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Multi swarm optimization algorithm with adaptive connectivity degree

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Abstract

Particle swarm optimization algorithms are very sensitive to their population topologies. In all PSO variants, each particle adjusts it flying velocity according to a set of attractor particles. The cardinality of this set is a feature of neighborhood topology. In order to investigate this property exclusively, this paper defines the concept of connectivity degree for the particles and presents an approach for its adaptive adjustment. The presented approach is based on cellular learning automata (CLA). The entire population of the particles is divided into several swarms, and each swarm is resided in one cell of a CLA. Each cell of the CLA also contains a group of learning automata. These learning automata are responsible for adjusting the connectivity degrees of their related particles. This task is achieved through periods of learning. In each period, the learning automata realize suitable connectivity degrees for the particles based on their experienced knowledge. The empirical studies on a divers set of problems with different characteristics show that the proposed multi swarm optimization method is quite effective in solving optimization problems.

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Author information

Authors and Affiliations

  1. Soft Computing Laboratory, Computer Engineering and Information Technology Department, Amirkabir University of Technology, Tehran, Iran

    Reza Vafashoar & Mohammad Reza Meybodi

Authors
  1. Reza Vafashoar
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  2. Mohammad Reza Meybodi
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Corresponding author

Correspondence to Mohammad Reza Meybodi.

Appendices

Appendix A

Statistical results of CLAMS-II under different settings of χ.

Table 7 Statistical results (average, median, min, max) of CLAMS-II on 30-D classical problems under different settings of χ
Full size table

Appendix B:

Statistical results of the peer methods on 30-D classical problems. The results include the average, median, best, and worst achieved fitness values in 30 independent runs as well as the Wilcoxon rank sum tests. In Wilcoxon tests each peer approach is compared with CLAMS-II. Some minor differences in the results presented in Table 8 are due to the precision limitations of the variables in the implementations. Accordingly, we have considered values less than 5.0e-15, 5.0e-15, 5.0e-16, 1.0e-28, 5.0e-16, and 5.0e-16, respectively for the benchmarks f 2, f 3, f 5, f 8, f 9, and f 13 as zero. Also, the results less than 0.1 of the obtained minimum in f 4 and f 15 are set to their minimum in the Wilcoxon tests.

Table 8 Statistical results of the peer methods on 30-D classical problems. Average, Median, Minimum, and Maximum of the final obtained solutions along with the Wilcoxon test results are reported on each benchmark problem
Full size table

Appendix C:

Statistical results on 50-D classical problems.

Table 9 Statistical results of the peer methods on 50-D classical problems. Average, Median, Minimum, and Maximum of the final obtained solutions along with the Wilcoxon test results are reported on each benchmark problem
Full size table

Appendix D:

Statistical results on CEC2013 benchmark set in terms of the average, median, minimum, maximum, and standard deviation of the error obtained in different runs (unavailable statistical information are identified with dashes).

Table 10 Statistical results on CEC2013 at 30D. Average, median, minimum, maximum, std of error in 51 independent runs
Full size table

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Vafashoar, R., Meybodi, M.R. Multi swarm optimization algorithm with adaptive connectivity degree. Appl Intell 48, 909–941 (2018). https://doi.org/10.1007/s10489-017-1039-4

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