Rechenberg’s (1+1)-Evolution Strategy (RES)

class, options)

Rechenberg’s (1+1)-Evolution Strategy with 1/5th success rule (RES).


RES is the first ES with self-adaptation of the global step-size, designed by Rechenberg, one recipient of IEEE Evolutionary Computation Pioneer Award 2002. As theoretically investigated in Rechenberg’s seminal PhD dissertation, the existence of narrow evolution window explains the necessarity of step-size adaptation to maximize local convergence progress, if possible.

Since there is only one parent and only one offspring for each generation, RES generally shows very limited exploration ability for large-scale black-box optimization (LSBBO). Therefore, it is highly recommended to first attempt more advanced ES variants (e.g. LMCMA, LMMAES) for LSBBO. Here we include it mainly for benchmarking and theoretical purpose.

AKA two-membered ES.

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

    • ’mean’ - initial (starting) point, aka mean of Gaussian search distribution (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’].

    • ’lr_sigma’ - learning rate of global step-size self-adaptation (float, default: 1.0/np.sqrt(problem[‘ndim_problem’] + 1.0)).


Use the optimizer to minimize the well-known test function Rosenbrock:

 1>>> import numpy
 2>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
 3>>> from import RES
 4>>> problem = {'fitness_function': rosenbrock,  # define problem arguments
 5...            'ndim_problem': 2,
 6...            'lower_boundary': -5*numpy.ones((2,)),
 7...            'upper_boundary': 5*numpy.ones((2,))}
 8>>> options = {'max_function_evaluations': 5000,  # set optimizer options
 9...            'seed_rng': 2022,
10...            'mean': 3*numpy.ones((2,)),
11...            'sigma': 0.1}  # the global step-size may need to be tuned for better performance
12>>> res = RES(problem, options)  # initialize the optimizer class
13>>> results = res.optimize()  # run the optimization process
14>>> # return the number of function evaluations and best-so-far fitness
15>>> print(f"RES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
16RES: 5000, 0.06701744137207027

For its correctness checking of coding, refer to this code-based repeatability report for more details.


learning rate of global step-size self-adaptation.




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




final global step-size, aka mutation strength.




Auger, A., Hansen, N., López-Ibáñez, M. and Rudolph, G., 2022. Tributes to Ingo Rechenberg (1934–2021). ACM SIGEVOlution, 14(4), pp.1-4.

Agapie, A., Solomon, O. and Giuclea, M., 2021. Theory of (1+1) ES on the RIDGE. IEEE Transactions on Evolutionary Computation, 26(3), pp.501-511.

Hansen, N., Arnold, D.V. and Auger, A., 2015. Evolution strategies. In Springer Handbook of Computational Intelligence (pp. 871-898). Springer, Berlin, Heidelberg.

Kern, S., Müller, S.D., Hansen, N., Büche, D., Ocenasek, J. and Koumoutsakos, P., 2004. Learning probability distributions in continuous evolutionary algorithms–a comparative review. Natural Computing, 3, pp.77-112.

Beyer, H.G. and Schwefel, H.P., 2002. Evolution strategies–A comprehensive introduction. Natural Computing, 1(1), pp.3-52.

Rechenberg, I., 2000. Case studies in evolutionary experimentation and computation. Computer Methods in Applied Mechanics and Engineering, 186(2-4), pp.125-140.

Rechenberg, I., 1989. Evolution strategy: Nature’s way of optimization. In Optimization: Methods and Applications, Possibilities and Limitations (pp. 106-126). Springer, Berlin, Heidelberg.

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.