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(n3/2). 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.


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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

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