Estimation of Multivariate Normal Algorithm (EMNA)

class pypop7.optimizers.eda.emna.EMNA(problem, options)

Estimation of Multivariate Normal Algorithm (EMNA).


EMNA learns the full covariance matrix of the Gaussian sampling distribution, resulting in a cubic time complexity w.r.t. each sampling. Therefore, it is rarely used for large-scale black-box optimization (LSBBO). It is highly recommended to first attempt other more advanced methods for LSBBO.

  • 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 (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, offspring population size (int, default: 200),

    • ’n_parents’ - number of parents, parental population size (int, default: int(options[‘n_individuals’]/2)).


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 pypop7.optimizers.eda.emna import EMNA
 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>>> emna = EMNA(problem, options)  # initialize the optimizer class
11>>> results = emna.optimize()  # run the optimization process
12>>> # return the number of function evaluations and best-so-far fitness
13>>> print(f"EMNA: {results['n_function_evaluations']}, {results['best_so_far_y']}")
14EMNA: 5000, 0.008375142194038284

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


number of offspring, aka offspring population size.




number of parents, aka parental population size.




Larrañaga, P. and Lozano, J.A. eds., 2002. Estimation of distribution algorithms: A new tool for evolutionary computation. Springer Science & Business Media.

Larranaga, P., Etxeberria, R., Lozano, J.A. and Pena, J.M., 2000. Optimization in continuous domains by learning and simulation of Gaussian networks. Technical Report, Department of Computer Science and Artificial Intelligence, University of the Basque Country.