import numpy as np # engine for numerical computing
from scipy.stats import levy_stable
from pypop7.optimizers.ep.cep import CEP
[docs]class LEP(CEP):
"""Lévy distribution based Evolutionary Programming (LEP).
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 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`),
* 'n_individuals' - number of offspring, aka offspring population size (`int`, default: `100`),
* 'q' - number of opponents for pairwise comparisons (`int`, default: `10`),
* 'tau' - learning rate of individual step-sizes self-adaptation (`float`, default:
`1.0/np.sqrt(2.0*np.sqrt(problem['ndim_problem']))`),
* 'tau_apostrophe' - learning rate of individual step-sizes self-adaptation (`float`, default:
`1.0/np.sqrt(2.0*problem['ndim_problem'])`.
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 # engine for numerical computing
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.ep.lep import LEP
>>> 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,
... 'sigma': 0.1}
>>> lep = LEP(problem, options) # initialize the optimizer class
>>> results = lep.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"LEP: {results['n_function_evaluations']}, {results['best_so_far_y']}")
LEP: 5000, 0.0359694938656471
For its correctness checking, refer to `this code-based repeatability report
<https://tinyurl.com/2dc6ym6j>`_ for more details.
Attributes
----------
n_individuals : `int`
number of offspring, aka offspring population size.
q : `int`
number of opponents for pairwise comparisons.
sigma : `float`
initial global step-size, aka mutation strength.
tau : `float`
learning rate of individual step-sizes self-adaptation.
tau_apostrophe : `float`
learning rate of individual step-sizes self-adaptation.
References
----------
Lee, C.Y. and Yao, X., 2004.
Evolutionary programming using mutations based on the Lévy probability distribution.
IEEE Transactions on Evolutionary Computation, 8(1), pp.1-13.
https://ieeexplore.ieee.org/document/1266370
"""
def __init__(self, problem, options):
CEP.__init__(self, problem, options)
def iterate(self, x=None, sigmas=None, y=None, xx=None, ss=None, yy=None, args=None):
for i in range(self.n_individuals):
if self._check_terminations():
return x, sigmas, y, xx, ss, yy
ss[i] = sigmas[i]*np.exp(self.tau_apostrophe*self.rng_optimization.standard_normal(
size=(self.ndim_problem,)) + self.tau*self.rng_optimization.standard_normal(
size=(self.ndim_problem,)))
xx[i] = x[i] + ss[i]*levy_stable.rvs(alpha=1.8, beta=1, size=(self.ndim_problem,),
random_state=self.rng_optimization)
yy[i] = self._evaluate_fitness(xx[i], args)
new_x = np.vstack((xx, x))
new_sigmas = np.vstack((ss, sigmas))
new_y = np.hstack((yy, y))
n_win = np.zeros((2*self.n_individuals,)) # number of win
for i in range(2*self.n_individuals):
for j in self.rng_optimization.choice([k for k in range(2*self.n_individuals) if k != i],
size=self.q, replace=False):
if new_y[i] < new_y[j]:
n_win[i] += 1
order = np.argsort(-n_win)[:self.n_individuals]
x[:self.n_individuals] = new_x[order]
sigmas[:self.n_individuals] = new_sigmas[order]
y[:self.n_individuals] = new_y[order]
self._n_generations += 1
return x, sigmas, y, xx, ss, yy