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)

Abstract

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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