Lyapunov Design of a Simple Step-Size Adaptation Strategy Based on Success

  • Claudia R. Correa
  • Elizabeth F. Wanner
  • Carlos M. Fonseca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9921)


A simple success-based step-size adaptation rule for single-parent 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 \((1\mathop {,}\limits ^{+}\lambda )\)-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 \(\lambda \). 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.


Step-size adaptation Evolution strategy Lyapunov function theory Convergence rate 



This work was partially supported by national funds through the Portuguese Foundation for Science and Technology (FCT) and by the European Regional Development Fund (FEDER) through COMPETE 2020 – Operational Program for Competitiveness and Internationalization (POCI). The authors also would like to thank the Brazilian funding agencies, CAPES, CNPq and FAPEMIG.


  1. 1.
    Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evol. Comput. 9(2), 159–195 (2001)CrossRefGoogle Scholar
  2. 2.
    Doerr, B., Doerr, C.: Optimal parameter choices through self-adjustment: applying the \(1/5\)-th rule in discrete settings. In: Proceedings of the 2015 ACM-GECCO Genetic and Evolutionary Computation Conference, pp. 1335–1342. ACM (2015)Google Scholar
  3. 3.
    Doerr, B., Doerr, C.: A tight runtime analysis of the \((1+(\lambda ,\lambda ))\) genetic algorithm on onemax. In: Proceedings of the 2015 ACM-GECCO Genetic and Evolutionary Computation Conference, pp. 1423–1430. ACM (2015)Google Scholar
  4. 4.
    Wanner, E.F., Fonseca, C.M., Cardoso, R.T.N., Takahashi, R.H.C.: Lyapunov stability analysis and adaptation law synthesis of a derandomized self-adaptive \((1,2)\)-ES. Under reviewGoogle Scholar
  5. 5.
    Semenov, M.A., Terkel, D.A.: Analysis of convergence of an evolutionary algorithm with self-adaptation using a stochastic Lyapunov function. Evol. Comput. 11(4), 363–379 (2003)CrossRefGoogle Scholar
  6. 6.
    Jägersküpper, J.: A blend of Markov-chain and drift analysis. In: Rudolph, G., Jansen, T., Lucas, S., Poloni, C., Beume, N. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 41–51. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    He, J., Yao, X.: Drift analysis and average time complexity of evolutionary algorithms. Artif. Intell. 127, 57–85 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    He, J., Yao, X.: A study of drift analysis for estimating computational time of evolutionary algorithms. Natural Comput. 3, 21–35 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64(4), 673–697 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Witt, C.: Tight bounds on the optimization time of a randomized search heuristic on linear function. Comb. Probab. Comput. 22(02), 294–318 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hart, W.E.: Rethinking the design of real-coded evolutionary algorithms: making discrete choices in continuous search domains. Soft Comput. J. 9, 225–235 (2002)CrossRefMATHGoogle Scholar
  12. 12.
    Lyapunov, A.M.: The general problem of stability of motion (reprint of the original paper of 1892). Int. J. Control 55(3), 531–773 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hahn, W.: Stability of Motion. Springer, Heidelberg (1967)CrossRefMATHGoogle Scholar
  14. 14.
    Janson, S., Luczak, T., Rucinski, A.: Random Graphs. Wiley, Hoboken (2000)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Claudia R. Correa
    • 1
  • Elizabeth F. Wanner
    • 2
  • Carlos M. Fonseca
    • 3
  1. 1.Post-Graduate Program in Mathematical and Computational ModelingCEFET-MGBelo HorizonteBrazil
  2. 2.Department of Computer EngineeringCEFET-MGBelo HorizonteBrazil
  3. 3.CISUC, Department of Informatics EngineeringUniversity of CoimbraCoimbraPortugal

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