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Convergence analysis of a new MaxMin-SOMO algorithm

  • Atlas KhanEmail author
  • Yan-Peng Qu
  • Zheng-Xue Li
Research Article

Abstract

The convergence analysis of MaxMin-SOMO algorithm is presented. The SOM-based optimization (SOMO) is an optimization algorithm based on the self-organizing map (SOM) in order to find a winner in the network. Generally, through a competitive learning process, the SOMO algorithm searches for the minimum of an objective function. The MaxMin-SOMO algorithm is the generalization of SOMO with two winners for simultaneously finding two winning neurons i.e., first winner stands for minimum and second one for maximum of the objective function. In this paper, the convergence analysis of the MaxMin-SOMO is presented. More specifically, we prove that the distance between neurons decreases at each iteration and finally converge to zero. The work is verified with the experimental results.

Keywords

Optimization self organizing map (SOM) SOM-based optimization (SOMO) algorithm particle swarm optimization (PSO) genetic algorithms (GAs) 

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Notes

Acknowledgements

The authors wish to thank the associate editor and the anonymous reviewers for their helpful and interesting comments. We are very grateful to professor Wu Wei for extremely helpful discussion and comments to improve the quality of the paper. Thanks to professor Ricardo Zorzetto Nicoliello Vêncio for his help to improve the text of the paper.

References

  1. [1]
    T. Kohonen. Analysis of a simple self-organizing process. Biological Cybernetics, vol. 44, no. 2, pp. 135–140, 1982.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    T. Kohonen. Self-organized formation of topologically correct feature maps. Biological Cybernetics, vol. 43, no. 1, pp. 59–69, 1982.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    T. Kohonen, E. Oja, O. Simula, A. Visa, J. Kangas. Engineering applications of the self-organizing map. Proceedings of the IEEE, vol. 84, no. 10, pp. 1358–1384, 1996.CrossRefGoogle Scholar
  4. [4]
    T. Kohonen. Essentials of the self-organizing map. Neural Networks, vol. 37, pp. 52–65, 2013.CrossRefGoogle Scholar
  5. [5]
    H. J. Yin. Nonlinear dimensionality reduction and data visualization: A review. International Journal of Automation and Computing, vol. 4, no. 3, pp. 294–303, 2007.CrossRefGoogle Scholar
  6. [6]
    X. F. Hu, Q. H. Weng. Estimating impervious surfaces from medium spatial resolution imagery using the self-organizing map and multi-layer perceptron neural networks. Remote Sensing of Environment, vol. 113, no. 10, pp. 2089–2102, 2009.CrossRefGoogle Scholar
  7. [7]
    R. Q. Huang, L. F. Xi, X. L. Li, C. R. Liu, H. Qiu, J. Lee. Residual life predictions for ball bearings based on selforganizing map and back propagation neural network methods. Mechanical Systems and Signal Processing, vol. 21, no. 1, pp. 193–207, 2007.CrossRefGoogle Scholar
  8. [8]
    D. R. Chen, R. F. Chang, Y. L. Huang. Breast cancer diagnosis using self-organizing map for sonography. Ultrasound in Medicine and Biology, vol. 26, no. 3, pp. 405–411, 2000.CrossRefGoogle Scholar
  9. [9]
    A. Skupin, J. R. Biberstine, K. Börner. Visualizing the topical structure of the medical sciences: A self-organizing map approach. PLoS ONE, vol. 8, no. 3, pp. e58779, 2013.CrossRefGoogle Scholar
  10. [10]
    J. H. Holland. Adaptation in Natural and Artificial Systems, Ann Arbor, USA: University of Michigan Press, 1975.Google Scholar
  11. [11]
    D. E. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning, Boston, USA: Addison-Wesley, 1989.zbMATHGoogle Scholar
  12. [12]
    L. J. Fogel. Evolutionary programming in perspective: The top-down view. Computational Intelligence: Imitating Life, J. M. Zurada, R. J. Marks II, C. J. Robinson, Eds., Piscataway, USA: IEEE Press, pp. 135–146, 1994.Google Scholar
  13. [13]
    I. Rechenberg. Evolution strategy. Computational Intelligence: Imitating Life, J. M. Zurada, R. J. Marks II, C. J. Robinson, Eds., Piscataway, USA: IEEE Press, pp. 147–159, 1994.Google Scholar
  14. [14]
    J. Kennedy, R. C. Eberhart, Y. H. Shi. Swarm Intelligence, New York, USA: Academic Press, 2001.Google Scholar
  15. [15]
    M. S. Armugam, M. V. C. Rao. On the optimal control of single-stage hybrid manufacturing systems via novel and different variants of particle swarm optimization algorithm. Discrete Dynamics in Nature and Society, vol. 2005, no. 3, pp. 257–279, 2005.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    R. Eberhart, J. Kennedy. A new optimizer using particle swarm theory. In Proceedings of the 6th International Symposium on Micro Machine and Human Science, IEEE, Nagoya, Japan, pp. 39–43, 1995.CrossRefGoogle Scholar
  17. [17]
    S. K. Goudos, J. N. Sahalos. Microwave absorber optimal design using multi-objective particle swarm optimization. Microwave and Optical Technology Letters, vol. 48, no. 8, pp. 1553–1558, 2006.CrossRefGoogle Scholar
  18. [18]
    H. Zhang, C. M. Tam, H. Li, J. J. Shi. Particle swarm optimization-supported simulation for construction operations. Journal of Construction Engineering and Management, vol. 132, no. 12, pp. 1267–1274, 2006.CrossRefGoogle Scholar
  19. [19]
    T. Kohonen. Self-organizing Maps, Berlin, Germany: Springer-Verlag, 1995.CrossRefzbMATHGoogle Scholar
  20. [20]
    B. Angéniol, G. de La Croix Vaubois, J. Y. Le Texier. Selforganizing feature maps and the travelling salesman problem. Neural Networks, vol. 1, no. 4, pp. 289–293, 1988.CrossRefGoogle Scholar
  21. [21]
    H. D. Jin, K. S. Leung, M. L. Wong, Z. B. Xu. An efficient self-organizing map designed by genetic algorithms for the traveling salesman problem. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 33, no. 6, pp. 877–888, 2003.CrossRefGoogle Scholar
  22. [22]
    J. H. Chen, L. R. Yang, M. C. Su. Comparison of SOMbased optimization and particle swarm optimization for minimizing the construction time of a secant pile wall. Automation in Construction, vol. 18, no. 6, pp. 844–848, 2009.CrossRefGoogle Scholar
  23. [23]
    M. C. Su, T. K. Liu, H. T. Chang. Improving the selforganizing feature map algorithm using an efficient initialization scheme. Tamkang Journal of Science and Engineering, vol. 5, no. 1, pp. 35–48, 2002.Google Scholar
  24. [24]
    J. H. Chen, L. R. Yang, M. C. Su, J. Z. Lin. Optimal construction sequencing for secant pile wall. In Proceedings of the 2008 IEEE International Conference on Industrial Engineering and Engineering Management, IEEE, Singapore, pp. 2142–2147, 2008.CrossRefGoogle Scholar
  25. [25]
    M. C. Su, Y. X. Zhao. A variant of the SOM algorithm and its interpretation in the viewpoint of social influence and learning. Neural Computing & Applications, vol. 18, no. 8, pp. 1043–1055, 2009.CrossRefGoogle Scholar
  26. [26]
    M. C. Su, Y. X. Zhao, J. Lee. SOM-based optimization. In Proceedings of the 2004 IEEE International Joint Conference on Neural Networks, IEEE, Budapest, Hungarg, pp. 781–786, 2004.Google Scholar
  27. [27]
    W. Wu, A. Khan. SOMO-m optimization algorithm with multiple winners. Discrete Dynamics in Nature and Society, Article ID 969104, 2012.Google Scholar
  28. [28]
    W. Wu, A. Khan. MaxMin-SOMO: An SOM optimization algorithm for simultaneously finding maximum and minimum of a function. In Proceedings of the 9th International Symposium on Neural Networks, Lecture Notes in Computer Science, Springer, Shenyang, China, pp. 598–606, 2012.Google Scholar
  29. [29]
    A. Khan, L. Z. Xue, W. Wei, Y. P. Qu, A. Hussain, R. Z. N. Vencio. Convergence analysis of a new self organizing map based optimization (SOMO) algorithm. Cognitive Computation, vol. 7, no. 4, pp. 477–486, 2015.CrossRefGoogle Scholar

Copyright information

© Institute of Automation, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics Dalian University of TechnologyDalianChina
  2. 2.School of Information Science and TechnologyDalian Maritime UniversityDalianChina
  3. 3.Department of Computing and Mathematics FFCLRPUniversity of Sao PauloRibeirao PretoBrazil

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