, Volume 81, Issue 2, pp 828–857 | Cite as

Reoptimization Time Analysis of Evolutionary Algorithms on Linear Functions Under Dynamic Uniform Constraints

  • Feng ShiEmail author
  • Martin Schirneck
  • Tobias Friedrich
  • Timo Kötzing
  • Frank Neumann
Part of the following topical collections:
  1. Special Issue on Theory of Genetic and Evolutionary Computation


Rigorous runtime analysis is a major approach towards understanding evolutionary computing techniques, and in this area linear pseudo-Boolean objective functions play a central role. Having an additional linear constraint is then equivalent to the NP-hard Knapsack problem, certain classes thereof have been studied in recent works. In this article, we present a dynamic model of optimizing linear functions under uniform constraints. Starting from an optimal solution with respect to a given constraint bound, we investigate the runtimes that different evolutionary algorithms need to recompute an optimal solution when the constraint bound changes by a certain amount. The classical \((1{+}1)\) EA and several population-based algorithms are designed for that purpose, and are shown to recompute efficiently. Furthermore, a variant of the \((1{+}(\lambda ,\lambda ) )\) GA for the dynamic optimization problem is studied, whose performance is better when the change of the constraint bound is small.


Evolutionary algorithm Runtime analysis Reoptimization time Dynamic constraint Uniform constraint 



The work leading up to this article has received funding from the German Research Foundation under Grant agreement FR2988 (TOSU) and the Australian Research Council under Grant agreements DP140103400 and DP160102401. We would like to thank the anonymous referees, whose comments and suggestions have greatly improved this paper.


  1. 1.
    Auger, A., Doerr, B.: Theory of Randomized Search Heuristics: Foundations and Recent Developments, vol. 1. World Scientific, Singapore (2011)zbMATHGoogle Scholar
  2. 2.
    Bienaymé, I.J.: Considérations à l’appui de la découverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés. Imprimerie de Mallet-Bachelier (1853)Google Scholar
  3. 3.
    Doerr, B., Doerr, C.: Optimal parameter choices through self-adjustment: Applying the 1/5-th rule in discrete settings. In: Proceedings of the 2015 Genetic and Evolutionary Computation Conference (GECCO), pp. 1335–1342 (2015)Google Scholar
  4. 4.
    Doerr, B., Doerr, C., Ebel, F.: From black-box complexity to designing new genetic algorithms. Theor. Comput. Sci. 567, 87–104 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64, 673–697 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Droste, S., Jansen, T., Wegener, I.: On the analysis of the (\(1+1\)) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Friedrich, T., He, J., Hebbinghaus, N., Neumann, F., Witt, C.: Approximating covering problems by randomized search heuristics using multi-objective models. Evol. Comput. 18, 617–633 (2010)CrossRefGoogle Scholar
  8. 8.
    Friedrich, T., Kötzing, T., Lagodzinski, J.A.G., Neumann, F., Schirneck, M.: Analysis of the (\(1+1\)) EA on subclasses of linear functions under uniform and linear constraints. In: Proceedings of the 14th Workshop on the Foundations of Genetic Algorithms (FOGA), pp. 45–54 (2017)Google Scholar
  9. 9.
    He, J., Yao, X.: A study of drift analysis for estimating computation time of evolutionary algorithms. Nat. Comput. 3, 21–35 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jansen, T.: Analyzing Evolutionary Algorithms. The Computer Science Perspective. Natural Computing Series. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kötzing, T., Lissovoi, A., Witt, C.: (\(1+1\)) EA on generalized dynamic OneMax. In: Proceedings of the 13th Workshop on the Foundations of Genetic Algorithms (FOGA), pp. 40–51 (2015)Google Scholar
  13. 13.
    Kratsch, S., Neumann, F.: Fixed-parameter evolutionary algorithms and the vertex cover problem. Algorithmica 65, 754–771 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mezura-Montes, E., Coello, C.A.C.: Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm Evol. Comput. 1, 173–194 (2011)CrossRefGoogle Scholar
  15. 15.
    Mühlenbein, H.: How genetic algorithms really work: Mutation and hillclimbing. In: Proceedings of the 2nd Conference on Parallel Problem Solving from Nature (PPSN), vol. 92, pp. 15–25 (1992)Google Scholar
  16. 16.
    Neumann, F., Wegener, I.: Minimum spanning trees made easier via multi-objective optimization. Nat. Comput. 5, 305–319 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization: Algorithms and Their Computational Complexity. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  18. 18.
    Shi, F., Schirneck, M., Friedrich, T., Kötzing, T., Neumann, F.: Reoptimization times of evolutionary algorithms on linear functions under dynamic uniform constraints. In: Proceedings of the 2017 Genetic and Evolutionary Computation Conference (GECCO), pp. 1407–1414 (2017)Google Scholar
  19. 19.
    Witt, C.: Tight bounds on the optimization time of a randomized search heuristic on linear functions. Comb. Probab. Comput. 22, 294–318 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhou, Y., He, J.: A runtime analysis of evolutionary algorithms for constrained optimization problems. IEEE Trans. Evol. Comput. 11, 608–619 (2007)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Information Science and EngineeringCentral South UniversityChangshaPeople’s Republic of China
  2. 2.Hasso Plattner InstitutePotsdamGermany
  3. 3.School of Computer ScienceThe University of AdelaideAdelaideAustralia

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