, Volume 59, Issue 3, pp 425–460 | Cite as

Log-Linear Convergence and Divergence of the Scale-Invariant (1+1)-ES in Noisy Environments

  • Mohamed Jebalia
  • Anne Auger
  • Nikolaus Hansen


Noise is present in many real-world continuous optimization problems. Stochastic search algorithms such as Evolution Strategies (ESs) have been proposed as effective search methods in such contexts. In this paper, we provide a mathematical analysis of the convergence of a (1+1)-ES on unimodal spherical objective functions in the presence of noise. We prove for a multiplicative noise model that for a positive expected value of the noisy objective function, convergence or divergence happens depending on the infimum of the support of the noise. Moreover, we investigate convergence rates and show that log-linear convergence is preserved in presence of noise. This result is a strong theoretical foundation of the robustness of ESs with respect to noise.


Numerical optimization Noisy optimization Stochastic optimization algorithms Evolution strategies Convergence Convergence rates Markov chains Borel-Cantelli lemma 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.TAO Team—INRIA Saclay-Île-de-FranceLRI—Paris-Sud UniversityOrsay CedexFrance

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