Markov Chain Analysis of Evolution Strategies on a Linear Constraint Optimization Problem

Alexandre Chotard 1, 2, * Anne Auger 1 Nikolaus Hansen 1
* Corresponding author
1 TAO - Machine Learning and Optimisation
CNRS - Centre National de la Recherche Scientifique : UMR8623, Inria Saclay - Ile de France, UP11 - Université Paris-Sud - Paris 11, LRI - Laboratoire de Recherche en Informatique
Abstract : This paper analyses a $(1,\lambda)$-Evolution Strategy, a randomised comparison-based adaptive search algorithm, on a simple constraint optimisation problem. The algorithm uses resampling to handle the constraint and optimizes a linear function with a linear constraint. Two cases are investigated: first the case where the step-size is constant, and second the case where the step-size is adapted using path length control. We exhibit for each case a Markov chain whose stability analysis would allow us to deduce the divergence of the algorithm depending on its internal parameters. We show divergence at a constant rate when the step-size is constant. We sketch that with step-size adaptation geometric divergence takes place. Our results complement previous studies where stability was assumed.
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Submitted on : Friday, December 5, 2014 - 2:44:17 PM
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  • HAL Id : hal-00977379, version 2
  • ARXIV : 1404.3023



Alexandre Chotard, Anne Auger, Nikolaus Hansen. Markov Chain Analysis of Evolution Strategies on a Linear Constraint Optimization Problem. IEEE Congress on Evolutionary Computation,, Jul 2014, Beijing, China. ⟨hal-00977379v2⟩



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