Advertisement

Journal of Optimization Theory and Applications

, Volume 170, Issue 2, pp 616–628 | Cite as

A Non-homogeneous Firefly Algorithm and Its Convergence Analysis

  • Ngaam J. Cheung
  • Xue-Ming Ding
  • Hong-Bin Shen
Article

Abstract

The firefly algorithm is a swarm-based search algorithm, in which fireflies cooperate with each other to look for the optimal solution to a given optimization problem in a provided search space. Even though the firefly algorithm has exhibited good performance, researchers have not adequately explained how it works and what effects of its control coefficients in terms of theory. Further, classical variants of the algorithm have unexpected parameter settings and limited update laws, notably the homogeneous rule is necessary to be improved in order to efficiently search the whole space as accurate as possible for the optimal solutions to various problems. This study analyzes the trajectory of a single firefly in both the traditional algorithm and an adaptive variant based on our previous study. Accordingly, these analyses lead to general models of the algorithm ? including a set of boundary conditions for selection of the control parameters, which can guarantee the convergence tendencies of all individuals. The numerical experiments on twelve well-suited benchmark functions show the implementation of the proposed adaptive algorithm, which is derived from the analyses, can enhance the search ability of each individual in looking for the optima.

Keywords

Convergence analysis Parameter selection Adaptive firefly algorithm NAdaFa 

Mathematics Subject Classification

93B40 

Notes

Acknowledgments

We thanks Professor Franco Giannessi, Professor David Hull and other anonymous reviewers for many constructive comments and suggestions. This work was supported by the China Scholarship Council and the Hujiang Foundation of China (C14002).

Compliance with ethical standards

Conflicts of interest

The authors declare no conflict of interest.

Supplementary material

10957_2016_875_MOESM1_ESM.pdf (136 kb)
Supplementary material 1 (pdf 136 KB)

References

  1. 1.
    Dye, C.Y., Hsieh, T.P.: A particle swarm optimization for solving lot-sizing problem with fluctuating demand and preservation technology cost under trade credit. J. Global Optim. 55(3), 655–679 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cheung, N.J., Xu, Z.K., Ding, X.M., Shen, H.B.: Modeling nonlinear dynamic biological systems with human-readable fuzzy rules optimized by convergent heterogeneous particle swarm. Eur. J. Oper. Res. 247(2), 349–358 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cheung, N.J., Ding, X.M., Shen, H.B.: Protein folds recognized by an intelligent predictor based-on evolutionary and structural information. J. Comput. Chem. 37(4), 426–436 (2016)CrossRefGoogle Scholar
  4. 4.
    Yang, X.S.: Nature-Inspired Metaheuristic Algorithms. Luniver Press, Bristol (2008)Google Scholar
  5. 5.
    Arora, S., Singh, S.: The firefly optimization algorithm: convergence analysis and parameter selection. Int. J. Comput. Appl. 69(3), 48–52 (2013)Google Scholar
  6. 6.
    Farahani, S.M., Abshouri, A., Nasiri, B., Meybodi, M.: A gaussian firefly algorithm. Int. J. Mach. Learn. Comput. 1(5), 21–32 (2011)Google Scholar
  7. 7.
    dos Santos Coelho, L., Mariani, V.C.: Improved firefly algorithm approach applied to chiller loading for energy conservation. Energy Build. 59, 273–278 (2013)CrossRefGoogle Scholar
  8. 8.
    Cheung, N.J., Ding, X.M., Shen, H.B.: Adaptive firefly algorithm: parameter analysis and its application. PLoS One 9(11), e112634 (2014)CrossRefGoogle Scholar
  9. 9.
    Yang, X.S.: Firefly algorithm, Levy flights and global optimization. In: Bramer, M., Ellis, R., Petridis, M. (eds.) Research and Development in Intelligent Systems XXVI, pp. 209–218. Springer, London (2010)Google Scholar
  10. 10.
    Fister, I., Jr, I.F., Yang, X.S., Brest, J.: A comprehensive review of firefly algorithms. Swarm Evolut. Comput. 13, 34–46 (2013)CrossRefGoogle Scholar
  11. 11.
    Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F.: Multimodal size, shape, and topology optimisation of truss structures using the firefly algorithm. Adv. Eng. Softw. 56, 23–37 (2013)CrossRefGoogle Scholar
  12. 12.
    Gandomi, A., Yang, X.S., Talatahari, S., Alavi, A.: Firefly algorithm with chaos. Commun. Nonlinear Sci. Numer. Simul. 18(1), 89–98 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Deng, J.L.: Introduction to grey system theory. J. Grey Syst. 1(1), 1–24 (1989)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lewis, A.L.S., Lucchetti, R.E.: Nonsmooth duality, sandwich, and squeeze theorems. SIAM J. Control Optim. 38(2), 613–626 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Leu, M.S., Yeh, M.F.: Grey particle swarm optimization. Appl. Soft Comput. 12(9), 2985–2996 (2012)CrossRefGoogle Scholar
  16. 16.
    Zhan, Z.H., Zhang, J., Li, Y., Chung, H.H.: Adaptive particle swarm optimization. IEEE Trans. Syst. Man Cybern. Part B Cybern. 39(6), 1362–1381 (2009)CrossRefGoogle Scholar
  17. 17.
    Clerc, M.: Standard Particle Swarm Optimisation. http://clerc.maurice.free.fr/pso/SPSO_descriptions (2012)
  18. 18.
    Fateen, S.E., Bonilla-Petriciolet, A.: Intelligent firefly algorithm for global optimization. In: Yang, X.S. (ed.) Cuckoo Search and Firefly Algorithm, Studies in Computational Intelligence, vol. 516, pp. 315–330. Springer, Berlin (2014)CrossRefGoogle Scholar
  19. 19.
    Yang, X.S.: Firefly algorithms for multimodal optimization. In: Stochastic Algorithms: Foundations and Applications, SAGA 2009, vol. 5792, pp. 169–178 (2009)Google Scholar
  20. 20.
    Derrac, J., García, S., Molina, D., Herrera, F.: A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evolut. Comput. 1(1), 3–18 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Ngaam J. Cheung
    • 1
    • 2
  • Xue-Ming Ding
    • 3
  • Hong-Bin Shen
    • 1
    • 4
  1. 1.Institute of Image Processing and Pattern RecognitionShanghai Jiao Tong UniversityShanghaiChina
  2. 2.James Franck InstituteThe University of ChicagoChicagoUSA
  3. 3.School of Optical-Electrical and Computer EngineeringUniversity of Shanghai for Science and TechnologyShanghaiChina
  4. 4.Key Laboratory of System Control and Information ProcessingMinistry of Education of ChinaShanghaiChina

Personalised recommendations