. A set of experiments is conducted to study the behavior of the proposed algorithm in different DOPs and to provide guidelines for setting the algorithm’s parameters in different problems. The results of CCPSO on a variety of moving peaks benchmark (MPB) functions are compared with those of several state-of-the-art PSO algorithms, indicating the efficiency of the proposed model.
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In this paper, we consider, without loss of generality, maximization problems.
Here, each objective function evaluation is regarded as a time step.
References
Akat, S. B., & Gazi, V. (2008). Particle swarm optimization with dynamic neighborhood topology: three neighborhood strategies and preliminary results. In Proceedings of the IEEE swarm intelligence symposium (SIS’08) (pp. 1–8). Piscataway: IEEE.
Angeline, P. J. (1997). Tracking extrema in dynamic environments. In Lecture notes in computer science: Vol. 1213. Evolutionary programming VI (pp. 335–345). Berlin: Springer.
Bird, S., & Li, X. (2006). Adaptively choosing niching parameters in a PSO. In M. Cattolico (Ed.), Proceedings of the genetic and evolutionary computation conference (GECCO’06) (pp. 3–10). New York: ACM.
Bird, S., & Li, X. (2007). Using regression to improve local convergence. In IEEE congress on evolutionary computation (pp. 592–599). Piscataway: IEEE.
Blackwell, T. M. (2007). Particle swarm optimization in dynamic environments. In S. Yang, Y.-S. Ong, & Y. Jin (Eds.), Evolutionary computation in dynamic and uncertain environments (pp. 29–49). Berlin: Springer.
Blackwell, T. M., & Bentley, P. (2002a). Don’t push me! collision avoiding swarms. In Proceedings of the IEEE congress on evolutionary computation (CEC’02) (Vol. 2, pp. 1691–1696). Piscataway: IEEE.
Blackwell, T. M., & Bentley, P. J. (2002b). Dynamic search with charged swarms. In Proceedings of the genetic and evolutionary computation conference (GECCO’02) (pp. 19–26). San Francisco: Morgan Kaufmann.
Blackwell, T. M., & Branke, J. (2004). Multi-swarm optimization in dynamic environments. In Lecture notes in computer science: Vol. 3005. Proceedings of the applications of evolutionary computing (pp. 489–500). Berlin: Springer.
Blackwell, T. M., & Branke, J. (2006). Multiswarms, exclusion, and anti-convergence in dynamic environments. IEEE Transactions on Evolutionary Computation, 10(4), 459–472.
Branke, J. (1999). Memory enhanced evolutionary algorithms for changing optimization problems. In Proceedings of the IEEE congress on evolutionary computation (CEC’99) (Vol. 3, pp. 1875–1882). Piscataway: IEEE.
Branke, J. (2001). Evolutionary optimization in dynamic environments. Norwell: Kluwer Academic.
Branke, J., & Schmeck, H. (2002). Designing evolutionary algorithms for dynamic optimization problems. In S. Tsutsui & A. Ghosh (Eds.), Theory and application of evolutionary computation: recent trends (pp. 239–262). Berlin: Springer.
Branke, J., Kaußler, T., Schmidt, C., & Schmeck, H. (2000). A multi-population approach to dynamic optimization problems. In Proceedings of the fourth international conference on adaptive computing in design and manufacture (ACDM 2000) (pp. 299–308). Berlin: Springer.
Brits, R., Engelbrecht, A. P., & van den Bergh, F. (2002a). Solving systems of unconstrained equations using particle swarm optimization. In Proceedings of the IEEE conference on systems, man and cybernetics (SMC’02) (Vol. 3, pp. 102–107). Piscataway: IEEE.
Brits, R., Engelbrecht, A. P., & van den Bergh, F. (2002b). A niching particle swarm optimizer. In Proceedings of the fourth Asia-pacific conference on simulated evolution and learning (SEAL’02) (Vol. 2, pp. 692–696). Singapore: Nanyang Technological University, School of Electrical and Electronic Engineering.
Carlisle, A., & Dozier, G. (2001). Tracking changing extrema with particle swarm optimizer. Technical report, Auburn University, Alabama.
De Jong, K. A., & Morrison, R. W. (1999). A test problem generator for non-stationary environments. In Proceedings of the IEEE congress on evolutionary computation (CEC’99) (Vol. 3, pp. 2047–2053). Piscataway: IEEE.
Eberhart, R. C., & Shi, Y. (2001). Tracking and optimizing dynamic systems with particle swarms. In Proceedings of the IEEE congress on evolutionary computation (CEC’01) (Vol. 1, pp. 94–100). Piscataway: IEEE.
Engelbrecht, A. P., & van Loggerenberg, L. N. H. (2007). Enhancing the NichePSO. In Proceedings of the IEEE congress on evolutionary computation (CEC’07) (pp. 2297–2302). Piscataway: IEEE.
Hu, X., & Eberhart, R. C. (2001). Tracking dynamic systems with PSO: where is the cheese? In Proceedings of the workshop on particle swarm optimization (pp. 80–83). Indianapolis: Purdue School of Engineering and Technology.
Hu, X., & Eberhart, R. C. (2002). Adaptive particle swarm optimisation: detection and response to dynamic systems. In Proceedings of the IEEE congress on evolutionary computation (CEC’02) (Vol. 2, pp. 1666–1670). Piscataway: IEEE.
Kennedy, J., & Eberhart, R. C. (1995). Particle swarm optimization. In Proceedings of the IEEE international conference on neural networks (pp. 1942–1948). Piscataway: IEEE.
Kennedy, J., & Mendes, R. (2002). Population structure and particle swarm performance. In Proceedings of the IEEE congress on evolutionary computation (CEC’02) (pp. 1671–1676). Piscataway: IEEE.
Krink, T., Vesterstrgm, J. S., & Riget, J. (2002). Particle swarm optimisation with spatial particle extension. In Proceedings of the IEEE congress on evolutionary computation (CEC’02) (Vol. 2, pp. 1474–1479). Piscataway: IEEE.
Li, C., Yang, S., & Pelta, D. A. (2011). A general framework of multi-population methods with clustering in undetectable dynamic environments. Technical report, Department of Information Systems and Computing, Brunel University, UK.
Li, X. (2004). Adaptively choosing neighborhood bests using species in a particle swarm optimizer for multimodal function optimization. In Lecture notes in computer science: Vol. 3103. Proceedings of the genetic and evolutionary computation conference (GECCO’04) (pp. 105–116). Berlin: Springer.
Li, X. (2010). Niching without niching parameters: particle swarm optimization using a ring topology. IEEE Transactions on Evolutionary Computation, 14(1), 150–169.
Li, X., Branke, J., & Blackwell, T. M. (2006). Particle swarm with speciation and adaptation in a dynamic environment. In M. Cattolico (Ed.), Proceedings of the genetic and evolutionary computation conference (GECCO’06) (pp. 51–58). New York: ACM.
Liang, J. J., Suganthan, P. N., & Deb, K. (2005). Novel composition test functions for numerical global optimization. In Proceedings of the IEEE international swarm intelligence symposium (SIS’05) (pp. 68–75). Piscataway: IEEE.
Liang, J. J., Qin, A. K., Suganthan, P. N., & Baskar, S. (2006). Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Transactions on Evolutionary Computation, 10(3), 281–295.
Liu, L., Yang, S., & Wang, D. (2010). Particle swarm optimization with composite particles in dynamic environments. IEEE Transactions on Systems, Man, and Cybernetics. Part B, 40(6), 1634–1648.
Nickabadi, A., Ebadzadeh, M. M., & Safabakhsh, R. (2008a). DNPSO: a dynamic niching particle swarm optimizer for multi-modal optimization. In Proceedings of the IEEE world congress on computational intelligence (WCCI’08) (pp. 26–32). Piscataway: IEEE.
Nickabadi, A., Ebadzadeh, M. M., & Safabakhsh, R. (2008b). Evaluating the performance of DNPSO in dynamic environments. In Proceedings of the IEEE international conference on systems, man, and cybernetics (SMC’08) (pp. 2640–2645). Piscataway: IEEE.
Nickabadi, A., Ebadzadeh, M. M., & Safabakhs, R. (2011). A novel particle swarm optimization algorithm with adaptive inertia weight. Applied Soft Computing, 11(4), 3658–3670.
Parrott, D., & Li, X. (2004). A particle swarm model for tracking multiple peaks in a dynamic environment using speciation. In Proceedings of the IEEE congress on evolutionary computation (CEC’04) (Vol. 1, pp. 98–103). Piscataway: IEEE.
Parrott, D., & Li, X. (2006). Locating and tracking multiple dynamic optima by a particle swarm model using speciation. IEEE Transactions on Evolutionary Computation, 10(4), 440–458.
Parsopoulos, K. E., & Vrahatis, M. N. (2002). Recent approaches to global optimization problems through particle swarm optimization. Natural Computing, 1(2–3), 235–306.
Passaro, A., & Starita, A. (2008). Particle swarm optimization for multimodal functions: a clustering approach. Journal of Artificial Evolution and Applications, 2008 1–15.
Schoeman, I., & Engelbrecht, A. P. (2006). Niching for dynamic environments using particle swarm optimization. In Lecture notes in computer science: Vol. 4247. Proceedings of the international conference on simulated evolution and learning (SEAL’06) (pp. 134–141). Berlin: Springer.
Schoeman, I. L., & Engelbrecht, A. P. (2005a). A parallel vector-based particle swarm optimizer. In Proceedings of the international conference on adaptive and natural computing algorithms (pp. 268–271). Vienna: Springer.
Schoeman, L., & Engelbrecht, A. P. (2005b). Containing particles inside niches when optimizing multimodal functions. In Proceedings of SAICSIT 2005, South African Institute for Computer Scientists and Information Technologists, Republic of South Africa (pp. 78–85).
Shi, Y. H., & Eberhart, R. C. (1998). A modified particle swarm optimizer. In Proceedings of the IEEE international conference on evolutionary computation (CEC’98) (pp. 69–73). Piscataway: IEEE.
Shi, Y. H., & Eberhart, R. C. (1999). Empirical study of particle swarm optimization. In Proceedings of the IEEE congress on evolutionary computation (CEC’99) (Vol. 3, pp. 1945–1950). Piscataway: IEEE.
van den Bergh, F., & Engelbrecht, A. P. (2002). A new locally convergent particle swarm optimizer. In Proceedings of the IEEE conference on systems, man and cybernetics (SMC’02) (Vol. 3, pp. 96–101). Piscataway: IEEE.
van den Bergh, F., & Engelbrecht, A. P. (2004). A cooperative approach to particle swarm optimization. IEEE Transactions on Evolutionary Computation, 8(3), 225–239.
Weicker, K. (2000). An analysis of dynamic severity and population size. In M. Schoenauer, K. Deb, G. Rudolph, X. Yao, E. Lutton, J. J. Merelo, & H.-P. Schwefel (Eds.), Parallel problem solving from nature (PPSN VI) (pp. 159–168). Berlin: Springer.
Yang, S., & Li, C. (2010). A clustering particle swarm optimizer for locating and tracking multiple optima in dynamic environments. IEEE Transactions on Evolutionary Computation, 14(6), 959–974.
Zhan, Z.-H., Zhang, J., Li, Y., & Chung, H. S.-H. (2009). Adaptive particle swarm optimization. IEEE Transactions on Systems, Man and Cybernetics, 39(6), 1362–1381.
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The authors would like to thank the anonymous referees and the editor-in-chief of the journal for their valuable and constructive comments which significantly improved the quality of the paper.
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Nickabadi, A., Ebadzadeh, M.M. & Safabakhsh, R. A competitive clustering particle swarm optimizer for dynamic optimization problems. Swarm Intell 6, 177–206 (2012). https://doi.org/10.1007/s11721-012-0069-0
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DOI: https://doi.org/10.1007/s11721-012-0069-0