Global Convergence Analysis of Cuckoo Search Using Markov Theory

  • Xing-Shi He
  • Fan Wang
  • Yan Wang
  • Xin-She YangEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 744)


The cuckoo search (CS) algorithm is a powerful metaheuristic algorithm for solving nonlinear global optimization problems. In this book chapter, we prove the global convergence of this algorithm using a Markov chain framework. By analyzing the state transition process of a population of cuckoos and the homogeneity of the constructed Markov chains, we can show that the constructed stochastic sequences can converge to the optimal state set. We also show that the algorithm structure of cuckoo search satisfies two convergence conditions and thus its global convergence is guaranteed. We then use numerical experiments to demonstrate that cuckoo search can indeed achieve global optimality efficiently.


Cuckoo search Convergence rate Global convergence Markov chain theory Optimization Swarm intelligence 



The authors would like to thank the financial support by Shaanxi Provincial Education Grant (12JK0744) and Shaanxi Provincial Soft Science Foundation (2012KRM58).


  1. 1.
    Ackley, D.H.: A Connectionist Machine For Genetic Hillclimbing. Kluwer Academic Publishers (1987)Google Scholar
  2. 2.
    Bhargava, V., Fateen, S.E.K., Bonilla-Petriciolet, A.: Cuckoo search: a new nature-inspired optimization method for phase equilibrium calculations. Fluid Phase. Equilib. 337, 191–200 (2013)CrossRefGoogle Scholar
  3. 3.
    Chandrasekaran, K., Simon, S.P.: Multi-objective scheduling problem: hybrid appraoch using fuzzy assisted cuckoo search algorithm. Swarm Evol. Comput. 5(1), 1–16 (2012)CrossRefGoogle Scholar
  4. 4.
    Clerc, M., Kennedy, J.: The particle swarm–explosion stability and convergence in a multidimensional complex space. IEEE Trans. Evol. Comput. 6(1), 58–73 (2002)Google Scholar
  5. 5.
    Dhivya, M., Sundarambal, M., Anand, L.N.: Energy efficient computation of data fusion in wireless sensor networks using cuckoo based particle approach (CBPA). Int. J. Commun. Netw. Syst. Sci. 4(4), 249–255 (2011)Google Scholar
  6. 6.
    Dhivya, M., Sundarambal, M.: Cuckoo search for data gathering in wireless sensor networks. Int. J. Mob. Commun. 9(4), 642–656 (2011)CrossRefGoogle Scholar
  7. 7.
    Durgun, I., Yildiz, A.R.: Structural design optimization of vehicle components using cuckoo search algorithm. Mater. Test. 3(3), 185–188 (2012)CrossRefGoogle Scholar
  8. 8.
    Gandomi, A.H., Yang, X.S., Alavi, A.H.: Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng. Comput. 29(1), 17–35 (2013)CrossRefGoogle Scholar
  9. 9.
    He, X.S., Yang, X.S., Karamanoglu, M., Zhao, Y.X.: Global convergence analysis of the flower pollination algorithm: a discrete-time Markov chain approach. Proced. Comput. Sci. 108(1), 1354–1363 (2017)CrossRefGoogle Scholar
  10. 10.
    Jiang, M., Luo, Y.P., Yang, S.Y.: Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm. Inf. Process. Lett. 102(1), 8–16 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks. Piscataway, NJ, pp. 1942–1948 (1995)Google Scholar
  12. 12.
    Koziel, S., Yang, X.S.: Computational Optimization Methods And Algorithms. Springer, Berlin (2011)Google Scholar
  13. 13.
    Moravej, Z., Akhlaghi, A.: A novel approach based on cuckoo search for DG allocation in distribution network. Electr. Power Energy Syst. 44(1), 672–679 (2013)CrossRefGoogle Scholar
  14. 14.
    Noghrehabadi, A., Ghalambaz, M., Vosough, A.: A hybrid power series–cuckoo search optimization algorithm to electrostatic deflection of micro fixed-fixed actuators. Int. J. Multidiscip. Sci. Eng. 2(4), 22–26 (2011)Google Scholar
  15. 15.
    Pavlyukevich, I.: Lévy flights, non-local search and simulated annealing. J. Comput. Phys. 226(2), 1830–1844 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ren, Z.H., Wang, J., Gao, Y.L.: The global convergence analysis of particle swarm optimization algorithm based on Markov chain. Control Theory Appl. 28(4), 462–466 (2011). (in Chinese)Google Scholar
  17. 17.
    Taweewat, P., Wutiwiwatchai, C.: Musical pitch estimation using a supervised single hidden layer feed-forward neural network. Expert Syst. Appl. 40(2), 575–589 (2013)CrossRefGoogle Scholar
  18. 18.
    Valian, E., Mohanna, S., Tavakoli, S.: Improved cuckoo search algorithm for feedforward neural network training. Int. J. Artif. Intell. Appl. 2(3), 36–43 (2011)Google Scholar
  19. 19.
    Valian, E., Tavakoli, S., Mohanna, S., Haghi, A.: Improved cuckoo search for reliability optimization problems. Comput. Ind. Eng. 64(1), 459–468 (2013)CrossRefGoogle Scholar
  20. 20.
    Vazquez, R.A.: Training spiking neural models using cuckoo search algorithm. In: IEEE Congress on Eovlutionary Computation (CEC’11), pp. 679–686 (2011)Google Scholar
  21. 21.
    Walton, S., Hassan, O., Morgan, K., Brown, M.R.: Modified cuckoo search: a new gradient free optimization algorithm. Chaos Solitons Fractals 44(9), 710–718 (2011)CrossRefGoogle Scholar
  22. 22.
    Wang, F., He, X.S., Wang, Y., Yang, S.M.: Markov model and convergence analysis of cuckoo search algorithm. Comput. Eng. 38(11), 180–185 (2012) (in Chinese)Google Scholar
  23. 23.
    Yang, X.S., Deb, S.: Cuckoo search via Lévy flights. In: Proceeding of World Congress on Nature and Biologically Inspired Computing (NaBic), pp. 210–214. IEEE Publications, Coimbatore, India, USA (2009)Google Scholar
  24. 24.
    Yang, X.S., Deb, S.: Engineering optimization by cuckoo search. Int. J. Math. Model. Numer. Optim. 1(4), 330–343 (2010)zbMATHGoogle Scholar
  25. 25.
    Yang, X.S., Deb, S.: Cuckoo search: recent advances and applications. Neural Comput. Appl. 24(1), 169–174 (2014)CrossRefGoogle Scholar
  26. 26.
    Yang, X.S.: Engineering Optimization: An Introduction With Metaheuristic Applications. Wiley (2010)Google Scholar
  27. 27.
    Yang, X.S.: A new metaheuristic bat-inspired algorithm. In: Nature-Inspired Cooperative Strategies for Optimization (NICSO), vol. 284, pp. 65–74. SCI, Springer (2010)Google Scholar
  28. 28.
    Yang, X.S.: Bat algorithm for multi-objective optimisation. Int. J. Bio-Inspir. Comput. 3(5), 267–274 (2011)CrossRefGoogle Scholar
  29. 29.
    Yang, X.S.: CUCKOO search and firefly algorithm: theory and applications. In: Studies in Computational Intelligence, vol. 516 Springer (2014)Google Scholar
  30. 30.
    Yang, X.S.: Nature-Inspired Optimization Algorithms. Elsevier Insight, London (2014)zbMATHGoogle Scholar
  31. 31.
    Yang, X.S., Deb, S.: Multiobjective cuckoo search for design optimization. Comput. Oper. Res. 40(6), 1616–1624 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yildiz, A.R.: Cuckoo search algorithm for the selection of optimal machine parameters in milling operations. Int. J. Adv. Manuf. Technol. 64(1), 55–61 (2013)CrossRefGoogle Scholar
  33. 33.
    Zheng, H.Q., Zhou, Y.: A novel cuckoo search optimization algorithm based on Gauss distribution. J. Comput. Inf. Syst. 8(10), 4193–4200 (2012)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.College of ScienceXi’an Polytechnic UniversityXi’anPeople’s Republic of China
  2. 2.School of Science and TechnologyMiddlesex UniversityLondonUK

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