Particle swarm optimization for single objective continuous space problems: A review

Evolutionary Computation

Volumn 25, Issue 1, 2017, Pages 1-54

  Bonyadi, Mohammad Reza ^a,b,c   Michalewicz, Zbigniew ^a,c,d,e  

^a UNIVERSITY OF ADELAIDE   (Australia)

^b UNIVERSITY OF QUEENSLAND   (Australia)

^c Data Science Group   (Australia)

^d INSTITUTE OF COMPUTER SCIENCE   (Poland)

^e POLISH JAPANESE INSTITUTE OF INFORMATION TECHNOLOGY   (Poland)

Author keywords

Constrained optimization; Invariance; Local convergence; Parameter selection; Particle swarm optimization; Stability analysis; Topology

Indexed keywords

CONSTRAINED OPTIMIZATION; INVARIANCE; OPTIMIZATION; TOPOLOGY;

CONTINUOUS SPACES; HIGH IMPACT; LOCAL CONVERGENCE; MODIFIED PSO ALGORITHMS; PARAMETER SELECTION; SINGLE OBJECTIVE; STABILITY ANALYSIS;

PARTICLE SWARM OPTIMIZATION (PSO);

ALGORITHM; AUTOMATED PATTERN RECOGNITION; COMPUTER SIMULATION; MARKOV CHAIN; THEORETICAL MODEL;

ALGORITHMS; COMPUTER SIMULATION; MODELS, THEORETICAL; PATTERN RECOGNITION, AUTOMATED; STOCHASTIC PROCESSES;

EID: 85014433250     PISSN: 10636560     EISSN: 15309304     Source Type: Journal    
DOI: 10.1162/EVCO_r_00180     Document Type: Review

References (170)

1. Agrafiotis, D. K., and Cedeno, W. (2002). Feature selection for structure-activity correlation using binary particle swarms. Journal of Medicinal Chemistry, 45 (5): 1098–1107.
2. Alatas, B., Akin, E., and Ozer, A. B. (2009). Chaos embedded particle swarm optimization algorithms. Chaos, Solutions and Fractals, 40 (4): 1715–1734.
3. Albert, R., and Barabási, A.-L. (2002). Statistical mechanics of complex networks. Reviews of Modern Physics, 74 (1): 47–97.
4. Angeline, P. J. (1998). Evolutionary optimization versus particle swarm optimization: Philosophy and performance differences. In Evolutionary Programming VII, pp. 601–610. Berlin: Springer.
5. Arabas, J., Michalewicz, Z., and Mulawka, J. (1994). GAVaPS-a genetic algorithm with varying population size. In IEEE World Congress on Computational Intelligence, pp. 73–78.
6. Arumugam, M. S., Rao, M., and Chandramohan, A. (2008). A new and improved version of particle swarm optimization algorithm with global–local best parameters. Knowledge and Information Systems, 16 (3): 331–357.
7. Arumugam, M. S., Rao, M.V. C., and Tan, A.W. C. (2009). Anovel and effective particle swarm optimization like algorithm with extrapolation technique. Applied Soft Computing, 9 (1): 308–320.
8. Auger, A., Hansen, N., Zerpa, J. P., Ros, R., and Schoenauer, M. (2009). Empirical comparisons of several derivative free optimization algorithms. In J. Vahrenhold (Ed.), 8th International Symposium on Experimental Algorithms, volume 5526 of LNCS, pp. 3–15. Berlin: Springer.
9. Baba, N. (1981). Convergence of a random optimization method for constrained optimization problems. Journal of Optimization Theory and Applications, 33 (4): 451–461.
10. Banks, A., Vincent, J., and Anyakoha, C. (2007). A review of particle swarm optimization. Part I: Background and development. Natural Computing, 6 (4): 467–484.
11. Banks, A., Vincent, J., and Anyakoha, C. (2008). A review of particle swarm optimization. Part II: Hybridisation, combinatorial, multicriteria and constrained optimization, and indicative applications. Natural Computing, 7 (1): 109–124.
12. Beyer, H.-G., and Schwefel, H.-P. (2002). Evolution strategies—A comprehensive introduction. Natural Computing, 1 (1): 3–52.
13. Birbil, S. I., Fang, S.-C., and Sheu, R.-L. (2004). On the convergence of a population-based global optimization algorithm. Journal of Global Optimization, 30 (2–3): 301–318.
14. Blackwell, T. (2012). A study of collapse in bare bones particle swarm optimization. IEEE Transactions on Evolutionary Computation, 16 (3): 354–372.
15. Bonyadi, M. R., Li, X., and Michalewicz, Z. (2013). Ahybrid particle swarm with velocity mutation for constraint optimization problems. In Genetic and Evolutionary Computation Conference (GECCO), pp. 1–8.
16. Bonyadi, M. R., Li, X., and Michalewicz, Z. (2014). A hybrid particle swarm with a timeadaptive topology for constrained optimization. Swarm and Evolutionary Computation, 18: 22–37.
17. Bonyadi, M. R., and Michalewicz, Z. (2012). A fast particle swarm optimization algorithm for the multidimensional knapsack problem. In IEEE Congress on Evolutionary Computation, pp. 1–8.
18. Bonyadi, M. R., and Michalewicz, Z. (2014a). A locally convergent rotationally invariant particle swarm optimization algorithm. Swarm Intelligence, 8 (3): 159–198.
19. Bonyadi, M. R., and Michalewicz, Z. (2014b). Locating potentially disjoint feasible regions of a search space with a particle swarm optimizer. In K. Deb and R. Datta (Eds.), Evolutionary constrained optimization, pp. 205–230. Berlin: Springer.
20. Bonyadi, M. R., and Michalewicz, Z. (2014c). On the edge of feasibility: Acase study of the particle swarm optimizer. In IEEE Congress on Evolutionary Computation, pp. 3059–3066.
21. Bonyadi, M. R., and Michalewicz, Z. (2014d). SPSO2011—Analysis of stability, local convergence, and rotation sensitivity. In Genetic and Evolutionary Computation Conference (GECCO), pp. 9–16.
22. Bonyadi, M. R., Michalewicz, Z., and Li, X. (2014). An analysis of the velocity updating rule of the particle swarm optimization algorithm. Journal of Heuristics, 20 (4): 417–452.
23. Bonyadi, M., and Michalewicz, Z. (2016). Analysis of stability, local convergence, and transformation sensitivity of a variant of particle swarm optimization algorithm. IEEE Transactions on Evolutionary Computation, 20 (3): 370–385.
24. Bonyadi, M., and Michalewicz, Z. (In Press). Stability analysis of the particle swarm optimization without stagnation assumption. IEEE Transactions on Evolutionary Computation, In Press.
25. Bonyadi, M. R., Michalewicz, Z., andWagner, M. (2014). Beyond the edge of feasibility: Analysis of bottlenecks. In G. Dick, W. Browne, P. Whigham, M. Zhang, L. Bui, H. Ishibuchi, Y. Jin, et al. (Eds.), Simulated evolution and learning, pp. 431–442. Berlin: Springer.
26. Bratton, D., and Kennedy, J. (2007). Defining a standard for particle swarm optimization. In IEEE Swarm Intelligence Symposium, pp. 120–127.
27. Caamao, P., Prieto, A., Becerra, J., Bellas, F., and Duro, R. (2010). Real-valued multimodal fitness landscape characterization for evolution. In K.Wong, B. Mendis, and A. Bouzerdoum (Eds.), Neural information processing: Theory and algorithms, pp. 567–574. Berlin: Springer.
28. Campana, E., Fasano, G., and Pinto, A. (2010). Dynamic analysis for the selection of parameters and initial population, in particle swarm optimization. Journal of Global Optimization, 48 (3): 347–397.
29. Carlisle, A., and Dozier, G. (2001). An off-the-shelf PSO. In Workshop on Particle Swarm Optimization, pp. 1–6. Indianapolis, Indiana: Purdue School of Engineering and Technology.
30. Chang-Huang, C., and Jia-Shing, S. (2009). Simple particle swarm optimization. In International Conference on Machine Learning and Cybernetics, pp. 460–466.
31. Changhe, L., Shengxiang, Y., and Trung Thanh, N. (2012). Aself-learning particle swarmoptimizer for global optimization problems. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 42 (3): 627–646.
32. Chatterjee, A., and Mukherjee, S. (2006). Unified concept of bottleneck. Technical Report W.P. No. 2006-05-01, Indian Institute of Management Ahmedabad, Research and Publication Department.
33. Chatterjee, A., and Siarry, P. (2006). Nonlinear inertia weight variation for dynamic adaptation in particle swarm optimization. Computers and Operations Research, 33 (3): 859–871.
34. Chen, W. N., Zhang, J., Chung, H. S. H., Zhong, W. L., Wu, W. G., and Shi, Y. H. (2010). A novel set-based particle swarm optimization method for discrete optimization problems. IEEE Transactions on Evolutionary Computation, 14 (2): 278–300.
35. Chen, D. B., and Zhao, C. X. (2009). Particle swarm optimization with adaptive population size and its application. Applied Soft Computing, 9 (1): 39–48.
36. Cleghorn, C. W., and Engelbrecht, A. P. (2014a). A generalized theoretical deterministic particle swarm model. Swarm intelligence, 8 (1): 35–59.
37. Cleghorn, C. W., and Engelbrecht, A. P. (2014b). Particle swarm convergence: An empirical investigation. In IEEE Congress on Evolutionary Computation, pp. 2524–2530.
38. Cleghorn, C. W., and Engelbrecht, A. P. (2014c). Particle swarm convergence: Standardized analysis and topological influence. In M. Dorigo, M. Birattari, S. Garnier, H. Hamann, M. Montes de Oca, C. Solnon, and T. Stützle (Eds.), Swarm intelligence, pp. 134–145. Berlin: Springer.
39. Cleghorn, C.W., and Engelbrecht, A. P. (2015). Particle swarmvariants: Standardized convergence analysis. Swarm Intelligence, 9 (2): 177–203.
40. Clerc, M. (1999). The swarm and the queen: Towards a deterministic and adaptive particle swarm optimization. In IEEE Congress on Evolutionary Computation, pp. 1951–1957.
41. Clerc, M. (2006). Particle swarm optimization. Newport Beach, CA: Wiley-ISTE.
42. Clerc, M. (2011). Standard particle swarm optimisation from 2006 to 2011. Technical Report hal-00764996.
43. Clerc, M., and Kennedy, J. (2002). The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation, 6 (1): 58–73.
44. Coath, G., and Halgamuge, S. K. (2003). A comparison of constraint-handling methods for the application of particle swarm optimization to constrained nonlinear optimization problems. In IEEE Congress on Evolutionary Computation, pp. 2419–2425.
45. Coello Coello, C. (2000). Use of a self-adaptive penalty approach for engineering optimization problems. Computers in Industry, 41 (2): 113–127.
46. Cooren, Y., Clerc, M., and Siarry, P. (2009). Performance evaluation of tribes: An adaptive particle swarm optimization algorithm. Swarm Intelligence, 3 (2): 149–178.
47. Dasgupta, S., Das, S., Biswas, A., and Abraham, A. (2009). On stability and convergence of the population-dynamics in differential evolution. AI Communications, 22 (1): 1–20.
48. Deng, J.-L. (1989). Introduction to grey system theory. The Journal of Grey System, 1 (1): 1–24.
49. Derrac, J., García, S., Molina, D., and Herrera, F. (2011). A practical tutorial on the use of nonparametric statistical tests as amethodology for comparing evolutionary and swarm intelligence algorithms. Swarm and Evolutionary Computation, 1 (1): 3–18.
50. Dorea, C. (1983). Expected number of steps of a random optimization method. Journal of Optimization Theory and Applications, 39 (2): 165–171.
51. Eberhart, R., andKennedy, J. (1995). Anew optimizer using particle swarm theory. In International Symposium on Micro Machine and Human Science, pp. 39–43.
52. Eberhart, R., and Shi, Y. (2001). Tracking and optimizing dynamic systems with particle swarms. In IEEE Congress on Evolutionary Computation, pp. 94–97.
53. Eberhart, R. C., and Shi, Y. (2000). Comparing inertia weights and constriction factors in particle swarm optimization. In IEEE Congress on Evolutionary Computation, pp. 84–88.
54. Eiben, A. E., Hinterding, R., and Michalewicz, Z. (1999). Parameter control in evolutionary algorithms. IEEE Transactions on Evolutionary Computation, 3 (2): 124–141.
55. Eiben, A. E., and Rudolph, G. (1999). Theory of evolutionary algorithms: A bird’s eye view. Theoretical Computer Science, 229 (1): 3–9.
56. Elsayed, S. M., Sarker, R. A., and Essam, D. L. (2012). Memetic multi-topology particle swarm optimizer for constrained optimization. In IEEE Congress on Evolutionary Computation, pp. 1–8.
57. Elsayed, S. M., Sarker, R. A., and Mezura-Montes, E. (2013). Particle swarm optimizer for constrained optimization. In IEEE Congress on Evolutionary Computation, pp. 2703–2711.
58. Emara, H. M. (2009). Adaptive clubs-based particle swarm optimization. In American Control Conference, pp. 5628–5634.
59. Engelbrecht, A. (2005). Fundamentals of computational swarm intelligence. Hoboken, NJ: Wiley.
60. Engelbrecht, A. (2012). Particle swarm optimization: Velocity initialization. In IEEE Congress on Evolutionary Computation, pp. 1–8.
61. Engelbrecht, A. (2013). Particle swarm optimization: Global best or local best. In IEEE Congress on Computational Intelligence, pp. 124–135.
62. García-Gonzalo, E., and Fernández-Martínez, J. L. (2014). Convergence and stochastic stability analysis of particle swarm optimization variants with generic parameter distributions. Applied Mathematics and Computation, 249: 286–302.
63. Ghosh, S., Das, S., Kundu, D., Suresh, K., Panigrahi, B. K., and Cui, Z. (2010). An inertia-adaptive particle swarm system with particlemobility factor for improved global optimization. Neural Computing and Applications, 21 (2): 237–250.
64. Gong, Y. J., and Zhang, J. (2013). Small-world particle swarm optimization with topology adaptation. In Genetic and Evolutionary Computation Conference, pp. 25–32.
65. Hansen, N. (2000). Invariance, self-adaptation and correlated mutations in evolution strategies. In Parallel Problem Solving from Nature, pp. 355–364.
66. Hansen, N. (2006). The CMA evolution strategy: A comparing review. In J. Lozano, P. Larrañaga, I. Inza, and E. Bengoetxea (Eds.), Towards a new evolutionary computation, volume 192 of Studies in Fuzziness and Soft Computing, pp. 75–102.
67. Hansen, N., and Ostermeier, A. (1996). Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation. In IEEE Congress on Evolutionary Computation, pp. 312–317.
68. Hansen, N., Ros, R., Mauny, N., Schoenauer, M., and Auger, A. (2008). PSO facing non-separable and ill-conditioned problems. Technical Report 6447, INRIA, France.
69. Hansen, N., Ros, R., Mauny, N., Schoenauer, M., and Auger, A. (2011). Impacts of invariance in search: When CMA-ES and PSO face ill-conditioned and non-separable problems. Applied Soft Computing, 11 (8): 5755–5769.
70. He, J., and Yao, X. (2002). From an individual to a population: An analysis of the first hitting time of population-based evolutionary algorithms. IEEE Transactions on Evolutionary Computation, 6 (5): 495–511.
71. He, Q., and Wang, L. (2007). An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Engineering Applications of Artificial Intelligence, 20 (1): 89–99.
72. Helwig, S., Branke, J., and Mostaghim, S. (2013). Experimental analysis of bound handling techniques in particle swarm optimization. IEEE Transactions on Evolutionary Computation, 17 (2): 259–271.
73. Helwig, S., and Wanka, R. (2007). Particle swarm optimization in high-dimensional bounded search spaces. In Swarm Intelligence Symposium, pp. 198–205.
74. Helwig, S., andWanka, R. (2008). Theoretical analysis of initial particle swarm behavior. In Parallel Problem Solving from Nature, pp. 889–898.
75. Hsieh, S. T., Sun, T. Y., Liu, C. C., and Tsai, S. J. (2009). Efficient population utilization strategy for particle swarm optimizer. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 39 (2): 444–456.
76. Hu, M., Wu, T., andWeir, J. D. (2012). An intelligent augmentation of particle swarm optimization with multiple adaptive methods. Information Sciences, 213: 68–83.
77. Hu, X., and Eberhart, R. (2002). Solving constrained nonlinear optimization problems with particle swarm optimization. In World Multiconference on Systemics, Cybernetics and Informatics, pp. 203–206.
78. Hu, X., Shi, Y., and Eberhart, R. (2004). Recent advances in particle swarm. In IEEE Congress on Evolutionary Computation, pp. 90–97.
79. Huang, H., Qin, H., Hao, Z., and Lim, A. (2010). Example-based learning particle swarm optimization for continuous optimization. Information Sciences, 182 (1): 125–138.
80. Jägersküpper, J. (2008). Lower bounds for randomized direct search with isotropic sampling. Operations Research Letters, 36 (3): 327–332.
81. Janson, S., and Middendorf, M. (2005). A hierarchical particle swarm optimizer and its adaptive variant. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 35 (6): 1272–1282.
82. Jiang, M., Luo, Y., and Yang, S. (2007a). Particle swarm optimization-stochastic trajectory analysis and parameter selection. In F. Chan and M. KumarTiwari (Eds.), Swarm intelligence, Focus on ant and particle swarm optimization, pp. 179–198. Wien: I-TECH Education and Publishing.
83. Jiang, M., Luo, Y. P., and Yang, S. Y. (2007b). Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm. Information Processing Letters, 102 (1): 8–16.
84. Kao, Y.-T., and Zahara, E. (2008). A hybrid genetic algorithm and particle swarm optimization for multimodal functions. Applied Soft Computing, 8 (2): 849–857.
85. Kaucic, M. (2013). A multi-start opposition-based particle swarm optimization algorithm with adaptive velocity for bound constrained global optimization. Journal of Global Optimization, 55 (1): 165–188.
86. Kelley, C. T. (1999). Iterative methods for optimization. Philadelphia: Society for Industrial and Applied Mathematics.
87. Kennedy, J. (1999). Small worlds and mega-minds: Effects of neighborhood topology on particle swarm performance. In IEEE Congress on Evolutionary Computation, pp. 1931–1938.
88. Kennedy, J. (2003). Bare bones particle swarms. In Swarm Intelligence Symposium, pp. 80–87.
89. Kennedy, J., and Eberhart, R. (1995). Particle swarm optimization. In International Conference on Neural Networks, pp. 1942–1948.
90. Kennedy, J., and Mendes, R. (2002). Population structure and particle swarm performance. In IEEE Congress on Evolutionary Computation, pp. 1671–1676.
91. Krink, T., VesterstrOm, J. S., and Riget, J. (2002). Particle swarm optimisation with spatial particle extension. In IEEE Congress on Evolutionary Computation, pp. 1474–1479.
92. Lehre, P. K., and Witt, C. (2013). Finite first hitting time versus stochastic convergence in particle swarm optimisation. In Advances in Metaheuristics, pp. 1–20.
93. Leu, M.-S., and Yeh, M.-F. (2012). Grey particle swarm optimization. Applied Soft Computing, 12 (9): 2985–2996.
94. Liang, J., Zhigang, S., and Zhihui, L. (2010). Coevolutionary comprehensive learning particle swarm optimizer. In IEEE Congress on Evolutionary Computation, pp. 1–8.
95. Liang, J. J., Qin, A. K., Suganthan, P. N., and Baskar, S. (2006). Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Transactions on Evolutionary Computation, 10 (3): 281–295.
96. Liu, Q. (2014). Order-2 stability analysis of particle swarm optimization. Evolutionary Computation, 23 (2): 187–216.
97. Mahor, A., Prasad, V., and Rangnekar, S. (2009). Economic dispatch using particle swarm optimization: A review. Renewable and Sustainable Energy Reviews, 13 (8): 2134–2141.
98. Malan, K., and Engelbrecht, A. (2009). Quantifying ruggedness of continuous landscapes using entropy. In IEEE Congress on Evolutionary Computation, pp. 1440–1447.
99. Mallipeddi, R., and Suganthan, P. (2010). Problem definitions and evaluation criteria for the CEC 2010 competition on constrained real-parameter optimization. Technical report, Nanyang Technological University, Singapore.
100. Matyas, J. (1965). Random optimization. Automation and Remote Control, 26 (2): 246–253.
101. Mendes, R. (2004). Population topologies and their influence in particle swarm performance. PhD thesis, Departmento de Informática, Universidade do Minho.
102. Mendes, R., Kennedy, J., and Neves, J. (2004). The fully informed particle swarm: Simpler, maybe better. IEEE Transactions on Evolutionary Computation, 8 (3): 204–210.
103. Michalewicz, Z. (1995). A survey of constraint handling techniques in evolutionary computation methods. In Annual Conference on Evolutionary Programming, pp. 135–155.
104. Michalewicz, Z., and Janikow, C. (1996). GENOCOP: A genetic algorithm for numerical optimization problems with linear constraints. Communications of the ACM, 39 (12): 223–240.
105. Michalewicz, Z., and Schoenauer, M. (1996). Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4 (1): 1–32.
106. Moler, C., and Van Loan, C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review, 45 (1): 3–49.
107. Montes de Oca, M. A., and Stützle, T. (2008). Towards incremental social learning in optimization and multiagent systems. In Genetic and Evolutionary Computation Conference, pp. 1939–1944.
108. Montes deOca, M. A., Stützle, T., Birattari, M., andDorigo, M. (2009). Frankenstein’s PSO: Acomposite particle swarm optimization algorithm. IEEE Transactions on Evolutionary Computation, 13 (5): 1120–1132.
109. Montes de Oca, M. A., Stützle, T., Van den Enden, K., and Dorigo, M. (2011). Incremental social learning in particle swarms. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 41 (2): 368–384.
110. Montgomery, D. C. (1984). Design and analysis of experiments. New York: Wiley.
111. Muller, C., Baumgartner, B., and Sbalzarini, I. F. (2009). Particle swarmCMAevolution strategy for the optimization of multi-funnel landscapes. In IEEE Congress on Evolutionary Computation, pp. 2685–2692.
112. Nasir, M., Das, S., Maity, D., Sengupta, S., Halder, U., and Suganthan, P. N. (2012). A dynamic neighborhood learning based particle swarm optimizer for global numerical optimization. Information Sciences, 209: 16–36.
113. Newman, M., Barabasi, A.-L., and Watts, D. J. (2006). The structure and dynamics of networks. Princeton: Princeton University Press.
114. Nguyen, T., and Yao, X. (2012). Continuous dynamic constrained optimisation—The challenges. IEEE Transactions on Evolutionary Computation, 16 (6): 769–786.
115. Nickabadi, A., Ebadzadeh, M. M., and Safabakhsh, R. (2011). A novel particle swarm optimization algorithm with adaptive inertia weight. Applied Soft Computing, 11 (4): 3658–3670.
116. Omidvar, M., Li, X., Mei, Y., and Yao, X. (2014). Cooperative co-evolution with differential grouping for large scale optimization. IEEE Transactions on Evolutionary Computation, 18 (3): 378–393.
117. Ozcan, E., and Mohan, C. K. (1999). Particle swarm optimization: Surfing the waves. In IEEE Congress on Evolutionary Computation, pp. 1939–1944.
118. Paquet, U., and Engelbrecht, A. (2003). A new particle swarm optimiser for linearly constrained optimisation. In IEEE Congress on Evolutionary Computation, pp. 227–233.
119. Paquet, U., and Engelbrecht, A. (2007). Particle swarms for linearly constrained optimisation. Fundamenta Informaticae, 76 (1): 147–170.
120. Parsopoulos, K., and Vrahatis, M. (2002a). Particle swarm optimization method for constrained optimization problems. Intelligent Technologies—Theory and Application: New Trends in Intelligent Technologies, 76: 214–220.
121. Parsopoulos, K., and Vrahatis, M. (2002b). Recent approaches to global optimization problems through particle swarm optimization. Natural Computing, 1 (2): 235–306.
122. Parsopoulos, K. E., and Vrahatis, M. N. (2004). On the computation of all global minimizers through particle swarm optimization. IEEE Transactions on Evolutionary Computation, 8 (3): 211–224.
123. Poli, R. (2008). Analysis of the publications on the applications of particle swarm optimisation. Journal of Artificial Evolution and Application, 2008 (3): 1–10.
124. Poli, R. (2009). Mean and variance of the sampling distribution of particle swarm optimizers during stagnation. IEEE Transactions on Evolutionary Computation, 13 (4): 712–721.
125. Poli, R., Brattonx, D., Blackwell, T., and Kennedy, J. (2007). Theoretical derivation, analysis and empirical evaluation of a simpler particle swarm optimiser. In IEEE Congress on Evolutionary Computation, pp. 1955–1962.
126. Poli, R., Kennedy, J., and Blackwell, T. (2007). Particle swarm optimization: An overview. Swarm Intelligence, 1 (1): 33–57.
127. PSO (2006). PSO source code (available online at PSO info). Retrieved from http://particleswarm. info/standard-PSO-2006.c
128. Pulido, G. T., and Coello, C. A. C. (2004). A constraint-handling mechanism for particle swarm optimization. In IEEE Congress on Evolutionary Computation, pp. 1396–1403.
129. Ratnaweera, A., Halgamuge, S. K., andWatson, H. C. (2004). Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Transactions on Evolutionary Computation, 8 (3): 240–255.
130. Rudolph, G. (1997). Local convergence rates of simple evolutionary algorithms with cauchy mutations. IEEE Transactions on Evolutionary Computation, 1 (4): 249–258.
131. Rudolph, G. (1998). Finite Markov chain results in evolutionary computation: A tour d’horizon. Fundamenta Informaticae, 35 (1–4): 67–89.
132. Rudolph, G. (2013). Stochastic convergence. In T. B. Rozenberg and J. Kok (Eds.), Handbook of natural computing, pp. 847–869. Berlin: Springer.
133. Salomon, R. (1995). Reevaluating genetic algorithm performance under coordinate rotation of benchmark functions—A survey of some theoretical and practical aspects of genetic algorithms. BioSystems, 39: 263–278.
134. Schmitt, M., andWanka, R. (2013). Particle swarm optimization almost surely finds local optima. In Genetic and Evolutionary Computation Conference, pp. 1629–1636.
135. Schoenauer, M., and Michalewicz, Z. (1996). Evolutionary computation at the edge of feasibility. Parallel Problem Solving from Nature, pp. 245–254.
136. Schoenauer, M., and Michalewicz, Z. (1997). Boundary operators for constrained parameter optimization problems. In Conference on Genetic Algorithms, pp. 322–329.
137. Sheskin, D. J. (2003). Handbook of parametric and nonparametric statistical procedures. NewYork: CRC Press.
138. Shi, Y., and Eberhart, R. (1998a). A modified particle swarm optimizer. In IEEEWorld Congress on Computational Intelligence, pp. 69–73.
139. Shi, Y., and Eberhart, R. (1998b). Parameter selection in particle swarm optimization. In V. Porto, N. Saravanan, D. Waagen, and A. Eiben (Eds.), Evolutionary programming VII, volume 1447 of LNCS, pp. 591–600. Berlin: Springer.
140. Shi, Y., and Eberhart, R. C. (2001). Fuzzy adaptive particle swarm optimization. In IEEE Congress on Evolutionary Computation, pp. 101–106.
141. Smith, L., Chinneck, J., and Aitken, V. (2013). Constraint consensus concentration for identifying disjoint feasible regions in nonlinear programmes. Optimization Methods and Software, 28 (2): 339–363.
142. Solis, F. J., and Wets, R. J.-B. (1981). Minimization by random search techniques. Mathematics of Operations Research, 6 (1): 19–30.
143. Song, M.-P., and Gu, G.-C. (2004). Research on particle swarm optimization: A review. In International Conference on Machine Learning and Cybernetics, pp. 2236–2241.
144. Spears, W., Green, D., and Spears, D. (2010). Biases in particle swarm optimization. International Journal of Swarm Intelligence Research, 1 (2): 34–57.
145. Storn, R., and Price, K. (1997). Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11 (4): 341–359.
146. Sun, C.-l., Zeng, J.-c., and Pan, J.-s. (2011). An improved vector particle swarm optimization for constrained optimization problems. Information Sciences, 181 (6): 1153–1163.
147. Takahama, T., and Sakai, S. (2005). Constrained optimization by constrained particle swarm optimizer with-level control. In A. Abraham, Y. Dote, T. Furuhashi, M. Köppen, A. Ohuchi, and Y. Ohsawa (Eds.), Soft computing as transdisciplinary science and technology, volume 29 of Advances in soft computing, pp. 1019–1029. Berlin: Springer.
148. Tang, Y., Wang, Z., and Fang, J.-a. (2011). Feedback learning particle swarm optimization. Applied Soft Computing, 11 (8): 4713–4725.
149. Trefethen, L. N., and Bau III, D. (1997). Numerical linear algebra. Philadelphia: Society for Industrial and Applied Mathematics.
150. Trelea, I. C. (2003). The particle swarm optimization algorithm: Convergence analysis and parameter selection. Information Processing Letters, 85 (6): 317–325.
151. Van den Bergh, F. (2002). An analysis of particle swarm optimizers. PhD thesis, Department of Computer Science, University of Pretoria.
152. Van den Bergh, F., and Engelbrecht, A. (2002). Anew locally convergent particle swarm optimiser. In Systems, Man and Cybernetics, pp. 96–101.
153. Van den Bergh, F., and Engelbrecht, A. (2004). A cooperative approach to particle swarm optimization. IEEE Transactions on Evolutionary Computation, 8 (3): 225–239.
154. Van den Bergh, F., and Engelbrecht, A. (2006). A study of particle swarm optimization particle trajectories. Information Sciences, 176 (8): 937–971.
155. Van den Bergh, F., and Engelbrecht, A. (2010). A convergence proof for the particle swarm optimiser. Fundamenta Informaticae, 105 (4): 341–374.
156. Voss, M. S. (2005). Principal component particle swarm optimization: A step towards topological swarm intelligence. In IEEE Congress on Evolutionary Computation, pp. 298–305.
157. Vrahatis, M., Androulakis, G., and Manoussakis, G. (1996). A new unconstrained optimization method for imprecise function and gradient values. Journal of Mathematical Analysis and Applications, 197 (2): 586–607.
158. Wang, H., Sun, H., Li, C., Rahnamayan, S., and Pan, J.-s. (2013). Diversity enhanced particle swarm optimization with neighborhood search. Information Sciences, 223: 119–135.
159. Wang, Y., Li, B., Weise, T., Wang, J., Yuan, B., and Tian, Q. (2011). Self-adaptive learning based particle swarm optimization. Information Sciences, 181 (20): 4515–4538.
160. Wilke, D. (2005). Analysis of the particle swarm optimization algorithm.Master’s thesis, Department of Mechanical and Aeronautical Engineering, University of Pretoria, South Africa.
161. Wilke, D., Kok, S., and Groenwold, A. (2007). Comparison of linear and classical velocity update rules in particle swarm optimization: Notes on scale and frame invariance. International Journal for Numerical Methods in Engineering, 70 (8): 985–1008.
162. Witt, C. (2009). Why standard particle swarm optimisers elude a theoretical runtime analysis. In Foundations of Genetic Algorithms, pp. 13–20.
163. Xinchao, Z. (2010). A perturbed particle swarm algorithm for numerical optimization. Applied Soft Computing, 10 (1): 119–124.
164. Yang, J.-M., Chen, Y.-P., Horng, J.-T., and Kao, C.-Y. (1997). Applying family competition to evolution strategies for constrained optimization. In P. Angeline, R. Reynolds, J. McDonnell, and R. Eberhart (Eds.), Evolutionary programming VI, volume 1213 of LNCS, pp. 201–211. Berlin: Springer.
165. Yao, X. (1999). Evolving artificial neural networks. Proceedings of the IEEE, 87 (9): 1423–1447.
166. Zhan, Z., Zhang, J., Li, Y., and Chung, H. (2009). Adaptive particle swarm optimization. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 39 (6): 1362–1381.
167. Zhan, Z.-H., Zhang, J., Li, Y., and Shi, Y.-h. (2011). Orthogonal learning particle swarm optimization. IEEE Transactions on Evolutionary Computation, 15 (6): 832–847.
168. Zhang, C., and Yi, Z. (2011). Scale-free fully informed particle swarm optimization algorithm. Information Sciences, 181 (20): 4550–4568.
169. Zheng, Y., Ma, L., Zhang, L., andQian, J. (2003). Empirical study of particle swarm optimizer with an increasing inertia weight. In IEEE Congress on Evolutionary Computation, pp. 221–226.
170. Zou, R., Kalivarapu, V., Winer, E., Oliver, J., and Bhattacharya, S. (2015). Particle swarm optimization-based source seeking. IEEE Transactions on Automation Science and Engineering, 12 (3): 865–875.