Example Landscapes to Support Analysis of Multimodal Optimisation

  • Thomas Jansen
  • Christine ZargesEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9921)


Theoretical analysis of all kinds of randomised search heuristics has been and keeps being supported and facilitated by the use of simple example functions. Such functions help us understand the working principles of complicated heuristics. If the function represents some properties of practical problem landscapes these results become practically relevant. While this has been very successful in the past for optimisation in unimodal landscapes there is a need for generally accepted useful simple example functions for situations where unimodal objective functions are insufficient: multimodal optimisation and investigation of diversity preserving mechanisms are examples. A family of example landscapes is defined that comes with a limited number of parameters that allow to control important features of the landscape while all being still simple in some sense. Different expressions of these landscapes are presented and fundamental properties are explored.


Search Space Fitness Landscape Search Point Peak Function Maximal Fitness 
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.


  1. 1.
    Branke, J.: Memory enhanced evolutionary algorithms for changing optimization problems. In: Proceedings of CEC, pp. 1875–1882. IEEE Press (1999)Google Scholar
  2. 2.
    Doerr, B., Hansen, N., Igel, C., Thiele, L.: Theory of evolutionary algorithms (Dagstuhl seminar 15211). Dagstuhl Rep. 5(5), 57–91 (2016)Google Scholar
  3. 3.
    Doerr, B., Winzen, C.: Ranking-based black-box complexity. Algorithmica 68(3), 571–609 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Droste, S., Jansen, T., Wegener, I.: A rigorous complexity analysis of the (1 + 1) evolutionary algorithm for linear functions with Boolean inputs. In: Proceedings of ICEC, pp. 499–504. IEEE Press (1998)Google Scholar
  5. 5.
    Fischer, S., Wegener, I.: The one-dimensional Ising model: mutation versus recombination. Theor. Comput. Sci. 344(2–3), 208–225 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Friedrich, T., Oliveto, P.S., Sudholt, D., Witt, C.: Analysis of diversity-preserving mechanisms for global exploration. Evol. Comput. 17(4), 455–476 (2009)CrossRefGoogle Scholar
  7. 7.
    He, J., Yao, X.: Drift analysis and average time complexity of evolutionary algorithms. Artif. Intell. 127, 57–85 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jansen, T.: Analyzing Evolutionary Algorithms. The Computer Science Perspective. Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Jansen, T., Zarges, C.: Performance analysis of randomised search heuristics operating with a fixed budget. Theor. Comput. Sci. 545, 39–58 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jong, K.D., Spears, W.M.: An analysis of the interacting roles of population size and crossover in genetic algorithms. In: Schwefel, H.-P., Manner, R. (eds.) Proceedings of PPSN, pp. 38–47. Springer, Heidelberg (1990)Google Scholar
  11. 11.
    Kennedy, J., Spears, W.M.: Matching algorithms to problems: an experimental test of the particle swarm and some genetic algorithms on the multimodal problem generator. In: Proceedings of WCCI, pp. 78–83. IEEE Press (1998)Google Scholar
  12. 12.
    Kötzing, T., Lissovoi, A., Witt, C.: (1+1) EA on generalized dynamic onemax. In: Proceedings of FOGA, pp. 40–51. ACM Press (2015)Google Scholar
  13. 13.
    Moraglio, A., Johnson, C.G.: Geometric generalization of the Nelder-Mead algorithm. In: Cowling, P., Merz, P. (eds.) EvoCOP 2010. LNCS, vol. 6022, pp. 190–201. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  15. 15.
    Mühlenbein, H.: How genetic algorithms really work: mutation and hillclimbing. In: Proceedings of PPSN, pp. 15–26. Elsevier (1992)Google Scholar
  16. 16.
    Oliveto, P.S., Sudholt, D., Zarges, C.: On the runtime analysis of fitness sharing mechanisms. In: Bartz-Beielstein, T., Branke, J., Filipič, B., Smith, J. (eds.) PPSN 2014. LNCS, vol. 8672, pp. 932–941. Springer, Heidelberg (2014)Google Scholar
  17. 17.
    Preuss, M.: Multimodal Optimization by Means of Evolutionary Algorithms. Springer, Heidelberg (2015)CrossRefzbMATHGoogle Scholar
  18. 18.
    Prügel-Bennett, A., Tayarani-Najaran, M.: Maximum satisfiability: anatomy of the fitness landscape for a hard combinatorial optimization problem. IEEE Trans. Evol. Comput. 16(3), 319–338 (2012)CrossRefGoogle Scholar
  19. 19.
    Shir, O.M.: Niching in evolutionary algorithms. In: Rozenberg, G., Bäck, T., Kok, J.N. (eds.) Handbook of Natural Computing, pp. 1035–1070. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  20. 20.
    Stadler, P.: Fitness landscapes. Biol. Evol. Stat. Phys. 585, 183–204 (2002)CrossRefGoogle Scholar
  21. 21.
    Sudholt, D.: Crossover is provably essential for the Ising model on trees. In: Proceedings of GECCO, pp. 1161–1167. ACM Press (2005)Google Scholar

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© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceAberystwyth UniversityAberystwythUK
  2. 2.School of Computer ScienceUniversity of BirminghamBirminghamUK

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