Evolution Strategies (ES)

class pypop7.optimizers.es.es.ES(problem, options)

Evolution Strategies (ES).

This is the abstract class for all ES classes. Please use any of its instantiated subclasses to optimize the black-box problem at hand.


ES are a well-established family of randomized population-based search algorithms, proposed by two German computer scientists Ingo Rechenberg and Hans-Paul Schwefel (two recipients of IEEE Evolutionary Computation Pioneer Award 2002). One key property of ES is adaptability of strategy parameters, which generally can significantly accelerate the (local) convergence rate. Recently, the theoretical foundation of its most representative (modern) version called CMA-ES has been well built on the Information-Geometric Optimization (IGO) framework via invariance principles (inspired by NES).

According to the latest Nature review, “the CMA-ES algorithm is widely regarded as the state of the art in numerical optimization”.

  • problem (dict) –

    problem arguments with the following common settings (keys):
    • ’fitness_function’ - objective function to be minimized (func),

    • ’ndim_problem’ - number of dimensionality (int),

    • ’upper_boundary’ - upper boundary of search range (array_like),

    • ’lower_boundary’ - lower boundary of search range (array_like).

  • options (dict) –

    optimizer options with the following common settings (keys):
    • ’max_function_evaluations’ - maximum of function evaluations (int, default: np.Inf),

    • ’max_runtime’ - maximal runtime to be allowed (float, default: np.Inf),

    • ’seed_rng’ - seed for random number generation needed to be explicitly set (int);

    and with the following particular settings (keys):
    • ’n_individuals’ - number of offspring/descendants, aka offspring population size (int),

    • ’n_parents’ - number of parents/ancestors, aka parental population size (int),

    • ’mean’ - initial (starting) point (array_like),

      • if not given, it will draw a random sample from the uniform distribution whose search range is bounded by problem[‘lower_boundary’] and problem[‘upper_boundary’].

    • ’sigma’ - initial global step-size, aka mutation strength (float).


initial (starting) point, aka mean of Gaussian search/sampling/mutation distribution.




number of offspring/descendants, aka offspring population size.




number of parents/ancestors, aka parental population size.




global step-size, aka mutation strength (i.e., overall std of Gaussian search distribution).





Ollivier, Y., Arnold, L., Auger, A. and Hansen, N., 2017. Information-geometric optimization algorithms: A unifying picture via invariance principles. Journal of Machine Learning Research, 18(18), pp.1-65. https://www.jmlr.org/papers/v18/14-467.html


Hansen, N., Arnold, D.V. and Auger, A., 2015. Evolution strategies. In Springer Handbook of Computational Intelligence (pp. 871-898). Springer, Berlin, Heidelberg. https://link.springer.com/chapter/10.1007%2F978-3-662-43505-2_44

Bäck, T., Foussette, C., & Krause, P. (2013). Contemporary evolution strategies. Berlin: Springer. https://link.springer.com/book/10.1007/978-3-642-40137-4


Beyer, H.G. and Schwefel, H.P., 2002. Evolution strategies–A comprehensive introduction. Natural Computing, 1(1), pp.3-52. https://link.springer.com/article/10.1023/A:1015059928466

Rechenberg, I., 2000. Case studies in evolutionary experimentation and computation. Computer Methods in Applied Mechanics and Engineering, 186(2-4), pp.125-140. https://www.sciencedirect.com/science/article/abs/pii/S0045782599003813

Rechenberg, I., 1989. Evolution strategy: Nature’s way of optimization. In Optimization: Methods and Applications, Possibilities and Limitations (pp. 106-126). Springer, Berlin, Heidelberg. https://link.springer.com/chapter/10.1007/978-3-642-83814-9_6

Schwefel, H.P., 1988. Collective intelligence in evolving systems. In Ecodynamics (pp. 95-100). Springer, Berlin, Heidelberg. https://link.springer.com/chapter/10.1007/978-3-642-73953-8_8

Schwefel, H.P., 1984. Evolution strategies: A family of non-linear optimization techniques based on imitating some principles of organic evolution. Annals of Operations Research, 1(2), pp.165-167. https://link.springer.com/article/10.1007/BF01876146

Rechenberg, I., 1984. The evolution strategy. A mathematical model of darwinian evolution. In Synergetics—from Microscopic to Macroscopic Order (pp. 122-132). Springer, Berlin, Heidelberg. https://link.springer.com/chapter/10.1007/978-3-642-69540-7_13