Volume 16 Number 4
August 2019
Article Contents
Atlas Khan, Yan-Peng Qu and Zheng-Xue Li. Convergence Analysis of a New MaxMin-SOMO Algorithm. International Journal of Automation and Computing, vol. 16, no. 4, pp. 534-542, 2019. doi: 10.1007/s11633-016-0996-0
Cite as: Atlas Khan, Yan-Peng Qu and Zheng-Xue Li. Convergence Analysis of a New MaxMin-SOMO Algorithm. International Journal of Automation and Computing, vol. 16, no. 4, pp. 534-542, 2019.

# Convergence Analysis of a New MaxMin-SOMO Algorithm

Author Biography:
• Yan-Peng Qu received the Ph. D. degree in computational mathematics from Dalian University of Technology, China in 2012. He is a lecturer with the Information Science and Technology College at Dalian Maritime University, China.
His research interests include rough and fuzzy set theory, pattern recognition, neural networks, classiflcation and feature selection.
E-mail: yanpengqu@dlmu.edu.cn

Zheng-Xue Li received the Ph. D. degree in mathematics from Jilin University, Changchun, China in 2001. He is currently an associate professor with Dalian University of Technology, China.
His research interests include nonlinear algorithm analysis and intelligent information processing.
E-mail: lizx@dlut.edu.cn

• Corresponding author: Atlas Khan received the B. Sc. and M. Sc. degrees in mathematics from Gomal University DI Khan Pakistan, in 2005 and 2007, respectively, and M. Phil. degree in mathematics from Quaid-i-Azam University, Pakistan in 2010. He obtained the Ph. D. degree from Department of Applied Mathematics, Dalian University of Technology, China in 2013. Since August 2013, he is doing post-docotor in bioinformatics with Department of Computing and Mathematics, University of Sao Paulo, Brazil. He has published a number of papers in international journals and conferences.
His research interests include bioinformatios, computational biology, neural networks and coding theory.
E-mail: atlas.khan@ficlrp.usp.br (Corresponding author)
ORCID iD: 0000-0002-6651-2725
• Accepted: 2015-07-03
• Published Online: 2017-02-21
• 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.
•  [1] T. Kohonen. Analysis of a simple self-organizing process. Biological Cybernetics, vol. 44, no. 2, pp. 135-140, 1982.  doi: 10.1007/BF00317973 [2] T. Kohonen. Self-organized formation of topologically correct feature maps. Biological Cybernetics, vol. 43, no. 1, pp. 59-69, 1982.  doi: 10.1007/BF00337288 [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.  doi: 10.1109/5.537105 [4] T. Kohonen. Essentials of the self-organizing map. Neural Networks, vol. 37, pp. 52-65, 2013.  doi: 10.1016/j.neunet.2012.09.018 [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.  doi: 10.1007/s11633-007-0294-y [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.  doi: 10.1016/j.rse.2009.05.014 [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 self-organizing map and back propagation neural network methods. Mechanical Systems and Signal Processing, vol. 21, no. 1, pp. 193-207, 2007.  doi: 10.1016/j.ymssp.2005.11.008 [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.  doi: 10.1016/S0301-5629(99)00156-8 [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.  doi: 10.1371/journal.pone.0058779 [10] J. H. Holland. Adaptation in Natural and Artificial Systems, Ann Arbor, USA: University of Michigan Press, 1975. [11] D. E. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning, Boston, USA: Addison-Wesley, 1989. [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. [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. [14] J. Kennedy, R. C. Eberhart, Y. H. Shi. Swarm Intelligence, New York, USA: Academic Press, 2001. [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.  doi: 10.1155/DDNS.2005.257 [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. [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.  doi: 10.1002/(ISSN)1098-2760 [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.  doi: 10.1061/(ASCE)0733-9364(2006)132:12(1267) [19] T. Kohonen. Self-organizing Maps, Berlin, Germany: Springer-Verlag, 1995. [20] B. Angéniol, G. de La Croix Vaubois, J. Y. Le Texier. Self-organizing feature maps and the travelling salesman problem. Neural Networks, vol. 1, no. 4, pp. 289-293, 1988.  doi: 10.1016/0893-6080(88)90002-0 [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.  doi: 10.1109/TSMCB.2002.804367 [22] J. H. Chen, L. R. Yang, M. C. Su. Comparison of SOM-based 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.  doi: 10.1016/j.autcon.2009.03.008 [23] M. C. Su, T. K. Liu, H. T. Chang. Improving the self-organizing feature map algorithm using an efficient initialization scheme. Tamkang Journal of Science and Engineering, vol. 5, no. 1, pp. 35-48, 2002. [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. [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. [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. [27] W. Wu, A. Khan. SOMO-m optimization algorithm with multiple winners. Discrete Dynamics in Nature and Society, Article ID 969104, 2012. [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. [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.  doi: 10.1007/s12559-014-9315-7
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## Convergence Analysis of a New MaxMin-SOMO Algorithm

• ###### Corresponding author:Atlas Khan received the B. Sc. and M. Sc. degrees in mathematics from Gomal University DI Khan Pakistan, in 2005 and 2007, respectively, and M. Phil. degree in mathematics from Quaid-i-Azam University, Pakistan in 2010. He obtained the Ph. D. degree from Department of Applied Mathematics, Dalian University of Technology, China in 2013. Since August 2013, he is doing post-docotor in bioinformatics with Department of Computing and Mathematics, University of Sao Paulo, Brazil. He has published a number of papers in international journals and conferences. His research interests include bioinformatios, computational biology, neural networks and coding theory. E-mail: atlas.khan@ficlrp.usp.br (Corresponding author) ORCID iD: 0000-0002-6651-2725

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.

Atlas Khan, Yan-Peng Qu and Zheng-Xue Li. Convergence Analysis of a New MaxMin-SOMO Algorithm. International Journal of Automation and Computing, vol. 16, no. 4, pp. 534-542, 2019. doi: 10.1007/s11633-016-0996-0
 Citation: Atlas Khan, Yan-Peng Qu and Zheng-Xue Li. Convergence Analysis of a New MaxMin-SOMO Algorithm. International Journal of Automation and Computing, vol. 16, no. 4, pp. 534-542, 2019.
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