Advertisement

Metaheuristic Optimization: Algorithm Analysis and Open Problems

  • Xin-She Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6630)

Abstract

Metaheuristic algorithms are becoming an important part of modern optimization. A wide range of metaheuristic algorithms have emerged over the last two decades, and many metaheuristics such as particle swarm optimization are becoming increasingly popular. Despite their popularity, mathematical analysis of these algorithms lacks behind. Convergence analysis still remains unsolved for the majority of metaheuristic algorithms, while efficiency analysis is equally challenging. In this paper, we intend to provide an overview of convergence and efficiency studies of metaheuristics, and try to provide a framework for analyzing metaheuristics in terms of convergence and efficiency. This can form a basis for analyzing other algorithms. We also outline some open questions as further research topics.

Keywords

Markov Chain Particle Swarm Optimization Random Walk Simulated Annealing Convergence Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Auger, A., Doerr, B.: Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific, Singapore (2010)zbMATHGoogle Scholar
  2. 2.
    Auger, A., Teytaud, O.: Continuous lunches are free plus the design of optimal optimization algorithms. Algorithmica 57(1), 121–146 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bell, W.J.: Searching Behaviour: The Behavioural Ecology of Finding Resources. Chapman & Hall, London (1991)Google Scholar
  4. 4.
    Bertsimas, D., Tsitsiklis, J.: Simulated annealing. Stat. Science 8, 10–15 (1993)CrossRefzbMATHGoogle Scholar
  5. 5.
    Blum, C., Roli, A.: Metaheuristics in combinatorial optimization: Overview and conceptural comparison. ACM Comput. Surv. 35, 268–308 (2003)CrossRefGoogle Scholar
  6. 6.
    Chatterjee, A., Siarry, P.: Nonlinear inertia variation for dynamic adapation in particle swarm optimization. Comp. Oper. Research 33, 859–871 (2006)CrossRefzbMATHGoogle Scholar
  7. 7.
    Clerc, M., Kennedy, J.: The particle swarm - explosion, stability, and convergence in a multidimensional complex space. IEEE Trans. Evolutionary Computation 6, 58–73 (2002)CrossRefGoogle Scholar
  8. 8.
    Dorigo, M.: Optimization, Learning and Natural Algorithms. PhD thesis. Politecnico di Milano, Italy (1992)Google Scholar
  9. 9.
    Engelbrecht, A.P.: Fundamentals of Computational Swarm Intelligence. Wiley, Chichester (2005)Google Scholar
  10. 10.
    Floudas, C.A., Pardalos, P.M.: A Collection of Test Problems for Constrained Global Optimization Algorithms. LNCS, vol. 455. Springer, Heidelberg (1990)zbMATHGoogle Scholar
  11. 11.
    Floudas, C.A., Pardolos, P.M.: Encyclopedia of Optimization, 2nd edn. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Gamerman, D.: Markov Chain Monte Carlo. Chapman & Hall/CRC (1997)Google Scholar
  13. 13.
    Geyer, C.J.: Practical Markov Chain Monte Carlo. Statistical Science 7, 473–511 (1992)CrossRefGoogle Scholar
  14. 14.
    Ghate, A., Smith, R.: Adaptive search with stochastic acceptance probabilities for global optimization. Operations Research Lett. 36, 285–290 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Granville, V., Krivanek, M., Rasson, J.P.: Simulated annealing: A proof of convergence. IEEE Trans. Pattern Anal. Mach. Intel. 16, 652–656 (1994)CrossRefGoogle Scholar
  16. 16.
    Gutowski, M.: Lévy flights as an underlying mechanism for global optimization algorithms. ArXiv Mathematical Physics e-Prints (June 2001)Google Scholar
  17. 17.
    Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proc. of IEEE International Conference on Neural Networks, Piscataway, NJ, pp. 1942–1948 (1995)Google Scholar
  18. 18.
    Kennedy, J., Eberhart, R.C.: Swarm intelligence. Academic Press, London (2001)Google Scholar
  19. 19.
    Holland, J.: Adaptation in Natural and Artificial systems. University of Michigan Press, Ann Anbor (1975)Google Scholar
  20. 20.
    Kirkpatrick, S., Gellat, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 670–680 (1983)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Matthews, C., Wright, L., Yang, X.S.: Sensitivity Analysis, Optimization, and Sampling Methodds Applied to Continous Models. National Physical Laboratory Report, UK (2009)Google Scholar
  22. 22.
    Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization: Algorithms and Their Computational Complexity. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  23. 23.
    Nolan, J.P.: Stable distributions: models for heavy-tailed data. American University (2009)Google Scholar
  24. 24.
    Pavlyukevich, I.: Lévy flights, non-local search and simulated annealing. J. Computational Physics 226, 1830–1844 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rebennack, S., Arulselvan, A., Elefteriadou, L., Pardalos, P.M.: Complexity analysis for maximum flow problems with arc reversals. J. Combin. Optimization 19(2), 200–216 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Reynolds, A.M., Rhodes, C.J.: The Lévy fligth paradigm: random search patterns and mechanisms. Ecology 90, 877–887 (2009)CrossRefGoogle Scholar
  27. 27.
    Steinhöfel, K., Albrecht, A.A., Wong, C.-K.: Convergence analysis of simulated annealing-based algorithms solving flow shop scheduling problems. In: Bongiovanni, G., Petreschi, R., Gambosi, G. (eds.) CIAC 2000. LNCS, vol. 1767, pp. 277–290. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  28. 28.
    Sobol, I.M.: A Primer for the Monte Carlo Method. CRC Press, Boca Raton (1994)zbMATHGoogle Scholar
  29. 29.
    Talbi, E.G.: Metaheuristics: From Design to Implementation. Wiley, Chichester (2009)CrossRefzbMATHGoogle Scholar
  30. 30.
    Viswanathan, G.M., Buldyrev, S.V., Havlin, S., da Luz, M.G.E., Raposo, E.P., Stanley, H.E.: Lévy flight search patterns of wandering albatrosses. Nature 381, 413–415 (1996)CrossRefGoogle Scholar
  31. 31.
    Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimisation. IEEE Transaction on Evolutionary Computation 1, 67–82 (1997)CrossRefGoogle Scholar
  32. 32.
    Yang, X.S.: Nature-Inspired Metaheuristic Algorithms. Luniver Press (2008)Google Scholar
  33. 33.
    Yang, X.S.: Engineering Optimization: An Introduction with Metaheuristic Applications. John Wiley & Sons, ChichesterGoogle Scholar
  34. 34.
    Yang, X.S.: Firefly algorithm, stochastic test functions and design optimisation. Int. J. Bio-Inspired Computation 2, 78–84 (2010)CrossRefGoogle Scholar
  35. 35.
    Yang, X.S., Deb, S.: Engineering optimization by cuckoo search. Int. J. Math. Modelling & Num. Optimization 1, 330–343 (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Xin-She Yang
    • 1
  1. 1.Mathematics and Scientific ComputingNational Physical LaboratoryTeddingtonUK

Personalised recommendations