Cross-Selection Based Evolution Strategies


  • Larysa O. Khilkova



Evolution Strategies, Cross-Selection Method, Optimization, Evolutionary Operators, Mutation, Selection, Recombination, Population, Generation, Fitness-Function


A search for an optimal value of a complex multi-dimensional continuous function is still one of the most pressing problems. The genetic algorithms (GA) and evolution strategies (ES) are methods to solving optimization problems that is based on natural selection, the process that drives biological evolution. Our goal was to use evolutionary optimization methods to find the global optimal value (minimum) of a non-smooth multi-dimensional function with a large number of local minimums. We took several test functions of different levels of complexity and used evolution strategies to solve the problem. The standard evolution strategies, which work well with smooth functions, gave us various points of local minimums as a solution, without finding the global minimum, for the complex function. In our work, we propose a new approach: the cross-selection method, which, in combination with previously developed methods - adaptive evolution strategies, gave a good result for the searth for the global minimum the complex function.


N. Hansen, D. Arnold, and A. Auger, “Evolution strategies,” in Springer Handbook of Computational Intelligence, 2015, pp. 871–898. [Online].


D. Simon, Evolutionary optimization algorithms. Biologically-Inspired and Population-Based Approaches to Comp. Intelligence. Wiley, 2020.

I. Rechenberg, Evolutionsstrategie: Optimierung technischer Systeme nach Prinzipien der biologischen Evolution, Dr.-Ing. Thesis. Technical University of Berlin, 1971.

H.-P. Schwefel, Evolutionsstrategie und numerische Optimierung, Dr.-Ing. Thesis. Technical University of Berlin, 1975.

H.-G. Beyer, “Toward a theory of evolution strategies: Some asymptotical results from the (1, +λ)-theory,” Evol. Comp., vol. 1, no. 2, pp. 165–188, 1993. [Online]. Available:

H.-P. Schwefel and G. Rudolph, “Contemporary evolution strategies,” in Advances in Artificial Life, vol. 929, 1995, pp. 891–907. [Online]. Available:

A. Bienvenüe and O. Francois, “Global convergence for evolution strategies in spherical problems: some simple proofs and difficulties,” Theor. Comp. Sci., vol. 306, no. 1, pp. 269–289, 2003. [Online]. Available:

D. Ackley, “An empirical study of bit vector function optimization,” in Genetic Algorithms and Simulated Annealing. Pitman Publishing, 1987, pp. 170–215.

J. Liang, P. Suganthan, and K. Deb, “Novel composition test functions for numerical global optimization,” in Swarm Intelligence Symposium, Pasadena, California, 2005, pp. 68–75. [Online]. Available:

H.-G. Beyer, “Some aspects of the ‘evolution strategy’ for solving tsp-like

optimization problems,” in Parallel Problem Solving from Nature, vol. 2,

, pp. 361–370.

M. Herdy, “Application of the ‘evolutions strategy’ to discrete optimization problems,” in Parallel Problem Solving from Nature, vol. 496. Springer-Verlag, Berlin, 1990, pp. 188–192.

H.-G. Beyer and H.-P. Schwefel, “Evolution strategies: A comprehensive introduction,” Natur. Comp., vol. 1, no. 1, pp. 3–52, 2002. [Online]. Available:

C. Kappler, “Are evolutionary algorithms improved by large mutations?” in Parallel Problem Solving from Nature, vol. 1141, 1996, pp. 346–355. [Online]. Available:

G. Rudolph, “Local convergence rates of simple evolutionary algorithms with cauchy mutations,” IEEE Trans. Evol. Comp., pp. 249–258.

X. Yao and Y. Liu, “Fast evolution strategies,” in Evolutionary Programming VI, 1997, pp. 151–161. [Online]. Available:

X. Yao, Y. Liu, and G. Lin, “Evolutionary programming made faster,” IEEE Trans. Evol. Comp., vol. 3, no. 2, pp. 82–102, 1999. [Online]. Available:

D. Wierstra and T. Schaul, “Natural evolution strategies,” J. of Machine Learning Research, vol. 15, pp. 949–980, 2014.

N. Hansen and A. Ostermeier, “Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation,” in IEEE Int. Conf. Evol. Comp., 1996, pp. 312–317. [Online]. Available:

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evol. Comp., vol. 9, no. 2, pp. 159–195, 2001.

N. Hansen and S. Kern, “Evaluating the cma evolution strategy on multimodal test functions,” in Parallel Problem Solving from Nature, vol. 3242, 2004, pp. 282–291. [Online]. Available:

S. Kern, S. Müller, N. Hansen, D. Büche, J. Ocenasek, and P. Koumoutsakos, “Learning probability distributions in continuous evolutionary algorithms – a comparative review,” Natur. Comp., vol. 3, no. 1, pp. 77–112, 2004. [Online]. Available:

H.-G. Beyer and K. Deb, “On self-adaptive features in real-parameter evolutionary algorithms,” IEEE Trans. Evol. Comp., vol. 5, no. 3, pp. 250–269, 2001.

I. Rechenberg, “Evolutionsstrategien,” in Simulationsmethoden in der Medizin und Biologie. Springer-Verlag, 1978, pp. 83–114.

M. Mühlenbein and D. Schlierkamp-Voosen, “Predictive models for the breeder genetic algorithm i. continuous parameter optimization,” Evol. Comp., vol. 1, no. 1, pp. 25–49, 1993.

D. Arnold, “Weighted multirecombination evolution strategies,” Theor. Comp. Sci., vol. 361, no. 1, pp. 18–37, 2006.




How to Cite

Khilkova, L. O. (2023). Cross-Selection Based Evolution Strategies. International Journal of Computing, 22(1), 69-77.