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Mathematical Analysis of Nature-Inspired Algorithms

  • Xin-She YangEmail author
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 744)

Abstract

Nature-inspired algorithms are a class of effective tools for solving optimization problems and these algorithms have good properties such as simplicity, flexibility and high efficiency. Despite their popularity in practice, a mathematical framework is yet to be developed to analyze these algorithms theoretically. This work intends to analyze nature-inspired algorithms both qualitatively and quantitatively. We briefly outline the links between self-organization and algorithms, and then analyze algorithms using Markov chain theory, dynamic system and other methods. This can serve as a basis for building a multidisciplinary framework for algorithm analysis.

Keywords

Algorithm Bat algorithm Cuckoo search Differential evolution Firefly algorithm Flower pollination algorithm Particle swarm optimization Metaheuristics Nature-inspired computation Optimization Self-organization Swarm intelligence 

References

  1. 1.
    Abdelaziz, A.Y., Ali, E.S., Abd Elazim, S.M.: Combined economic and emission dispatch solution using flower pollination algorithm. Int. J. Electr. Power Energy Syst. 80(2), 264–274 (2016)CrossRefGoogle Scholar
  2. 2.
    Alam, D.F., Yousri, D.A., Eteiba, M.B.: Flower pollination algorithm based solar PV parameter estimation. Energy Convers. Manage. 101(2), 410–422 (2015)CrossRefGoogle Scholar
  3. 3.
    Ashby, W.A.: Principles of the self-organizing system. In: Von Foerster, H., Zopf Jr., G.W. (eds.) Principles of Self-Organization: Transactions of the University of Illinois Symposium, pp. 255–278. Pergamon Press, London, UK (1962)Google Scholar
  4. 4.
    Bekdas, G., Nigdeli, S.M., Yang, X.S.: Sizing optimization of truss structures using flower pollination algorithm. Appl. Soft Comput. 37(1), 322–331 (2015)CrossRefGoogle Scholar
  5. 5.
    Blum, C., Roli, A.: Metaheuristics in combinatorial optimization: overview and conceptural comparision. ACM Comput. Survey 35(2), 268–308 (2003)CrossRefGoogle Scholar
  6. 6.
    Bonabeau, E., Dorigo, M., Theraulaz, G.: Swarm Intelligence: From Natural to Artificial Systems. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  7. 7.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Univeristy Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Dhivya, M., Sundarambal, M.: Cuckoo search for data gathering in wireless sensor networks. Int. J. Mobile Commun. 9(4), 642–656 (2011)CrossRefGoogle Scholar
  10. 10.
    Durgun, I., Yildiz, A.R.: Structural design optimization of vehicle components using cuckoo search algorithm. Mater. Test. 3(3), 185–188 (2012)CrossRefGoogle Scholar
  11. 11.
    Eiben, A.E., Smit, S.K.: Parameter tuning for configuring and analyzing evolutionary aglorithms. Swarm Evol. Comput. 1(1), 19–31 (2011)CrossRefGoogle Scholar
  12. 12.
    Fishman, G.S.: Monte Carlo: Concepts, Algorithms and Applications. Springer, New York (1995)zbMATHGoogle Scholar
  13. 13.
    Fisher, L.: The Perfect Swarm: The Science of Complexity in Everyday Life. Basic Books (2009)Google Scholar
  14. 14.
    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
  15. 15.
    Ghate, A., Smith, R.: Adaptive search with stochastic acceptance probabilities for global optimization. Oper. Res. Lett. 36(3), 285–290 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimisation and Machine Learning. Reading, Mass, Addison Wesley (1989)zbMATHGoogle Scholar
  17. 17.
    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. Procedia Comput. Sci. 108(1), 1354–1363 (2017)CrossRefGoogle Scholar
  18. 18.
    Holland, J.: Adaptation in Natural and Arficial Systems. University of Michigan Press, Ann Arbor, USA (1975)Google Scholar
  19. 19.
    Keller, E.F.: Organisms, machines, and thunderstorms: a history of self-organization, part two. Complexity, emergence, and stable attractors. Hist. Stud. Nat. Sci. 39(1), 1–31 (2009)CrossRefGoogle Scholar
  20. 20.
    Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks, pp. 1942–1948. Piscataway, NJ (1995)Google Scholar
  21. 21.
    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
  22. 22.
    Pavlyukevich, I.: Lévy flights, non-local search and simulated annealing. J. Comput. Phys. 226(2), 1830–1844 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Reynolds, A.M., Rhodes, C.J.: The Lévy fligth paradigm: random search patterns and mechanisms. Ecology 90(4), 877–887 (2009)CrossRefGoogle Scholar
  24. 24.
    Rodrigues, D., Silva, G.F.A., Papa, J.P., Marana, A.N., Yang, X.S.: EEG-based person identificaiton through binary flower pollination algorithm. Expert Syst. Appl. 62(1), 81–90 (2016)CrossRefGoogle Scholar
  25. 25.
    Storn, R., Price, K.: Differential evolution: a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11(4), 341–59 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Süli, E., Mayer, D.: An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  27. 27.
    Surowiecki, J.: The Wisdom of Crowds. Anchor Books (2004)Google Scholar
  28. 28.
    Suzuki, J.A.: A Markov chain analysis on simple genetic algorithms. IEEE Trans. Sys. Man Cybern. 25(4), 655–9 (1995)CrossRefGoogle Scholar
  29. 29.
    Villalobos-Arias, M., Colleo, C.A.C., Hernández-Lerma, O.: Asypmotic convergence of metaheuristics for multiobjective optimization problems. Soft Comput. 10(11), 1001–5 (2005)CrossRefGoogle Scholar
  30. 30.
    Wolpert, D.H., Macready, W.G.: No free lunch theorem for optimization. IEEE Trans. Evol. Comput. 1(1), 67–82 (1997)CrossRefGoogle Scholar
  31. 31.
    Wolpert, D.H., Macready, W.G.: Coevolutionary free lunches. IEEE Trans. Evol. Comput. 9(6), 721–735 (2005)CrossRefGoogle Scholar
  32. 32.
    Yang, X.S.: Firefly algorithm, stochastic test functions and design optimisation. Int. J. Bio-Inspired Comput. 2(2), 78–84 (2010)CrossRefGoogle Scholar
  33. 33.
    Yang, X.S.: Engineering Optimization: An Introduction with Metaheuristic Applications. Wiley, Hoboken, NJ (2010)CrossRefGoogle Scholar
  34. 34.
    Yang, X.S.: A new metaheuristic bat-inspired algorithm. In: Nature-Inspired Cooperative Strategies for Optimization (NICSO 2010), pp. 65–74, SCI vol. 284. Springer (2010)Google Scholar
  35. 35.
    Yang, X.S.: Bat algorithm for multi-objective optimisation. Int. J. Bio-Inspired Comput. 3(5), 267–274 (2011)CrossRefGoogle Scholar
  36. 36.
    Yang, X.S.: Cuckoo Search and Firefly Algorithm: Theory and Applications. Studies in Computational Intelligence, vol. 516. Springer (2014)Google Scholar
  37. 37.
    Yang, X.S.: Nature-Inspired Optimization Algorithms. Elsevier Insight, London (2014)zbMATHGoogle Scholar
  38. 38.
    Yang, X.S., Chien, S.F., Ting, T.O.: Bio-Inspired Computation in Telecommunications. Morgan Kaufmann, Waltham (2015)Google Scholar
  39. 39.
    Yang, X.S., Deb, S.: Cuckoo search via Lévy flights. In: Proceedings of World Congress on Nature & Biologically Inspired Computing (NaBic 2009), Coimbatore, India, pp. 210–214. IEEE Publications, USA (2009)Google Scholar
  40. 40.
    Yang, X.S., Deb, S.: Engineering optimization by cuckoo search. Int. J. Math. Model. Num. Optim. 1(4), 330–343 (2010)zbMATHGoogle Scholar
  41. 41.
    Yang, X.S., Deb, S.: Multiobjective cuckoo search for design optimization. Comput. Oper. Res. 40(6), 1616–1624 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Yang, X.S., Deb, S.: Cuckoo search: recent advances and applications. Neural Comput. Appl. 24(1), 169–174 (2014)CrossRefGoogle Scholar
  43. 43.
    Yang, X.S., Deb, S., Loomes, M., Karamanoglu, M.: A framework for self-tuning optimization algorithm. Neural Comput. Appl. 23(7–8), 2051–2057 (2013)CrossRefGoogle Scholar
  44. 44.
    Yang, X.S., Karamanoglu, M., He, X.S.: Flower pollination algorithm: a novel approach for multiobjective optimization. Eng. Optim. 46(9), 1222–1237 (2014)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Yang, X.S., Papa, J.P.: Bio-Inspired Computation and Applications in Image Processing. Academic Press, London (2016)Google Scholar
  46. 46.
    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
  47. 47.
    Zaharie, D.: Influence of crossover on the behaviour of the differential evolution algorithm. Appl. Soft Comput. 9(3), 1126–38 (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.School of Science and TechnologyMiddlesex UniversityLondonUK

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