### Abstract

A simple success-based step-size adaptation rule for singleparent Evolution Strategies is formulated, and the setting of the corresponding parameters is considered. Theoretical convergence on the class of strictly unimodal functions of one variable that are symmetric around the optimum is investigated using a stochastic Lyapunov function method developed by Semenov and Terkel [5] in the context of martingale theory. General expressions for the conditional expectations of the next values of step size and distance to the optimum under (1 +, λ)-selection are analytically derived, and an appropriate Lyapunov function is constructed. Convergence rate upper bounds, as well as adaptation parameter values, are obtained through numerical optimization for increasing values of λ. By selecting the number of offspring that minimizes the bound on the convergence rate with respect to the number of function evaluations, all strategy parameter values result from the analysis.

Original language | English |
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Title of host publication | Parallel Problem Solving from Nature – PPSN XIV |

Subtitle of host publication | 14th International Conference, Edinburgh, UK, September 17-21, 2016, Proceedings |

Editors | Julia Handl, Emma Hart, Peter R. Lewis, et al |

Place of Publication | Cham (CH) |

Publisher | Springer |

Pages | 101-110 |

Number of pages | 10 |

ISBN (Electronic) | 978-3-319-45823-6 |

ISBN (Print) | 978-3-319-45822-9 |

DOIs | |

Publication status | E-pub ahead of print - 31 Aug 2016 |

Event | 14th International Conference on Parallel Problem Solving from Nature - Edinburgh, United Kingdom Duration: 17 Sep 2016 → 21 Sep 2016 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer |

Volume | 9921 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 14th International Conference on Parallel Problem Solving from Nature |
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Abbreviated title | PPSN 2016 |

Country | United Kingdom |

City | Edinburgh |

Period | 17/09/16 → 21/09/16 |

### Fingerprint

### Keywords

- convergence rate
- evolution strategy
- Lyapunov function theory
- step-size adaptation

### Cite this

*Parallel Problem Solving from Nature – PPSN XIV: 14th International Conference, Edinburgh, UK, September 17-21, 2016, Proceedings*(pp. 101-110). (Lecture Notes in Computer Science; Vol. 9921). Cham (CH): Springer. https://doi.org/10.1007/978-3-319-45823-6_10

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*Parallel Problem Solving from Nature – PPSN XIV: 14th International Conference, Edinburgh, UK, September 17-21, 2016, Proceedings.*Lecture Notes in Computer Science, vol. 9921, Springer, Cham (CH), pp. 101-110, 14th International Conference on Parallel Problem Solving from Nature, Edinburgh, United Kingdom, 17/09/16. https://doi.org/10.1007/978-3-319-45823-6_10

**Lyapunov design of a simple step-size adaptation strategy based on success.** / Correa, Claudia R.; Wanner, Elizabeth F.; Fonseca, Carlos M.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Lyapunov design of a simple step-size adaptation strategy based on success

AU - Correa, Claudia R.

AU - Wanner, Elizabeth F.

AU - Fonseca, Carlos M.

PY - 2016/8/31

Y1 - 2016/8/31

N2 - A simple success-based step-size adaptation rule for singleparent Evolution Strategies is formulated, and the setting of the corresponding parameters is considered. Theoretical convergence on the class of strictly unimodal functions of one variable that are symmetric around the optimum is investigated using a stochastic Lyapunov function method developed by Semenov and Terkel [5] in the context of martingale theory. General expressions for the conditional expectations of the next values of step size and distance to the optimum under (1 +, λ)-selection are analytically derived, and an appropriate Lyapunov function is constructed. Convergence rate upper bounds, as well as adaptation parameter values, are obtained through numerical optimization for increasing values of λ. By selecting the number of offspring that minimizes the bound on the convergence rate with respect to the number of function evaluations, all strategy parameter values result from the analysis.

AB - A simple success-based step-size adaptation rule for singleparent Evolution Strategies is formulated, and the setting of the corresponding parameters is considered. Theoretical convergence on the class of strictly unimodal functions of one variable that are symmetric around the optimum is investigated using a stochastic Lyapunov function method developed by Semenov and Terkel [5] in the context of martingale theory. General expressions for the conditional expectations of the next values of step size and distance to the optimum under (1 +, λ)-selection are analytically derived, and an appropriate Lyapunov function is constructed. Convergence rate upper bounds, as well as adaptation parameter values, are obtained through numerical optimization for increasing values of λ. By selecting the number of offspring that minimizes the bound on the convergence rate with respect to the number of function evaluations, all strategy parameter values result from the analysis.

KW - convergence rate

KW - evolution strategy

KW - Lyapunov function theory

KW - step-size adaptation

UR - http://link.springer.com/chapter/10.1007%2F978-3-319-45823-6_10

UR - http://www.scopus.com/inward/record.url?scp=84988485731&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-45823-6_10

DO - 10.1007/978-3-319-45823-6_10

M3 - Conference contribution

AN - SCOPUS:84988485731

SN - 978-3-319-45822-9

T3 - Lecture Notes in Computer Science

SP - 101

EP - 110

BT - Parallel Problem Solving from Nature – PPSN XIV

A2 - Handl, Julia

A2 - Hart, Emma

A2 - Lewis, Peter R.

A2 - et al,

PB - Springer

CY - Cham (CH)

ER -