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


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.


Convergence analysis Parameter selection Adaptive firefly algorithm NAdaFa 

Mathematics Subject Classification




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)


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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

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