# Estimation of Multivariate Normal Algorithm (EMNA)¶

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

Estimation of Multivariate Normal Algorithm (EMNA).

Note

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.

Parameters:
• 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)).

Examples

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.

n_individuals

number of offspring, aka offspring population size.

Type:

int

n_parents

number of parents, aka parental population size.

Type:

int

References

Larrañaga, P. and Lozano, J.A. eds., 2002. Estimation of distribution algorithms: A new tool for evolutionary computation. Springer Science & Business Media. https://link.springer.com/book/10.1007/978-1-4615-1539-5

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. https://tinyurl.com/3bw6n3x4