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January  2019, 15(1): 177-198. doi: 10.3934/jimo.2018038

Improved particle swarm optimization and neighborhood field optimization by introducing the re-sampling step of particle filter

Qifeng Cheng 1,2,, , Xue Han 2, , Tingting Zhao 2, and  V S Sarma Yadavalli 3,

1. 

Research and Development Center, China Academy of Launch Vehicle Technology, Fengtai District, Beijing 100076, China
2. 

Institute of Intelligent Engineering and Mathematics, Liaoning Technical University, Fuxin City, Liaoning Province 123000, China
3. 

Department of Industrial and Systems Engineering, University of Pretoria, Pretoria 0002, South Africa

* Corresponding author: Qifeng Cheng

Received  May 2017 Revised  October 2017 Published  April 2018

A technique of introducing the re-sampling step of particle filter is proposed to improve the particle swarm optimization (PSO) algorithm, a typical global search algorithm. The re-sampling step can decrease particles with low weights and duplicate particles with high weights, given that we define a type of suitable weights for the particles. To prevent the identity of particles, the re-sampling step is followed by the existing method of particle variation. Through this technique, the local search capability is enhanced greatly in the later searching stage of PSO algorithm. More interesting, this technique can also be employed to improve another algorithm of which the philosophy is "learning from neighbors", i.e., the neighborhood field optimization (NFO) algorithm. The improved algorithms (PSO-resample and NFO-resample) are compared with other metaheuristic algorithms through extensive simulations. The experiments show that the improved algorithms are superior in terms of convergence rate, search accuracy and robustness. Our results also suggest that the proposed technique can be general in the sense that it can probably improve other particle-based intelligent algorithms.

Keywords: Re-sampling step, particle filter, PSO, neighborhood field optimization, particle variation.
Mathematics Subject Classification: Primary: 90C59; Secondary: 68T20, 65K05.
Citation: Qifeng Cheng, Xue Han, Tingting Zhao, V S Sarma Yadavalli. Improved particle swarm optimization and neighborhood field optimization by introducing the re-sampling step of particle filter. Journal of Industrial & Management Optimization, 2019, 15 (1) : 177-198. doi: 10.3934/jimo.2018038
References:
[1]

K. Amin and M. Guerrero-Zapata, A hybrid multiobjective RBF-PSO method for mitigating dos attacks in named data networking, Neurocomputing, 151 (2015), 1262-1282.

[2]

K. Amin and M. Guerrero-Zapata, A fuzzy anomaly detection system based on hybrid PSO-Kmeans algorithm in content-centric networks, Neurocomputing, 149 (2015), 1253-1269.

[3]

M. S. Arulampalam, S. Maskell, N. Gordon and T. clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Transactions on Signal Processing, 50 (2002), 174-188. doi: 10.1109/78.978374.

[4]

T. M. Blackwell and P. Bentley, Don'T push me! Collision-avoiding swarms, Proceedings of the 2002 Congress on Evolutionary Computation, 2 (2002), 1691-1696. doi: 10.1109/CEC.2002.1004497.

[5]

J. Carpenter, P. Clifford and F. Fearnhead, An improved particle filter for non-linear problems, IEE Proceedings-Radar, Sonar and Navigation, 146 (1999), 2-7.

[6]

M. Clerc and J. Kennedy, The particle swarm-explosion, stability, and convergence in a multidimensional complex space, IEEE Transactions on Evolutionary Computation, 6 (2002), 58-73. doi: 10.1109/4235.985692.

[7]

D. Crisan and A. Doucet, A survey of convergence result on particle filtering methods for practitioners, IEEE Transactions on Signal Processing, 50 (2002), 736-746. doi: 10.1109/78.984773.

[8]

A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling method for Bayesian filtering, Statistics and Computing, 10 (2000), 197-208.

[9] A. Doucet, J. Freitas and N. Gordon, Sequential Monte Carlo Methods in Practice, Springer-Verlag, 2001.
[10]

R. C. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, Proceedings of the 6th International Symposium on Micro machine and Human Science, (1995), 39-43. doi: 10.1109/MHS.1995.494215.

[11]

F. Glover, Tabu search-part Ⅱ, ORSA Journal on Computing, 2 (1990), 4-32. doi: 10.1287/ijoc.2.1.4.

[12] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, New York, 1989.
[13]

N. J. Gordon, D. J. Salmond and A. F. M. Smith, Novel approach to nonlinear/non-Gaussian Bayesian state estimation, Radar and Signal Processing, IEE Proceedings F, 140 (1993), 107-113. doi: 10.1049/ip-f-2.1993.0015.

[14]

R. Greiner, POLO: A probabilistic hill-climbing algorithm, Artificial Intelligence, 84 (1996), 177-208. doi: 10.1016/0004-3702(95)00040-2.

[15]

J. Grobler, A. P. Engelbrecht, G. Kendall and S. Yadavalli, Heuristic space diversity control for improved meta-hyper-heuristic performance, Information Sciences, 300 (2015), 49-62.

[16]

J. Grobler and A. P. Engelbrecht, Headless chicken particle swarm optimization algorithms, Tan Y., Shi Y., Niu B. (eds) Advances in Swarm Intelligence. ICSI 2016. Lecture Notes in Computer Science, 9712 (2016), 350-357. doi: 10.1007/978-3-319-41000-5_35.

[17]

J. Grobler and A. P. Engelbrecht, A scalability analysis of particle swarm optimization roaming behaviour, Advances in Swarm Intelligence. ICSI 2017. Lecture Notes in Computer Science, 10385 (2017), 119-130. doi: 10.1007/978-3-319-61824-1_13.

[18]

J. D. Hol, T. B. Schon and F. Gustafsson, On resampling algorithms for particle filters, IEEE Nonlinear Statistical Signal Processing Workshop, (2006), 79-82. doi: 10.1109/NSSPW.2006.4378824.

[19]

B. J. Jain, H. Pohlheim and W. Joachim, On termination criteria of evolutionary algorithms, Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation, (2001), 768-775.

[20]

J. Kennedy and R. C. Eberhart, Particle swarm optimization, Proceedings of the IEEE International Conference on Neural Networks, (1995), 1942-1948. doi: 10.1109/ICNN.1995.488968.

[21]

J. Kennedy and R. Mendes, Population structure and particle swarm performance, Proceedings of the 2002 Congress on Evolutionary Computation, 2 (2002), 1671-1676. doi: 10.1109/CEC.2002.1004493.

[22]

S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671.

[23]

G. Kitagawa, Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, Journal of Computational and Graphical Statistics, 5 (1996), 1-25.

[24]

J. Lampinen and R. Storn, Differential evolution, Onwubolu G, Babu BV (eds) New Optimization Techniques in Engineering, (2004), 123-166. doi: 10.1007/978-3-540-39930-8_6.

[25]

J. J. Liang, A. K. Qin, P. N. Suganthan and S. Baskar, Comprehensive learning particle swarm optimizer for global optimization of multimodal functions, IEEE Transactions on Evolutionary Computation, 10 (2006), 281-295. doi: 10.1109/TEVC.2005.857610.

[26]

J. S. Liu and R. Chen, Sequential Monte Carlo methods for dynamic systems, Journal of the American Statistical Association, 93 (1998), 1032-1044. doi: 10.1080/01621459.1998.10473765.

[27]

R. Mendes, J. Kennedy and J. Neves, The fully informed particle swarm: simpler, maybe better, IEEE Transactions on Evolutionary Computation, 8 (2004), 204-210. doi: 10.1109/TEVC.2004.826074.

[28]

K. E. Parsopoulos and M. N. Vrahatis, UPSO-A unified particle swarm optimization scheme, Lecture Series on Computational Sciences, 1 (2004), 868-873.

[29]

T. Peram, K. Veeramachaneni and C. K. Mohan, Fitness-distance-ratio based particle swarm optimization, Proceedings of the 2003 IEEE on Swarm Intelligence Symposium, (2003), 174-181. doi: 10.1109/SIS.2003.1202264.

[30]

Y. Shi and R. C. Eberhart, A modified particle swarm optimizer, IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence, (1998), 69-73. doi: 10.1109/ICEC.1998.699146.

[31]

R. Storn and K. Price, Differential evolutional simple and efficient heuristic for global optimization over continuous space, Journal of Global Optimization, 11 (1997), 341-359. doi: 10.1023/A:1008202821328.

[32]

P. N. Suganthan, Particle swarm optimiser with neighborhood operator, Proceedings of the 1999 Congress on Evolutionary Computation, 3 (1999), 1958-1962.

[33] M. D. Vose, Simple Genetic Algorithm: Foundation and Theory, MIT Press, MI, 1999.
[34]

Z. Wu and T. W. S. Chow, A local multiobjective optimization algorithm using neighborhood field, Structural and Multidisciplinary Optimization, 46 (2012), 853-870. doi: 10.1007/s00158-012-0800-x.

[35]

Z. Wu and T. W. S. Chow, Neighborhood field for cooperative optimization, Soft Computing, 17 (2013), 819-834. doi: 10.1007/s00500-012-0955-9.

[36]

Z. Wu and T. W. S. Chow, Binary neighborhood field optimization for unit commitment problems, IET Generation Transmission and Distribution, 7 (2013), 299-308.

[37]

C. Yang, L. Gu and W. Gui, Particle swarm optimization algorithm with adaptive mutation, Computer Engineering, 34 (2008), 188-190.

[38]

X. Yao, Y. Liu and G. M. Lin, Evolutionary programming made faster, IEEE Transactions on Evolutionary Computation, 3 (1999), 82-102.

[39]

B. Yao, B. Yu, P. Hu, J. Gao and M. H. Zhang, An improved particle swarm optimization for carton heterogeneous vehicle routing problem with a collection depot, Annals of Operations Research, 242 (2016), 303-320. doi: 10.1007/s10479-015-1792-x.

[40]

T. T. Zhao, Q. F. Cheng and Z. F. Wang, Nonlinear model predictive control optimization with improved particle swarm algorithm, Liaoning Gongcheng Jishu Daxue Xuebao (Ziran Kexue Ban)/Journal of Liaoning Technical University (Natural Science Edition), 34 (2015), 517-522.

show all references

References:
[1]

K. Amin and M. Guerrero-Zapata, A hybrid multiobjective RBF-PSO method for mitigating dos attacks in named data networking, Neurocomputing, 151 (2015), 1262-1282.

[2]

K. Amin and M. Guerrero-Zapata, A fuzzy anomaly detection system based on hybrid PSO-Kmeans algorithm in content-centric networks, Neurocomputing, 149 (2015), 1253-1269.

[3]

M. S. Arulampalam, S. Maskell, N. Gordon and T. clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Transactions on Signal Processing, 50 (2002), 174-188. doi: 10.1109/78.978374.

[4]

T. M. Blackwell and P. Bentley, Don'T push me! Collision-avoiding swarms, Proceedings of the 2002 Congress on Evolutionary Computation, 2 (2002), 1691-1696. doi: 10.1109/CEC.2002.1004497.

[5]

J. Carpenter, P. Clifford and F. Fearnhead, An improved particle filter for non-linear problems, IEE Proceedings-Radar, Sonar and Navigation, 146 (1999), 2-7.

[6]

M. Clerc and J. Kennedy, The particle swarm-explosion, stability, and convergence in a multidimensional complex space, IEEE Transactions on Evolutionary Computation, 6 (2002), 58-73. doi: 10.1109/4235.985692.

[7]

D. Crisan and A. Doucet, A survey of convergence result on particle filtering methods for practitioners, IEEE Transactions on Signal Processing, 50 (2002), 736-746. doi: 10.1109/78.984773.

[8]

A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling method for Bayesian filtering, Statistics and Computing, 10 (2000), 197-208.

[9] A. Doucet, J. Freitas and N. Gordon, Sequential Monte Carlo Methods in Practice, Springer-Verlag, 2001.
[10]

R. C. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, Proceedings of the 6th International Symposium on Micro machine and Human Science, (1995), 39-43. doi: 10.1109/MHS.1995.494215.

[11]

F. Glover, Tabu search-part Ⅱ, ORSA Journal on Computing, 2 (1990), 4-32. doi: 10.1287/ijoc.2.1.4.

[12] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, New York, 1989.
[13]

N. J. Gordon, D. J. Salmond and A. F. M. Smith, Novel approach to nonlinear/non-Gaussian Bayesian state estimation, Radar and Signal Processing, IEE Proceedings F, 140 (1993), 107-113. doi: 10.1049/ip-f-2.1993.0015.

[14]

R. Greiner, POLO: A probabilistic hill-climbing algorithm, Artificial Intelligence, 84 (1996), 177-208. doi: 10.1016/0004-3702(95)00040-2.

[15]

J. Grobler, A. P. Engelbrecht, G. Kendall and S. Yadavalli, Heuristic space diversity control for improved meta-hyper-heuristic performance, Information Sciences, 300 (2015), 49-62.

[16]

J. Grobler and A. P. Engelbrecht, Headless chicken particle swarm optimization algorithms, Tan Y., Shi Y., Niu B. (eds) Advances in Swarm Intelligence. ICSI 2016. Lecture Notes in Computer Science, 9712 (2016), 350-357. doi: 10.1007/978-3-319-41000-5_35.

[17]

J. Grobler and A. P. Engelbrecht, A scalability analysis of particle swarm optimization roaming behaviour, Advances in Swarm Intelligence. ICSI 2017. Lecture Notes in Computer Science, 10385 (2017), 119-130. doi: 10.1007/978-3-319-61824-1_13.

[18]

J. D. Hol, T. B. Schon and F. Gustafsson, On resampling algorithms for particle filters, IEEE Nonlinear Statistical Signal Processing Workshop, (2006), 79-82. doi: 10.1109/NSSPW.2006.4378824.

[19]

B. J. Jain, H. Pohlheim and W. Joachim, On termination criteria of evolutionary algorithms, Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation, (2001), 768-775.

[20]

J. Kennedy and R. C. Eberhart, Particle swarm optimization, Proceedings of the IEEE International Conference on Neural Networks, (1995), 1942-1948. doi: 10.1109/ICNN.1995.488968.

[21]

J. Kennedy and R. Mendes, Population structure and particle swarm performance, Proceedings of the 2002 Congress on Evolutionary Computation, 2 (2002), 1671-1676. doi: 10.1109/CEC.2002.1004493.

[22]

S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671.

[23]

G. Kitagawa, Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, Journal of Computational and Graphical Statistics, 5 (1996), 1-25.

[24]

J. Lampinen and R. Storn, Differential evolution, Onwubolu G, Babu BV (eds) New Optimization Techniques in Engineering, (2004), 123-166. doi: 10.1007/978-3-540-39930-8_6.

[25]

J. J. Liang, A. K. Qin, P. N. Suganthan and S. Baskar, Comprehensive learning particle swarm optimizer for global optimization of multimodal functions, IEEE Transactions on Evolutionary Computation, 10 (2006), 281-295. doi: 10.1109/TEVC.2005.857610.

[26]

J. S. Liu and R. Chen, Sequential Monte Carlo methods for dynamic systems, Journal of the American Statistical Association, 93 (1998), 1032-1044. doi: 10.1080/01621459.1998.10473765.

[27]

R. Mendes, J. Kennedy and J. Neves, The fully informed particle swarm: simpler, maybe better, IEEE Transactions on Evolutionary Computation, 8 (2004), 204-210. doi: 10.1109/TEVC.2004.826074.

[28]

K. E. Parsopoulos and M. N. Vrahatis, UPSO-A unified particle swarm optimization scheme, Lecture Series on Computational Sciences, 1 (2004), 868-873.

[29]

T. Peram, K. Veeramachaneni and C. K. Mohan, Fitness-distance-ratio based particle swarm optimization, Proceedings of the 2003 IEEE on Swarm Intelligence Symposium, (2003), 174-181. doi: 10.1109/SIS.2003.1202264.

[30]

Y. Shi and R. C. Eberhart, A modified particle swarm optimizer, IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence, (1998), 69-73. doi: 10.1109/ICEC.1998.699146.

[31]

R. Storn and K. Price, Differential evolutional simple and efficient heuristic for global optimization over continuous space, Journal of Global Optimization, 11 (1997), 341-359. doi: 10.1023/A:1008202821328.

[32]

P. N. Suganthan, Particle swarm optimiser with neighborhood operator, Proceedings of the 1999 Congress on Evolutionary Computation, 3 (1999), 1958-1962.

[33] M. D. Vose, Simple Genetic Algorithm: Foundation and Theory, MIT Press, MI, 1999.
[34]

Z. Wu and T. W. S. Chow, A local multiobjective optimization algorithm using neighborhood field, Structural and Multidisciplinary Optimization, 46 (2012), 853-870. doi: 10.1007/s00158-012-0800-x.

[35]

Z. Wu and T. W. S. Chow, Neighborhood field for cooperative optimization, Soft Computing, 17 (2013), 819-834. doi: 10.1007/s00500-012-0955-9.

[36]

Z. Wu and T. W. S. Chow, Binary neighborhood field optimization for unit commitment problems, IET Generation Transmission and Distribution, 7 (2013), 299-308.

[37]

C. Yang, L. Gu and W. Gui, Particle swarm optimization algorithm with adaptive mutation, Computer Engineering, 34 (2008), 188-190.

[38]

X. Yao, Y. Liu and G. M. Lin, Evolutionary programming made faster, IEEE Transactions on Evolutionary Computation, 3 (1999), 82-102.

[39]

B. Yao, B. Yu, P. Hu, J. Gao and M. H. Zhang, An improved particle swarm optimization for carton heterogeneous vehicle routing problem with a collection depot, Annals of Operations Research, 242 (2016), 303-320. doi: 10.1007/s10479-015-1792-x.

[40]

T. T. Zhao, Q. F. Cheng and Z. F. Wang, Nonlinear model predictive control optimization with improved particle swarm algorithm, Liaoning Gongcheng Jishu Daxue Xuebao (Ziran Kexue Ban)/Journal of Liaoning Technical University (Natural Science Edition), 34 (2015), 517-522.

Figure 1.  Flowchart related to the re-sampling step
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Figure 2.  Flowchart of the PSO-resample (NFO-resample)
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Figure 3.  Comparison of the convergence rate using different test functions ($D = 20, N = 30$)
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Figure 4.  Comparison of the convergence rate under eight algorithms ($D = 20, N = 30$)
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Table 1.  Parameter settings
Algorithm parameters
Improved PSO $c_{1}=c_{2}=2$, $w=0.7298$
Standard PSO $c_{1}=c_{2}=2$, $w=0.7298$
Improved NFO $\alpha=1.3$, $Cr=0.1$
Standard NFO $\alpha=1.3$, $Cr=0.1$
PSO-cf-local $c_{1}=c_{2}=2$, $w=0.7298$
UPSO $c_{1}=c_{2}=2$, $w=0.7298$, $u=0.1$
FDRPSO $c_{1}=c_{2}=1, \ c_3=2$
CLPSO $c_{1}=c_{2}=2$, $w:0.9\sim0.4$, $m$=0.7
LDWPSO $c_{1}=c_{2}=2$, $w:0.9\sim0.4$
DE $F=0.5$, $CR$=0.9
Table Options

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Algorithm parameters
Improved PSO $c_{1}=c_{2}=2$, $w=0.7298$
Standard PSO $c_{1}=c_{2}=2$, $w=0.7298$
Improved NFO $\alpha=1.3$, $Cr=0.1$
Standard NFO $\alpha=1.3$, $Cr=0.1$
PSO-cf-local $c_{1}=c_{2}=2$, $w=0.7298$
UPSO $c_{1}=c_{2}=2$, $w=0.7298$, $u=0.1$
FDRPSO $c_{1}=c_{2}=1, \ c_3=2$
CLPSO $c_{1}=c_{2}=2$, $w:0.9\sim0.4$, $m$=0.7
LDWPSO $c_{1}=c_{2}=2$, $w:0.9\sim0.4$
DE $F=0.5$, $CR$=0.9
Table 2.  Benchmark functions
$f_i$ Range of $x_i$ Theoretical optimum
$f_{1}=\sum\limits^{D}\limits_{i=1}x_{i}^{2}$ [-100,100] 0
$f_{2}=\sum\limits^{D-1}\limits_{i=1}|x_{i}|+\prod\limits^{D-1}\limits_{i=1}|x_{i}|$ [-10, 10] 0
$f_{3}=\sum\limits^{D}\limits_{i=1}(\sum\limits^{i}\limits_{j=1}x_{j})^{2}$ [-100,100] 0
$f_{4}=\max\limits_{i}\{x_{i}, 1\leq i\leq D\}$ [-100,100] 0
$f_{5}=\sum\limits^{D-1}\limits_{i=1}(100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2})$ [-100,100] 0
$f_{6}=\sum\limits^{D}\limits_{i=1}(\lfloor x_{i}+0.5\rfloor)^{2}$ [-100,100] 0
$f_{7}=\sum\limits^{D}\limits_{i=1}ix_{i}^{4}+random[0, 1)$ [-1.28, 1.28] 0
$ f_{8}=\sum\limits^{D}\limits_{i=1}(x_{i}^{2}-10\cos(2\pi x_{i})+10)$ [-5.12, 5.12] 0
$f_{9}=20+e-20exp(-0.2\sqrt{\frac{1}{D}\sum\limits^{D}\limits_{i=1}x_{i}^{2}})-exp(\frac{1}{D}\sum\limits^{D}\limits_{i=1}\cos2\pi x_{i})$ [-30, 30] 0
$f_{10}=\sum\limits^{D}\limits_{i=1}\frac{1}{4000}x_{i}^{2}-\prod\limits^{D}\limits_{i=1}\cos(\frac{x_{i}}{\sqrt{i}})+1$ [-600,600] 0
$ f_{11}=\frac{\pi}{30}\{10sin^{2}(\pi y_{1}) + \sum\limits^{D-1}\limits_{i=1}(y_{i}-1)^{2} \cdot[1+10sin^{2}(\pi y_{i+1})]\\\;\;\;\;\;\;\;\;+(y_{D}-1)^{2}\} + \sum\limits^{D}\limits_{i=1} u(x_{i}, 10,100, 4) $ [-50, 50] 0
$f_{12} = 0.1\{10sin^{2}(\pi 3x_{1}) + \sum\limits^{D-1}\limits_{i=1}(y_{i}-1)^{2} \cdot[1+sin^{2}(3 \pi x_{i+1})]\\ \;\;\;\;\;\;\;\;+(x_{D}-1)^{2}[1+sin^{2}(2 \pi x_{D})] \}+\sum\limits^{D}\limits_{i=1} u(x_{i},5,100,4)\\\;\;\;\;\;\;u({x_i},a,k,m) = \left\{ \begin{array}{l} k{({x_i} - a)^m},\;\;\;\;\;{{x_{ i} > a}}\\ {{0,}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - a \le {x_i} \le a\\ k{( - {x_i} - a)^m},\;\;\;\;{{x_{ i} }} < {{ - a}} \end{array} \right.$ [-50, 50] 0
$f_{13}=[\frac{1}{500}+\sum\limits^{25}\limits_{j=1}(1/{j+\sum\limits^{2}\limits_{i=1} (x_{i}-a_{ij})^6})]^{-1} $ [-65.536, 65.536] 1
$f_{14}=\sum\limits^{11}\limits_{i=1}[a_{i}-(x_{1}(b_i^2+b_i x_2)/(b_i^2+b_i x_3+x_4 ))]^2$ [-5, 5] 0.0003075
$f_{15}=4x_{1}^2-2.1x_{1}^4+\frac{1}{3}x_{1}^6+x_{1}x_{2}-4x_{2}^2$ [-5, 5] -1.03163
$f_{16}=(x_{2}-\frac{5.1}{4\pi^2}x_{1}^2+\frac{5}{\pi}x_{1}-6)^2+10(1-\frac{1}{8\pi})\cos x_{1}+10$ $[-5, 10]\times [0, 15]$ 0.398
$f_{17}=[1+(x_1+x_2+1)^2(19-14x_1\\\;\;\;\;\;\;\;\;+3x_1^2 -14x_2+6x_1x_2+3x_2^2)]\times[30+(2x_1-3x_2)^2 \\ \;\;\;\;\;\;\times(18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2)] $ $[-2, 2]$ 3
$f_{18}=-\sum\limits_i\limits^4 c_i exp[-\sum\limits_j\limits^6 a_{ij}(x_j-a_{ij})]$ $[0, 1]$ -3.32
$f_{19}=-\sum\limits_i\limits^5 [(x-a_i)(x-a_i)^T]^{-1}$ $[0, 10]$ $-10$
$f_{20}=-\sum\limits_i\limits^7 [(x-a_i)(x-a_i)^T]^{-1}$ $[0, 10]$ $-10$
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$f_i$ Range of $x_i$ Theoretical optimum
$f_{1}=\sum\limits^{D}\limits_{i=1}x_{i}^{2}$ [-100,100] 0
$f_{2}=\sum\limits^{D-1}\limits_{i=1}|x_{i}|+\prod\limits^{D-1}\limits_{i=1}|x_{i}|$ [-10, 10] 0
$f_{3}=\sum\limits^{D}\limits_{i=1}(\sum\limits^{i}\limits_{j=1}x_{j})^{2}$ [-100,100] 0
$f_{4}=\max\limits_{i}\{x_{i}, 1\leq i\leq D\}$ [-100,100] 0
$f_{5}=\sum\limits^{D-1}\limits_{i=1}(100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2})$ [-100,100] 0
$f_{6}=\sum\limits^{D}\limits_{i=1}(\lfloor x_{i}+0.5\rfloor)^{2}$ [-100,100] 0
$f_{7}=\sum\limits^{D}\limits_{i=1}ix_{i}^{4}+random[0, 1)$ [-1.28, 1.28] 0
$ f_{8}=\sum\limits^{D}\limits_{i=1}(x_{i}^{2}-10\cos(2\pi x_{i})+10)$ [-5.12, 5.12] 0
$f_{9}=20+e-20exp(-0.2\sqrt{\frac{1}{D}\sum\limits^{D}\limits_{i=1}x_{i}^{2}})-exp(\frac{1}{D}\sum\limits^{D}\limits_{i=1}\cos2\pi x_{i})$ [-30, 30] 0
$f_{10}=\sum\limits^{D}\limits_{i=1}\frac{1}{4000}x_{i}^{2}-\prod\limits^{D}\limits_{i=1}\cos(\frac{x_{i}}{\sqrt{i}})+1$ [-600,600] 0
$ f_{11}=\frac{\pi}{30}\{10sin^{2}(\pi y_{1}) + \sum\limits^{D-1}\limits_{i=1}(y_{i}-1)^{2} \cdot[1+10sin^{2}(\pi y_{i+1})]\\\;\;\;\;\;\;\;\;+(y_{D}-1)^{2}\} + \sum\limits^{D}\limits_{i=1} u(x_{i}, 10,100, 4) $ [-50, 50] 0
$f_{12} = 0.1\{10sin^{2}(\pi 3x_{1}) + \sum\limits^{D-1}\limits_{i=1}(y_{i}-1)^{2} \cdot[1+sin^{2}(3 \pi x_{i+1})]\\ \;\;\;\;\;\;\;\;+(x_{D}-1)^{2}[1+sin^{2}(2 \pi x_{D})] \}+\sum\limits^{D}\limits_{i=1} u(x_{i},5,100,4)\\\;\;\;\;\;\;u({x_i},a,k,m) = \left\{ \begin{array}{l} k{({x_i} - a)^m},\;\;\;\;\;{{x_{ i} > a}}\\ {{0,}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - a \le {x_i} \le a\\ k{( - {x_i} - a)^m},\;\;\;\;{{x_{ i} }} < {{ - a}} \end{array} \right.$ [-50, 50] 0
$f_{13}=[\frac{1}{500}+\sum\limits^{25}\limits_{j=1}(1/{j+\sum\limits^{2}\limits_{i=1} (x_{i}-a_{ij})^6})]^{-1} $ [-65.536, 65.536] 1
$f_{14}=\sum\limits^{11}\limits_{i=1}[a_{i}-(x_{1}(b_i^2+b_i x_2)/(b_i^2+b_i x_3+x_4 ))]^2$ [-5, 5] 0.0003075
$f_{15}=4x_{1}^2-2.1x_{1}^4+\frac{1}{3}x_{1}^6+x_{1}x_{2}-4x_{2}^2$ [-5, 5] -1.03163
$f_{16}=(x_{2}-\frac{5.1}{4\pi^2}x_{1}^2+\frac{5}{\pi}x_{1}-6)^2+10(1-\frac{1}{8\pi})\cos x_{1}+10$ $[-5, 10]\times [0, 15]$ 0.398
$f_{17}=[1+(x_1+x_2+1)^2(19-14x_1\\\;\;\;\;\;\;\;\;+3x_1^2 -14x_2+6x_1x_2+3x_2^2)]\times[30+(2x_1-3x_2)^2 \\ \;\;\;\;\;\;\times(18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2)] $ $[-2, 2]$ 3
$f_{18}=-\sum\limits_i\limits^4 c_i exp[-\sum\limits_j\limits^6 a_{ij}(x_j-a_{ij})]$ $[0, 1]$ -3.32
$f_{19}=-\sum\limits_i\limits^5 [(x-a_i)(x-a_i)^T]^{-1}$ $[0, 10]$ $-10$
$f_{20}=-\sum\limits_i\limits^7 [(x-a_i)(x-a_i)^T]^{-1}$ $[0, 10]$ $-10$
Table 3.  Means and variances under PSO-resample, with different $\delta$ and $k_1$
$f$ $k_1$ $ \delta=10$ $ \delta=20$ $ \delta=30$ $ \delta=40$
$f_{1}$ 80 mean 1.1697e-23 1.4034e-22 1.3997e-19 1.5248e-17
variance 3.1624e-45 1.7331e-43 1.9198e-37 9.7892e-34
100 mean 1.2345e-24 3.4313e-23 6.7734e-20 1.3077e-17
variance 1.2889e-47 4.0997e-45 5.7511e-38 1.7539e-33
150 mean 4.8036e-24 5.5423e-22 4.7328e-19 1.5124e-17
variance 5.3061e-46 1.6406e-42 1.4786e-36 4.8891e-34
200 mean 3.4374e-23 2.2118e-21 8.3626e-19 9.8416e-17
variance 2.0994e-44 1.6926e-41 3.6846e-36 5.8052e-32
300 mean 3.7652e-23 8.6217e-21 9.6451e-18 1.7330e-16
variance 1.6413e-44 3.3834e-40 6.4054e-34 8.3178e-32
$f_{5}$ 80 mean 23.0984 15.5308 15.6512 17.5839
variance 1.3502e+03 0.2934 0.8921 132.7182
100 mean 15.4714 15.4349 15.4707 15.5330
variance 0.3979 0.4585 1.4279 0.9254
150 mean 15.5411 15.5168 15.5302 15.8413
variance 0.4272 0.9433 0.7471 1.0840
200 mean 15.7879 15.5266 15.9301 18.0431
variance 0.4044 1.6317 1.9228 132.4126
300 mean 15.9560 15.5901 17.9298 19.8994
variance 1.3168 0.5490 119.4753 236.7976
$f_{8}$ 80 mean 14.2808 13.4628 14.0816 15.6759
variance 18.3861 70.4926 30.3199 46.6280
100 mean 14.1719 11.2533 12.9326 15.5054
variance 19.8901 21.2570 36.3528 79.5550
150 mean 14.8875 12.1143 13.0902 14.8153
variance 20.5411 29.8877 37.4915 30.4357
200 mean 15.5786 13.5650 13.6945 15.0470
variance 37.2106 26.6829 59.1946 39.4785
300 mean 16.3959 14.1838 15.1895 15.7031
variance 35.1016 37.6114 42.1339 69.3613
$f_{10}$ 80 mean 0.0093 0.0089 0.0164 0.0165
variance 5.5534e-04 3.3844e-04 8.0860e-04 5.2949e-04
100 mean 0.0092 0.0064 0.0121 0.0156
variance 4.5978e-04 2.2374e-04 5.6635e-04 5.1330e-04
150 mean 0.0165 0.0157 0.0153 0.0191
variance 0.0018 5.5528e-04 4.3368e-04 0.0025
200 mean 0.0328 0.0179 0.0245 0.0305
variance 0.0098 5.0790e-04 0.0010 0.0019
300 mean 0.1185 0.0489 0.0608 0.0676
variance 0.0615 0.0124 0.0142 0.0109
$f_{13}$ 80 mean 1.1567 1.2361 1.0774 1.2361
variance 0.2897 0.4157 0.1512 0.4157
100 mean 1.3155 1.0774 0.1512 1.3155
variance 0.5291 0.1512 0.4157 0.5291
150 mean -1.1567 1.4680 1.1567 1.2361
variance 0.2897 3.7509 0.2897 0.4157
200 mean 1.0774 1.1567 1.3155 1.0774
variance 0.1512 0.2897 0.5291 0.1512
300 mean 1.2361 1.3948 1.2361 1.2755
variance 0.4157 0.6299 0.4157 0.5908
$f_{15}$ 80 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 3.6682e-31 7.8886e-32 1.0847e-31 1.2030e-31
100 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 7.4942e-32 1.0255e-31 1.0255e-31 9.0719e-32
150 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 5.5220e-32 7.2970e-32 8.4803e-32 1.3805e-31
200 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 5.3248e-32 9.6635e-32 9.6635e-32 1.1438e-31
300 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 9.0719e-32 1.3213e-31 1.4397e-31 1.2622e-31
$f_{18}$ 80 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 6.4394e-21 9.9421e-28 2.7926e-30 3.6524e-30
100 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 9.0777e-21 8.4809e-28 3.0844e-30 3.5420e-30
150 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 2.0992e-24 4.3654e-28 2.8872e-30 3.2895e-30
200 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 8.4359e-28 2.8557e-30 3.2028e-30 3.7944e-30
300 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 3.2422e-30 3.4394e-30 3.5814e-30 4.2914e-30
$f_{19}$ 80 mean -7.2340 -7.7273 -7.2340 -8.7433
variance 9.9440 7.8747 9.9440 6.6108
100 mean -8.4358 -8.4421 -7.9412 -7.1354
variance 6.4802 7.9474 9.2356 10.5886
150 mean -8.6379 -7.7325 -8.4424 -8.0363
variance 6.1125 7.8573 7.9354 8.4419
200 mean -7.6363 -6.8345 -7.5391 -6.9396
variance 10.1076 10.9456 9.3057 11.8352
300 mean -7.3312 -7.9391 -7.2340 -7.9373
variance 9.2654 9.2577 9.9440 9.2681
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$f$ $k_1$ $ \delta=10$ $ \delta=20$ $ \delta=30$ $ \delta=40$
$f_{1}$ 80 mean 1.1697e-23 1.4034e-22 1.3997e-19 1.5248e-17
variance 3.1624e-45 1.7331e-43 1.9198e-37 9.7892e-34
100 mean 1.2345e-24 3.4313e-23 6.7734e-20 1.3077e-17
variance 1.2889e-47 4.0997e-45 5.7511e-38 1.7539e-33
150 mean 4.8036e-24 5.5423e-22 4.7328e-19 1.5124e-17
variance 5.3061e-46 1.6406e-42 1.4786e-36 4.8891e-34
200 mean 3.4374e-23 2.2118e-21 8.3626e-19 9.8416e-17
variance 2.0994e-44 1.6926e-41 3.6846e-36 5.8052e-32
300 mean 3.7652e-23 8.6217e-21 9.6451e-18 1.7330e-16
variance 1.6413e-44 3.3834e-40 6.4054e-34 8.3178e-32
$f_{5}$ 80 mean 23.0984 15.5308 15.6512 17.5839
variance 1.3502e+03 0.2934 0.8921 132.7182
100 mean 15.4714 15.4349 15.4707 15.5330
variance 0.3979 0.4585 1.4279 0.9254
150 mean 15.5411 15.5168 15.5302 15.8413
variance 0.4272 0.9433 0.7471 1.0840
200 mean 15.7879 15.5266 15.9301 18.0431
variance 0.4044 1.6317 1.9228 132.4126
300 mean 15.9560 15.5901 17.9298 19.8994
variance 1.3168 0.5490 119.4753 236.7976
$f_{8}$ 80 mean 14.2808 13.4628 14.0816 15.6759
variance 18.3861 70.4926 30.3199 46.6280
100 mean 14.1719 11.2533 12.9326 15.5054
variance 19.8901 21.2570 36.3528 79.5550
150 mean 14.8875 12.1143 13.0902 14.8153
variance 20.5411 29.8877 37.4915 30.4357
200 mean 15.5786 13.5650 13.6945 15.0470
variance 37.2106 26.6829 59.1946 39.4785
300 mean 16.3959 14.1838 15.1895 15.7031
variance 35.1016 37.6114 42.1339 69.3613
$f_{10}$ 80 mean 0.0093 0.0089 0.0164 0.0165
variance 5.5534e-04 3.3844e-04 8.0860e-04 5.2949e-04
100 mean 0.0092 0.0064 0.0121 0.0156
variance 4.5978e-04 2.2374e-04 5.6635e-04 5.1330e-04
150 mean 0.0165 0.0157 0.0153 0.0191
variance 0.0018 5.5528e-04 4.3368e-04 0.0025
200 mean 0.0328 0.0179 0.0245 0.0305
variance 0.0098 5.0790e-04 0.0010 0.0019
300 mean 0.1185 0.0489 0.0608 0.0676
variance 0.0615 0.0124 0.0142 0.0109
$f_{13}$ 80 mean 1.1567 1.2361 1.0774 1.2361
variance 0.2897 0.4157 0.1512 0.4157
100 mean 1.3155 1.0774 0.1512 1.3155
variance 0.5291 0.1512 0.4157 0.5291
150 mean -1.1567 1.4680 1.1567 1.2361
variance 0.2897 3.7509 0.2897 0.4157
200 mean 1.0774 1.1567 1.3155 1.0774
variance 0.1512 0.2897 0.5291 0.1512
300 mean 1.2361 1.3948 1.2361 1.2755
variance 0.4157 0.6299 0.4157 0.5908
$f_{15}$ 80 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 3.6682e-31 7.8886e-32 1.0847e-31 1.2030e-31
100 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 7.4942e-32 1.0255e-31 1.0255e-31 9.0719e-32
150 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 5.5220e-32 7.2970e-32 8.4803e-32 1.3805e-31
200 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 5.3248e-32 9.6635e-32 9.6635e-32 1.1438e-31
300 mean -1.0316 -1.0316 -1.0316 -1.0316
variance 9.0719e-32 1.3213e-31 1.4397e-31 1.2622e-31
$f_{18}$ 80 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 6.4394e-21 9.9421e-28 2.7926e-30 3.6524e-30
100 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 9.0777e-21 8.4809e-28 3.0844e-30 3.5420e-30
150 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 2.0992e-24 4.3654e-28 2.8872e-30 3.2895e-30
200 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 8.4359e-28 2.8557e-30 3.2028e-30 3.7944e-30
300 mean -3.8628 -3.8628 -3.8628 -3.8628
variance 3.2422e-30 3.4394e-30 3.5814e-30 4.2914e-30
$f_{19}$ 80 mean -7.2340 -7.7273 -7.2340 -8.7433
variance 9.9440 7.8747 9.9440 6.6108
100 mean -8.4358 -8.4421 -7.9412 -7.1354
variance 6.4802 7.9474 9.2356 10.5886
150 mean -8.6379 -7.7325 -8.4424 -8.0363
variance 6.1125 7.8573 7.9354 8.4419
200 mean -7.6363 -6.8345 -7.5391 -6.9396
variance 10.1076 10.9456 9.3057 11.8352
300 mean -7.3312 -7.9391 -7.2340 -7.9373
variance 9.2654 9.2577 9.9440 9.2681
Table 4.  Convergence steps under PSO-resample
$\delta$ $f_1$ $f_2$ $f_3$ $f_4$ $f_5$ $f_6$ $f_7$ $f_8$ $f_9$ $f_{10}$ $f_{11}$ $f_{12}$ $f_{13}$ $f_{14}$ $f_{15}$ $f_{16}$ $f_{17}$ $f_{18}$ $f_{19}$ $f_{20}$
10 348 487 646 78 - 253 322 - 550 624 1683 - 109 146 46 41 53 87 105 108
20 432 537 794 75 - 348 87 - 648 746 - - 160 149 44 46 55 97 107 93
30 449 549 908 76 - 391 255 - 1201 782 - - 156 147 44 47 51 77 94 99
40 491 570 1022 76 - 374 282 - 780 830 - - 81 146 43 49 56 89 86 84
  *-implies the algorithm does not converge.
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$\delta$ $f_1$ $f_2$ $f_3$ $f_4$ $f_5$ $f_6$ $f_7$ $f_8$ $f_9$ $f_{10}$ $f_{11}$ $f_{12}$ $f_{13}$ $f_{14}$ $f_{15}$ $f_{16}$ $f_{17}$ $f_{18}$ $f_{19}$ $f_{20}$
10 348 487 646 78 - 253 322 - 550 624 1683 - 109 146 46 41 53 87 105 108
20 432 537 794 75 - 348 87 - 648 746 - - 160 149 44 46 55 97 107 93
30 449 549 908 76 - 391 255 - 1201 782 - - 156 147 44 47 51 77 94 99
40 491 570 1022 76 - 374 282 - 780 830 - - 81 146 43 49 56 89 86 84
  *-implies the algorithm does not converge.
Table 5.  Means and variances on $D = 20$ test functions for $N = 30$ ($25$ runs)
Function PSO-resample Standard PSO NFO-resample Standard NFO
$f_{1}$ 3.4313e-23(4.0997e-45) 7.5386(24.3743) 5.0244e-38(6.0588e-74) 1.0370e-33(2.5810e-65)
$f_{2}$ 3.1292e-13(4.2561e-25) 2.5043(4.7289) 2.2622e-23(1.2282e-44) 1.1769e-20(3.3240e-39)
$f_{3}$ 4.8851e-11(8.4971e-21) 49.8877(710.3746) 1.3053e-07(4.0837e-13) 5.7124(747.2053)
$f_{4}$ 1.5284e-94(5.2777e-187) 1.5390e-54(2.3745e-107) 6.3508e-154(9.6797e-306) 2.8710e-35(1.9782e-68)
$f_{5}$ 15.4349(0.4585) 4.4216e+03(5.1577e+07) 2.3160(12.4777) 16.3890(2.6798e+03)
$f_{6}$ 0(0) 68.8400(2.1942e+03) 0(0) 0(0)
$f_{7}$ 0.5155(0.0432) 0.6040(0.0987) 0.1368(0.0049) 0.2147(0.0059)
$f_{8}$ 11.2533(21.2570) 27.3944(104.6841) 0(0) 7.1054e-17(1.2117e-31)
$f_{9}$ 1.0509e-11(2.3499e-22) 4.3043(2.0710) -6.0396e-16(1.9387e-30) 1.3465e-14(4.8468e-30)
$f_{10}$ 0.0064(3.7421e-04) 0.8194(0.0562) 0(0) 0(0)
$f_{11}$ 1.6987(5.8474) 5.5496(11.4830) 0.8247(1.1210e-15) 0.8247(1.9722e-31)
$f_{12}$ -0.2002(0.1663) 17.6488(186.4889) -1.1504(3.6713e-12) -1.1504(1.9722e-31)
$f_{13}$ 1.0774(0.1512) 1.3155(0.5282) 0.9980(3.5922e-22) 0.9980(0)
$f_{14}$ 0.0035(3.6803e-05) 0.0041(5.3083e-05) 9.5850e-04(9.2612e-11) 0.0044(1.0963e-10)
$f_{15}$ -1.0316(1.0255e-31) -1.0316(1.9722e-31) -1.0316(2.0116e-31) -1.0316(1.9722e-31)
$f_{16}$ 0.3979(0) 0.3979(0) 0.3979(1.7588e-14) 0.3979(0)
$f_{17}$ 3.0000(8.1600e-29) 3.0000(2.0905e-30) 3.0000(1.5108e-09) 3.0000(3.1554e-30)
$f_{18}$ -3.2789(0.0032) -3.2694(0.0035) -3.3215(1.0546e-10) -3.3215(5.9953e-31)
$f_{19}$ -8.4421(7.9474) -6.8474(12.3691) -10.1532(6.5141e-17) -10.1532(2.9579e-09)
$f_{20}$ -9.8865(3.1789) -8.2623(11.7828) -10.4029(2.9297e-12) -10.4029(1.0758e-11)
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Function PSO-resample Standard PSO NFO-resample Standard NFO
$f_{1}$ 3.4313e-23(4.0997e-45) 7.5386(24.3743) 5.0244e-38(6.0588e-74) 1.0370e-33(2.5810e-65)
$f_{2}$ 3.1292e-13(4.2561e-25) 2.5043(4.7289) 2.2622e-23(1.2282e-44) 1.1769e-20(3.3240e-39)
$f_{3}$ 4.8851e-11(8.4971e-21) 49.8877(710.3746) 1.3053e-07(4.0837e-13) 5.7124(747.2053)
$f_{4}$ 1.5284e-94(5.2777e-187) 1.5390e-54(2.3745e-107) 6.3508e-154(9.6797e-306) 2.8710e-35(1.9782e-68)
$f_{5}$ 15.4349(0.4585) 4.4216e+03(5.1577e+07) 2.3160(12.4777) 16.3890(2.6798e+03)
$f_{6}$ 0(0) 68.8400(2.1942e+03) 0(0) 0(0)
$f_{7}$ 0.5155(0.0432) 0.6040(0.0987) 0.1368(0.0049) 0.2147(0.0059)
$f_{8}$ 11.2533(21.2570) 27.3944(104.6841) 0(0) 7.1054e-17(1.2117e-31)
$f_{9}$ 1.0509e-11(2.3499e-22) 4.3043(2.0710) -6.0396e-16(1.9387e-30) 1.3465e-14(4.8468e-30)
$f_{10}$ 0.0064(3.7421e-04) 0.8194(0.0562) 0(0) 0(0)
$f_{11}$ 1.6987(5.8474) 5.5496(11.4830) 0.8247(1.1210e-15) 0.8247(1.9722e-31)
$f_{12}$ -0.2002(0.1663) 17.6488(186.4889) -1.1504(3.6713e-12) -1.1504(1.9722e-31)
$f_{13}$ 1.0774(0.1512) 1.3155(0.5282) 0.9980(3.5922e-22) 0.9980(0)
$f_{14}$ 0.0035(3.6803e-05) 0.0041(5.3083e-05) 9.5850e-04(9.2612e-11) 0.0044(1.0963e-10)
$f_{15}$ -1.0316(1.0255e-31) -1.0316(1.9722e-31) -1.0316(2.0116e-31) -1.0316(1.9722e-31)
$f_{16}$ 0.3979(0) 0.3979(0) 0.3979(1.7588e-14) 0.3979(0)
$f_{17}$ 3.0000(8.1600e-29) 3.0000(2.0905e-30) 3.0000(1.5108e-09) 3.0000(3.1554e-30)
$f_{18}$ -3.2789(0.0032) -3.2694(0.0035) -3.3215(1.0546e-10) -3.3215(5.9953e-31)
$f_{19}$ -8.4421(7.9474) -6.8474(12.3691) -10.1532(6.5141e-17) -10.1532(2.9579e-09)
$f_{20}$ -9.8865(3.1789) -8.2623(11.7828) -10.4029(2.9297e-12) -10.4029(1.0758e-11)
Table 6.  Average computational time (seconds) for each run on $D = 20$ test functions for $N = 30$
FunctionPSO-resample Standard PSO NFO-resample Standard NFO
$f_{1}$ 1.20 0.49 23.65 25.31
$f_{2}$ 1.30 0.52 25.61 25.26
$f_{3}$ 4.35 1.77 666.50 643.08
$f_{4}$ 2.94 0.91 50.79 54.41
$f_{5}$ 1.71 0.62 31.57 31.53
$f_{6}$ 1.49 0.51 26.43 24.78
$f_{7}$ 4.02 1.30 76.40 73.21
$f_{8}$ 1.85 0.74 33.34 35.71
$f_{9}$ 2.46 1.05 60.64 50.17
$f_{10}$ 2.57 1.01 52.85 57.37
$f_{11}$ 6.12 2.58 158.61 149.46
$f_{12}$ 4.42 2.22 129.69 127.31
$f_{13}$ 5.32 2.49 184.21 172.35
$f_{14}$ 1.71 0.72 42.70 42.96
$f_{15}$ 1.27 0.54 56.11 28.77
$f_{16}$ 1.06 0.47 22.90 20.07
$f_{17}$ 1.29 0.49 28.37 25.47
$f_{18}$ 2.09 0.84 47.97 50.29
$f_{19}$ 21.98 9.84 549.53 551.62
$f_{20}$ 2.59 1.16 133.86 120.27
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FunctionPSO-resample Standard PSO NFO-resample Standard NFO
$f_{1}$ 1.20 0.49 23.65 25.31
$f_{2}$ 1.30 0.52 25.61 25.26
$f_{3}$ 4.35 1.77 666.50 643.08
$f_{4}$ 2.94 0.91 50.79 54.41
$f_{5}$ 1.71 0.62 31.57 31.53
$f_{6}$ 1.49 0.51 26.43 24.78
$f_{7}$ 4.02 1.30 76.40 73.21
$f_{8}$ 1.85 0.74 33.34 35.71
$f_{9}$ 2.46 1.05 60.64 50.17
$f_{10}$ 2.57 1.01 52.85 57.37
$f_{11}$ 6.12 2.58 158.61 149.46
$f_{12}$ 4.42 2.22 129.69 127.31
$f_{13}$ 5.32 2.49 184.21 172.35
$f_{14}$ 1.71 0.72 42.70 42.96
$f_{15}$ 1.27 0.54 56.11 28.77
$f_{16}$ 1.06 0.47 22.90 20.07
$f_{17}$ 1.29 0.49 28.37 25.47
$f_{18}$ 2.09 0.84 47.97 50.29
$f_{19}$ 21.98 9.84 549.53 551.62
$f_{20}$ 2.59 1.16 133.86 120.27
Table 7.  Means and variances on $D = 40$ test functions for $N = 30$ ($25$ runs)
Function PSO-resample Standard PSO NFO-resample Standard NFO
$f_{1}$ 1.5796e-13(5.5281e-26) 185.5670(5.1824e+03) 3.1264e-20(2.3457e-38) 1.2589e-14(3.8034e-27)
$f_{2}$ 6.0313e-09(3.9308e-17) 15.9041(29.1664)1.5768e-13(5.9668e-25) 4.2327e-10(4.2998e-18)
$f_{3}$ 5.7985e-04(4.9215e-07) 1.7963e+03(4.6628e+05) 1.0969e-04(1.9506e-07) 890.1068(7.2077e+06)
$f_{4}$ 3.6273e-92(3.1490e-182) 1.9099e-53(3.6487e-105) 3.0444e-161(2.2248e-320) 7.1526e-33(1.2278e-63)
$f_{5}$ 36.4677(0.5677) 8.1082e+05(3.6749e+11) 22.7611(25.6539) 27.7804(146.3549)
$f_{6}$ 0(0) 950.9600(7.5934e+04) 0(0) 0(0)
$f_{7}$ 0.6291(0.0868) 0.6804(0.1075) 0.2750(0.0084) 0.3944(0.0121)
$f_{8}$ 53.1581(129.8373) 85.4913(724.0781) 5.0148(185.7615) 80.0579(33.3354)
$f_{9}$ 0.0042(4.1661e-04) 7.3142(2.9770) 6.6771e-12(1.0702e-21) 6.6946e-09(1.0756e-15)
$f_{10}$ 7.7593e-04(6.0020e-06) 2.1590(0.6044) 0(0) 1.4020e-13(4.7177e-25)
$f_{11}$ 5.7450(10.1131) 16.2519(46.5782) 0.4123(1.6140e-13) 0.4123(1.4388e-27)
$f_{12}$ 11.0551(421.2096) 1.9596e+03(3.1327e+07) -1.1504(1.5460e-12) -1.1504(3.3189e-27)
$f_{13}$ 1.0774(0.1512) 1.4333(1.2554) 0.9980(5.3883e-21) 0.9980(0)
$f_{14}$ 0.0038(3.3734e-05) 0.0052(6.1041e-05) 9.6003e-04(3.1558e-10) 0.0044(1.0963e-10)
$f_{15}$ -1.0316(1.0255e-31) -1.0316(1.9722e-31) -1.0316(7.2889e-16) -1.0316(1.6763e-31)
$f_{16}$ 0.3979(0) 0.3979(0) 0.3979(6.3308e-11) 0.3979(0)
$f_{17}$ 3.0000(9.1551e-28) 3.0000(2.0984e-30) 3.0000(6.0843e-19) 3.0000(3.1554e-30)
$f_{18}$ -3.2741(0.0034) -3.2647(0.0035) -3.3215(8.4559e-12) -3.3215(1.9722e-31)
$f_{19}$ -8.2382(8.2475) -6.7459(11.4480) -10.1532(6.6789e-13) -10.1532(1.2204e-19)
$f_{20}$ -8.6646(9.7556) -7.2678(12.8782) -10.4029(5.2199e-09) -10.4029(3.8379e-15)
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Function PSO-resample Standard PSO NFO-resample Standard NFO
$f_{1}$ 1.5796e-13(5.5281e-26) 185.5670(5.1824e+03) 3.1264e-20(2.3457e-38) 1.2589e-14(3.8034e-27)
$f_{2}$ 6.0313e-09(3.9308e-17) 15.9041(29.1664)1.5768e-13(5.9668e-25) 4.2327e-10(4.2998e-18)
$f_{3}$ 5.7985e-04(4.9215e-07) 1.7963e+03(4.6628e+05) 1.0969e-04(1.9506e-07) 890.1068(7.2077e+06)
$f_{4}$ 3.6273e-92(3.1490e-182) 1.9099e-53(3.6487e-105) 3.0444e-161(2.2248e-320) 7.1526e-33(1.2278e-63)
$f_{5}$ 36.4677(0.5677) 8.1082e+05(3.6749e+11) 22.7611(25.6539) 27.7804(146.3549)
$f_{6}$ 0(0) 950.9600(7.5934e+04) 0(0) 0(0)
$f_{7}$ 0.6291(0.0868) 0.6804(0.1075) 0.2750(0.0084) 0.3944(0.0121)
$f_{8}$ 53.1581(129.8373) 85.4913(724.0781) 5.0148(185.7615) 80.0579(33.3354)
$f_{9}$ 0.0042(4.1661e-04) 7.3142(2.9770) 6.6771e-12(1.0702e-21) 6.6946e-09(1.0756e-15)
$f_{10}$ 7.7593e-04(6.0020e-06) 2.1590(0.6044) 0(0) 1.4020e-13(4.7177e-25)
$f_{11}$ 5.7450(10.1131) 16.2519(46.5782) 0.4123(1.6140e-13) 0.4123(1.4388e-27)
$f_{12}$ 11.0551(421.2096) 1.9596e+03(3.1327e+07) -1.1504(1.5460e-12) -1.1504(3.3189e-27)
$f_{13}$ 1.0774(0.1512) 1.4333(1.2554) 0.9980(5.3883e-21) 0.9980(0)
$f_{14}$ 0.0038(3.3734e-05) 0.0052(6.1041e-05) 9.6003e-04(3.1558e-10) 0.0044(1.0963e-10)
$f_{15}$ -1.0316(1.0255e-31) -1.0316(1.9722e-31) -1.0316(7.2889e-16) -1.0316(1.6763e-31)
$f_{16}$ 0.3979(0) 0.3979(0) 0.3979(6.3308e-11) 0.3979(0)
$f_{17}$ 3.0000(9.1551e-28) 3.0000(2.0984e-30) 3.0000(6.0843e-19) 3.0000(3.1554e-30)
$f_{18}$ -3.2741(0.0034) -3.2647(0.0035) -3.3215(8.4559e-12) -3.3215(1.9722e-31)
$f_{19}$ -8.2382(8.2475) -6.7459(11.4480) -10.1532(6.6789e-13) -10.1532(1.2204e-19)
$f_{20}$ -8.6646(9.7556) -7.2678(12.8782) -10.4029(5.2199e-09) -10.4029(3.8379e-15)
Table 8.  Means and variances under different algorithms for $D=20$ and $N = 30$ (25 runs)
Function $f_{1}$ $f_{2}$ $f_{3}$ $f_{4}$
PSO-resample 3.4313e-23(4.0997e-45) 3.1292e-13(4.2561e-25) 4.8851e-11(8.4971e-21) 1.5284e-94(5.2777e-187)
NFO-resample 5.0244e-38(6.0588e-74) 2.2622e-23(1.2282e-44) 1.3053e-07 (4.0837e-13) 6.3508e-154(9.6797e-306)
PSO$_{-}$cf$_{-}$local 3.1842(5.2344) 1.5076(0.5649) 12.8833(67.5186) 1.7914e-46(3.8620e-91)
UPSO 2.0381e-15(8.3665e-30) 3.7556e-11(2.9156e-20) 1.7202(2.2547) 2.8064e-95(1.1021e-188)
FDRPSO 3.0682e-09(3.2419e-18) 8.1237e-06(8.0225e-12) 1.9837 (5.4554) 1.4365e-74(4.8423e-147)
CLPSO 3.3930e-06(5.9273e-12) 1.5475e-04(1.8278e-09) 786.7243(3.3263e+04) 2.0995e-66(5.6095e-131)
LDWPSO 1.2282(0.9361) 0.5367(0.0501) 49.3415(1.3333e+03) 1.0966e-92(2.6342e-183)
DE 99.0340(1.0350e+05) 1.6477(16.3922) 625.9111(9.9835e+05) 3.1044e-183(0)
Function $f_{5}$ $f_{6}$ $f_{7}$ $f_{8}$
PSO-resample 15.4349(0.4585) 0(0) 0.5155( 0.0432) 11.2533(21.2570)
NFO-resample 2.3160(12.4777) 0(0) 0.1368(0.0049) 0(0)
PSO$_{-}$cf$_{-}$local 1.2137e+03(1.0794e+06) 41.7600(706.5824) 0.4164(0.0697) 26.6187(95.9579)
UPSO 21.4613(478.5440) 0(0) 0.5336(0.0577) 20.7462(41.9989)
FDRPSO 47.6522(1.4556e+04) 0(0) 0.5480(0.0660) 31.3348(228.5036)
CLPSO 87.8288(1.4948e+03) 0(0) 0.6045(0.1044) 0.5915(0.3101)
LDWPSO 1.1821e+03(2.1126e+06) 14.1200(36.3456) 0.4455(0.0878) 29.9535(68.6640)
DE 3.3685e+06(1.4229e+14) 2428(0) 0.2584(0.0022) 50.3008(175.7736)
Function $f_{9}$ $f_{10}$ $f_{11}$ $f_{12}$
PSO-resample 1.0509e-11(2.3499e-22) 0.0084(3.7421e-04) 1.6987( 5.8474) -0.3243(0.1740)
NFO-resample -6.0396e-16(1.9387e-30) 0(0) 0.8247(1.1210e-15) -1.1504(3.6713e-12)
PSO$_{-}$cf$_{-}$local 3.8709(0.4796) 0.8778(0.0276) 2.0460(1.3128) 3.2971(20.3606)
UPSO 1.1981e-08(7.7256e-17) 0.0037(4.7966e-05) 0.8247(1.0699e-31) -1.1504(6.9108e-29)
FDRPSO 1.6932e-05(3.0526e-11) 0.0398(0.0012) 0.8247(9.1812e-26) -1.1500(4.6357e-06)
CLPSO 0.0011(1.4550e-07) 0.0012(1.8683e-06) 0.8247(7.1869e-16) -1.1504(3.5925e-12)
LDWPSO 2.7815(0.4828) 0.5950(0.0336) 3.0484(4.9952) 0.5622(1.1895)
DE 8.4865(0.0087) 3.9331(29.0800) 7.6613(0.3984) 29.9478(384.2202)
Function $f_{13}$ $f_{14}$ $f_{15}$ $f_{16}$
PSO-resample 1.0774(0.1512) 0.0035(3.6803e-05) -1.0316(1.0255e-31) 0.3979(0)
NFO-resample 0.9980(3.5922e-22) 9.5850e-04(9.2612e-11) -1.0316(2.0116e-31) 0.3979(1.7588e-14)
PSO$_{-}$cf$_{-}$local 1.5132(1.1064) 0.0035(3.6843e-05) -1.0316(1.9722e-31) 0.3979(0)
UPSO 0.9980(0) 7.5611e-04(4.6700e-08) -1.0316(1.9722e-31) 0.3979(0)
FDRPSO 0.9980(1.9722e-33) 0.0045(3.7597e-05) -1.0316(1.2735e-18) 0.3979(2.8926e-14)
CLPSO 0.9980(0) 0.0044(6.1811e-10) -1.0316( 6.7053e-32) 0.3979(2.1405e-28)
LDWPSO 1.0774(0.1512) 0.0021(1.5442e-05) -1.0316(1.9722e-31) 0.3979(0)
DE 0.9980(0) 0.0044(8.4259e-37) -1.0316(1.9722e-31) 0.3979(0)
Function $f_{17}$ $f_{18}$ $f_{19}$ $f_{20}$
PSO-resample 3.0000(8.1600e-29) -3.2789( 0.0032) -8.4421(7.9474) -9.8865(3.1789)
NFO-resample 3.0000(1.5108e-09) -3.3215(1.0546e-10) -10.1532(6.5141e-17) -10.4029( 2.9297e-12)
PSO$_{-}$cf$_{-}$local 3.0000(2.2719e-30) -3.2836(0.0030) -8.0573(11.2958) -9.4854 (6.1741 )
UPSO 3.0000(3.1554e-30) -3.3215(6.4687e-31) -9.9511( 0.9802) -10.4029(4.6701e-30)
FDRPSO 3.0000(2.6269e-30) -3.2931(0.0026) -7.1588(13.4501) -10.4029(7.6993e-30)
CLPSO 3.0000(6.0537e-29) -3.3215(9.4155e-15) -10.1532(5.9021e-17) -10.4029(4.4227e-09)
LDWPSO 3.0000(2.6111e-30) -3.2741(0.0034) -8.2319( 6.7778) -8.1481(11.1390)
DE 3.0000(3.1554e-30) -3.1982(5.4264e-05) -8.1072(6.0146) -10.1427(0.7160)
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Function $f_{1}$ $f_{2}$ $f_{3}$ $f_{4}$
PSO-resample 3.4313e-23(4.0997e-45) 3.1292e-13(4.2561e-25) 4.8851e-11(8.4971e-21) 1.5284e-94(5.2777e-187)
NFO-resample 5.0244e-38(6.0588e-74) 2.2622e-23(1.2282e-44) 1.3053e-07 (4.0837e-13) 6.3508e-154(9.6797e-306)
PSO$_{-}$cf$_{-}$local 3.1842(5.2344) 1.5076(0.5649) 12.8833(67.5186) 1.7914e-46(3.8620e-91)
UPSO 2.0381e-15(8.3665e-30) 3.7556e-11(2.9156e-20) 1.7202(2.2547) 2.8064e-95(1.1021e-188)
FDRPSO 3.0682e-09(3.2419e-18) 8.1237e-06(8.0225e-12) 1.9837 (5.4554) 1.4365e-74(4.8423e-147)
CLPSO 3.3930e-06(5.9273e-12) 1.5475e-04(1.8278e-09) 786.7243(3.3263e+04) 2.0995e-66(5.6095e-131)
LDWPSO 1.2282(0.9361) 0.5367(0.0501) 49.3415(1.3333e+03) 1.0966e-92(2.6342e-183)
DE 99.0340(1.0350e+05) 1.6477(16.3922) 625.9111(9.9835e+05) 3.1044e-183(0)
Function $f_{5}$ $f_{6}$ $f_{7}$ $f_{8}$
PSO-resample 15.4349(0.4585) 0(0) 0.5155( 0.0432) 11.2533(21.2570)
NFO-resample 2.3160(12.4777) 0(0) 0.1368(0.0049) 0(0)
PSO$_{-}$cf$_{-}$local 1.2137e+03(1.0794e+06) 41.7600(706.5824) 0.4164(0.0697) 26.6187(95.9579)
UPSO 21.4613(478.5440) 0(0) 0.5336(0.0577) 20.7462(41.9989)
FDRPSO 47.6522(1.4556e+04) 0(0) 0.5480(0.0660) 31.3348(228.5036)
CLPSO 87.8288(1.4948e+03) 0(0) 0.6045(0.1044) 0.5915(0.3101)
LDWPSO 1.1821e+03(2.1126e+06) 14.1200(36.3456) 0.4455(0.0878) 29.9535(68.6640)
DE 3.3685e+06(1.4229e+14) 2428(0) 0.2584(0.0022) 50.3008(175.7736)
Function $f_{9}$ $f_{10}$ $f_{11}$ $f_{12}$
PSO-resample 1.0509e-11(2.3499e-22) 0.0084(3.7421e-04) 1.6987( 5.8474) -0.3243(0.1740)
NFO-resample -6.0396e-16(1.9387e-30) 0(0) 0.8247(1.1210e-15) -1.1504(3.6713e-12)
PSO$_{-}$cf$_{-}$local 3.8709(0.4796) 0.8778(0.0276) 2.0460(1.3128) 3.2971(20.3606)
UPSO 1.1981e-08(7.7256e-17) 0.0037(4.7966e-05) 0.8247(1.0699e-31) -1.1504(6.9108e-29)
FDRPSO 1.6932e-05(3.0526e-11) 0.0398(0.0012) 0.8247(9.1812e-26) -1.1500(4.6357e-06)
CLPSO 0.0011(1.4550e-07) 0.0012(1.8683e-06) 0.8247(7.1869e-16) -1.1504(3.5925e-12)
LDWPSO 2.7815(0.4828) 0.5950(0.0336) 3.0484(4.9952) 0.5622(1.1895)
DE 8.4865(0.0087) 3.9331(29.0800) 7.6613(0.3984) 29.9478(384.2202)
Function $f_{13}$ $f_{14}$ $f_{15}$ $f_{16}$
PSO-resample 1.0774(0.1512) 0.0035(3.6803e-05) -1.0316(1.0255e-31) 0.3979(0)
NFO-resample 0.9980(3.5922e-22) 9.5850e-04(9.2612e-11) -1.0316(2.0116e-31) 0.3979(1.7588e-14)
PSO$_{-}$cf$_{-}$local 1.5132(1.1064) 0.0035(3.6843e-05) -1.0316(1.9722e-31) 0.3979(0)
UPSO 0.9980(0) 7.5611e-04(4.6700e-08) -1.0316(1.9722e-31) 0.3979(0)
FDRPSO 0.9980(1.9722e-33) 0.0045(3.7597e-05) -1.0316(1.2735e-18) 0.3979(2.8926e-14)
CLPSO 0.9980(0) 0.0044(6.1811e-10) -1.0316( 6.7053e-32) 0.3979(2.1405e-28)
LDWPSO 1.0774(0.1512) 0.0021(1.5442e-05) -1.0316(1.9722e-31) 0.3979(0)
DE 0.9980(0) 0.0044(8.4259e-37) -1.0316(1.9722e-31) 0.3979(0)
Function $f_{17}$ $f_{18}$ $f_{19}$ $f_{20}$
PSO-resample 3.0000(8.1600e-29) -3.2789( 0.0032) -8.4421(7.9474) -9.8865(3.1789)
NFO-resample 3.0000(1.5108e-09) -3.3215(1.0546e-10) -10.1532(6.5141e-17) -10.4029( 2.9297e-12)
PSO$_{-}$cf$_{-}$local 3.0000(2.2719e-30) -3.2836(0.0030) -8.0573(11.2958) -9.4854 (6.1741 )
UPSO 3.0000(3.1554e-30) -3.3215(6.4687e-31) -9.9511( 0.9802) -10.4029(4.6701e-30)
FDRPSO 3.0000(2.6269e-30) -3.2931(0.0026) -7.1588(13.4501) -10.4029(7.6993e-30)
CLPSO 3.0000(6.0537e-29) -3.3215(9.4155e-15) -10.1532(5.9021e-17) -10.4029(4.4227e-09)
LDWPSO 3.0000(2.6111e-30) -3.2741(0.0034) -8.2319( 6.7778) -8.1481(11.1390)
DE 3.0000(3.1554e-30) -3.1982(5.4264e-05) -8.1072(6.0146) -10.1427(0.7160)
Table 9.  Convergence steps under the eight algorithms with $D = 20, N = 30$
Function $f_{1}$ $f_{2}$ $f_{3}$ $f_{4}$ $f_{5}$ $f_{6}$ $f_{7}$ $f_{8}$ $f_{9}$ $f_{10}$ $f_{11}$ $f_{12}$ $f_{13}$ $f_{14}$ $f_{15}$ $f_{16}$ $f_{17}$ $f_{18}$ $f_{19}$ $f_{20}$
PSO-resample 432 537 794 75 - 348 87 - 648 746 - - 160 149 44 46 55 97 107 93
NFO-resample 252 237 54 35 509 182 53 85 257 292 309 249 41 25 17 39 39 73 74 72
PSO$_{-}$cf$_{-}$local - 1026 - 76 - 357 361 - 1059 1145 - 1408 86 158 52 53 65 85 105 119
UPSO 706 745 - 63 - 519 1700 1230 877 877 530 810 204 138 44 52 58 89 122 121
FDRPSO 1579 1712 - 116 - 1294 735 137 1737 - 1289 1550 203 152 106 126 162 182 538 552
CLPSO 1638 1654 - 133 - 1201 820 - 1921 1888 1358 1617 263 351 144 200 348 589 861 737
LDWPSO 1050 930 1296 171 1468 676 1092 1068 919 973 1077 1072 169 208 82 72 110 179 278 286
DE 17 25 38 28 19 27 500 19 41 49 29 18 19 22 17 32 22 25 27 35
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Function $f_{1}$ $f_{2}$ $f_{3}$ $f_{4}$ $f_{5}$ $f_{6}$ $f_{7}$ $f_{8}$ $f_{9}$ $f_{10}$ $f_{11}$ $f_{12}$ $f_{13}$ $f_{14}$ $f_{15}$ $f_{16}$ $f_{17}$ $f_{18}$ $f_{19}$ $f_{20}$
PSO-resample 432 537 794 75 - 348 87 - 648 746 - - 160 149 44 46 55 97 107 93
NFO-resample 252 237 54 35 509 182 53 85 257 292 309 249 41 25 17 39 39 73 74 72
PSO$_{-}$cf$_{-}$local - 1026 - 76 - 357 361 - 1059 1145 - 1408 86 158 52 53 65 85 105 119
UPSO 706 745 - 63 - 519 1700 1230 877 877 530 810 204 138 44 52 58 89 122 121
FDRPSO 1579 1712 - 116 - 1294 735 137 1737 - 1289 1550 203 152 106 126 162 182 538 552
CLPSO 1638 1654 - 133 - 1201 820 - 1921 1888 1358 1617 263 351 144 200 348 589 861 737
LDWPSO 1050 930 1296 171 1468 676 1092 1068 919 973 1077 1072 169 208 82 72 110 179 278 286
DE 17 25 38 28 19 27 500 19 41 49 29 18 19 22 17 32 22 25 27 35
[1]

Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337
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