Optimizing Monotone Functions Can Be Difficult

  • Benjamin Doerr
  • Thomas Jansen
  • Dirk Sudholt
  • Carola Winzen
  • Christine Zarges
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6238)


Extending previous analyses on function classes like linear functions, we analyze how the simple (1+1) evolutionary algorithm optimizes pseudo-Boolean functions that are strictly monotone. Contrary to what one would expect, not all of these functions are easy to optimize. The choice of the constant c in the mutation probability p(n) = c/n can make a decisive difference.

We show that if c < 1, then the (1+1) EA finds the optimum of every such function in Θ(n logn) iterations. For c = 1, we can still prove an upper bound of O(n 3/2). However, for c > 33, we present a strictly monotone function such that the (1+1) EA with overwhelming probability does not find the optimum within 2Ω(n) iterations. This is the first time that we observe that a constant factor change of the mutation probability changes the run-time by more than constant factors.


Evolutionary Algorithm Monotone Function Mutation Probability Search Point Unimodal Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mühlenbein, H.: How genetic algorithms really work. Mutation and hillclimbing. In: Proceedings of PPSN II, pp. 15–25. North-Holland, Amsterdam (1992)Google Scholar
  2. 2.
    Droste, S., Jansen, T., Wegener, I.: On the optimization of unimodal functions with the (1+1) evolutionary algorithm. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 13–22. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. 3.
    Horn, J., Goldberg, D., Deb, K.: Long path problems. In: Davidor, Y., Männer, R., Schwefel, H.-P. (eds.) PPSN 1994. LNCS, vol. 866, pp. 149–158. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  4. 4.
    Igel, C., Toussaint, M.: A no-free-lunch theorem for non-uniform distributions of target functions. J. of Mathematical Modelling and Algorithms 3, 313–322 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Oliveto, P., He, J., Yao, X.: Time complexity of evolutionary algorithms for combinatorial optimization: A decade of results. International Journal of Automation and Computing 4, 281–293Google Scholar
  6. 6.
    Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theoretical Computer Science 276, 51–81 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    He, J., Yao, X.: Drift analysis and average time complexity of evolutionary algorithms. Artificial Intelligence 127, 57–85 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    He, J., Yao, X.: Erratum to [7]. Artificial Intelligence 140, 245–248 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Jägersküpper, J.: A blend of Markov-chain and drift analysis. In: Rudolph, G., Jansen, T., Lucas, S., Poloni, C., Beume, N. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 41–51. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Doerr, B., Fouz, M., Witt, C.: Quasirandom evolutionary algorithms. In: Proceedings of GECCO 2010, pp. 1457–1464. ACM, New York (2010)Google Scholar
  11. 11.
    Doerr, B., Johannsen, D., Winzen, C.: Drift analysis and linear functions revisited. In: Proceedings of CEC 2010, pp. 1967–1974. IEEE, Los Alamitos (2010)Google Scholar
  12. 12.
    Doerr, B., Goldberg, L.: Adaptive drift analysis. In: Schaefer, R., et al. (eds.) PPSN XI, Part I. LNCS, vol. 6238, pp. 32–41. Springer, Heidelberg (2010)Google Scholar
  13. 13.
    Jansen, T., Wegener, I.: On the choice of the mutation probability for the (1+1) EA. In: Deb, K., Rudolph, G., Lutton, E., Merelo, J.J., Schoenauer, M., Schwefel, H.-P., Yao, X. (eds.) PPSN 2000. LNCS, vol. 1917, pp. 89–98. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  14. 14.
    Bäck, T., Fogel, D., Michalewicz, Z. (eds.): Handbook of Evolutionary Computation. Oxford University Press, Oxford (1997)zbMATHGoogle Scholar
  15. 15.
    Ochoa, G.: Setting the mutation rate: Scope and limitations of the 1/L heuristic. In: Proceedings of GECCO 2002, pp. 495–502. Morgan Kaufmann, San Francisco (2002)Google Scholar
  16. 16.
    Cervantes, J., Stephens, C.R.: Limitations of existing mutation rate heuristics and how a rank GA overcomes them. IEEE Trans. on Evol. Comp. 13, 369–397 (2009)CrossRefGoogle Scholar
  17. 17.
    Jansen, T.: On the brittleness of evolutionary algorithms. In: Stephens, C.R., Toussaint, M., Whitley, L.D., Stadler, P.F. (eds.) FOGA 2007. LNCS, vol. 4436, pp. 54–69. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Alon, N., Spencer, J.H.: The Probabilistic Method, 2nd edn. Wiley, Chichester (2000)zbMATHCrossRefGoogle Scholar
  19. 19.
    Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  20. 20.
    Oliveto, P.S., Witt, C.: Simplified drift analysis for proving lower bounds in evolutionary computation. Algorithmica (2010),

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Benjamin Doerr
    • 1
  • Thomas Jansen
    • 2
  • Dirk Sudholt
    • 3
  • Carola Winzen
    • 1
  • Christine Zarges
    • 4
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.University College CorkCorkIreland
  3. 3.International Computer Science InstituteBerkeleyUSA
  4. 4.Technische Universität DortmundDortmundGermany

Personalised recommendations