import numpy as np # engine for numerical computing
# class of Univariate Marginal Distribution Algorithm for normal models (UMDA)
from pypop7.optimizers.eda.umda import UMDA
[docs]class EMNA(UMDA):
"""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, now it is **rarely** used for large-scale black-box optimization
(LBO). It is **highly recommended** to first attempt other more advanced
optimization methods for LBO.
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 <http://en.wikipedia.org/wiki/Rosenbrock_function>`_:
.. code-block:: python
:linenos:
>>> import numpy
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.eda.emna import EMNA
>>> problem = {'fitness_function': rosenbrock, # define problem arguments
... 'ndim_problem': 2,
... 'lower_boundary': -5*numpy.ones((2,)),
... 'upper_boundary': 5*numpy.ones((2,))}
>>> options = {'max_function_evaluations': 5000, # set optimizer options
... 'seed_rng': 2022}
>>> emna = EMNA(problem, options) # initialize the optimizer class
>>> results = emna.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"EMNA: {results['n_function_evaluations']}, {results['best_so_far_y']}")
EMNA: 5000, 0.008375142194038284
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/2p8xksyy>`_ for more details.
Attributes
----------
n_individuals : `int`
number of offspring, aka offspring population size.
n_parents : `int`
number of parents, aka parental population size.
References
----------
Larrañaga, P. and Lozano, J.A. eds., 2002.
`Estimation of distribution algorithms: A new tool for evolutionary computation.
<https://link.springer.com/book/10.1007/978-1-4615-1539-5>`_
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. Spain.
(Unfortunately, to our knowledge this online document is not openly accessible now.)
"""
def __init__(self, problem, options):
UMDA.__init__(self, problem, options)
def iterate(self, x=None, y=None, args=None):
order = np.argsort(y)[:self.n_parents]
mean = np.mean(x[order], axis=0)
cov = np.cov(np.transpose(x[order]))
for i in range(self.n_individuals):
if self._check_terminations():
break
x[i] = self.rng_optimization.multivariate_normal(mean, cov)
y[i] = self._evaluate_fitness(x[i], args)
return x, y