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Evolution Strategies

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Contemporary Evolution Strategies

Part of the book series: Natural Computing Series ((NCS))

Abstract

Prior to introducing the particular algorithms in Sect. 2.2, the more general foundations of evolution strategies are introduced in Sect. 2.1. To start with, the definition of an optimization task as used throughout this book is given in Sect. 2.1.1. Following [58], Sect. 2.1.2 presents a discussion of evolution strategy metaheuristics as a special case of evolutionary algorithms.

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Notes

  1. 1.

    This statement, however, is not meant to support the myth mentioned explicitly by Rudolph [58]: “Since early theoretical publications mainly analyzed simple ES without recombination, somehow the myth arose that ES put more emphasis on mutation than on recombination: This is a fatal misconception! Recombination has been an important ingredient of ES from the early beginning and this is still valid today.”

  2. 2.

    See Sect. 12.2.1 in [17] for the definition of a distance measure.

  3. 3.

    In the case of the (1+1)-ES the strategy parameters may be assigned to the algorithm itself instead of the individual, because only one set of strategy parameters is needed. This also holds for any strategy parameters which are not needed on the individual level (for example the covariance matrix of the CMA-ES).

  4. 4.

    Algorithm 3 in [58].

  5. 5.

    The normal distribution achieves maximum entropy among the distributions on the real domain. (See [64] for more details.)

  6. 6.

    A symmetric matrix \(\mathbf{A} \in {\mathbb{R}}^{n\times n}\) is positive definite iff \({\mathbf{x}}^{T}\mathbf{A}\mathbf{x} > 0\) for all \(\mathbf{x} \in {\mathbb{R}}^{n}\setminus \{\mathbf{0}\}\) [17].

  7. 7.

    For an orthogonal matrix A, \(\mathbf{A}{\mathbf{A}}^{T} ={ \mathbf{A}}^{T}\mathbf{A} = \mathbf{I}\) holds.

  8. 8.

    See Sect. 6.2.2.3 in [17].

  9. 9.

    The rectangular corridor model according to [8]: \(f_{1}(\mathbf{x}) = c_{0} + c_{1} \cdot x_{1}\) if the constraints \(g_{j}(\mathbf{x}): x_{j} \leq b\) with \(b \in {\mathbb{R}}^{+}\) for \(j \in \{2,\ldots,n\}\) are fulfilled, f 1(x) = ∞ otherwise.

  10. 10.

    The sphere model according to [8]: \(f_{2}(\mathbf{x}) = c_{0} + c_{1} \cdot \sum _{n}^{i=1}{(x_{i} - x_{i}^{{\ast}})}^{2}\).

  11. 11.

    The exact values are 0.184 and 0.2025 for the corridor and sphere models, respectively [8].

  12. 12.

    MSC is an abbreviation of mutative self-adaptation of covariances.

  13. 13.

    In the original publication it is called (1,λ)-ES with derandomized mutative step size.

  14. 14.

    This way, adapting the step size by a factor ξ requires at least 1∕β > 1 generations.

  15. 15.

    In the original paper, the algorithm is called (1,λ)-ES with derandomized mutative step size control using accumulated information.

  16. 16.

    The column vectors of the matrix B form a so-called generating set, which motivates the terminology generating set adaptation.

  17. 17.

    According to [32], the suggestion to use weighted recombination within the CMA-ES is due to Ingo Rechenberg, based on personal communication in 1998.

  18. 18.

    See [17]: \(\Gamma (n) =\int _{ 0}^{\infty }{x}^{n-1}\exp (-x)\,\mbox{ d}x\).

  19. 19.

    With the additional condition for A to consist of at least m = n 2 tuples.

  20. 20.

    Compare Sect. 19.2.1.2 in [17].

  21. 21.

    The term active is motivated by the fact that specifically the bad offspring individuals play an active role, although they would normally not be taken into account after selection has been applied.

  22. 22.

    This is explicitly avoided due to the occurrence of numerical instabilities for certain objective functions; see [40].

  23. 23.

    Population sizes μ and λ are not counted.

  24. 24.

    Instead of the term symmetrical, this is called mirrored in the context of this strategy.

  25. 25.

    In [26] the eNES are called exact natural evolution strategies.

  26. 26.

    The aforementioned techniques self-adaptation (see Sect. 2.2.1.2) or cumulative step size adaptation (see Sect. 2.2.2.1) are suitable.

  27. 27.

    See [70] for literature references on these topics as well as the Kriging modeling method.

  28. 28.

    For λ def the standard setting of a (μ W ,λ)-CMA-ES with \(\lambda _{\mathit{def }} = 4 + \lfloor 3\log n\rfloor \) is used.

  29. 29.

    In principal, any modeling technique can be used to establish the relationship between the exogenous parameters and the performance measure.

  30. 30.

    For example the NP-hard Traveling Salesman Problem.

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Authors and Affiliations

  1. Leiden University, Leiden, The Netherlands

    Thomas Bäck

  2. divis intelligent solutions GmbH, Dortmund, Germany

    Christophe Foussette & Peter Krause

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Bäck, T., Foussette, C., Krause, P. (2013). Evolution Strategies. In: Contemporary Evolution Strategies. Natural Computing Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40137-4_2

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