754 lines
30 KiB
Python
754 lines
30 KiB
Python
|
"""
|
||
|
|
basinhopping: The basinhopping global optimization algorithm
|
||
|
|
"""
|
||
|
|
import numpy as np
|
||
|
|
import math
|
||
|
|
import inspect
|
||
|
|
import scipy.optimize
|
||
|
|
from scipy._lib._util import check_random_state
|
||
|
|
|
||
|
|
__all__ = ['basinhopping']
|
||
|
|
|
||
|
|
|
||
|
|
_params = (inspect.Parameter('res_new', kind=inspect.Parameter.KEYWORD_ONLY),
|
||
|
|
inspect.Parameter('res_old', kind=inspect.Parameter.KEYWORD_ONLY))
|
||
|
|
_new_accept_test_signature = inspect.Signature(parameters=_params)
|
||
|
|
|
||
|
|
|
||
|
|
class Storage:
|
||
|
|
"""
|
||
|
|
Class used to store the lowest energy structure
|
||
|
|
"""
|
||
|
|
def __init__(self, minres):
|
||
|
|
self._add(minres)
|
||
|
|
|
||
|
|
def _add(self, minres):
|
||
|
|
self.minres = minres
|
||
|
|
self.minres.x = np.copy(minres.x)
|
||
|
|
|
||
|
|
def update(self, minres):
|
||
|
|
if minres.success and (minres.fun < self.minres.fun
|
||
|
|
or not self.minres.success):
|
||
|
|
self._add(minres)
|
||
|
|
return True
|
||
|
|
else:
|
||
|
|
return False
|
||
|
|
|
||
|
|
def get_lowest(self):
|
||
|
|
return self.minres
|
||
|
|
|
||
|
|
|
||
|
|
class BasinHoppingRunner:
|
||
|
|
"""This class implements the core of the basinhopping algorithm.
|
||
|
|
|
||
|
|
x0 : ndarray
|
||
|
|
The starting coordinates.
|
||
|
|
minimizer : callable
|
||
|
|
The local minimizer, with signature ``result = minimizer(x)``.
|
||
|
|
The return value is an `optimize.OptimizeResult` object.
|
||
|
|
step_taking : callable
|
||
|
|
This function displaces the coordinates randomly. Signature should
|
||
|
|
be ``x_new = step_taking(x)``. Note that `x` may be modified in-place.
|
||
|
|
accept_tests : list of callables
|
||
|
|
Each test is passed the kwargs `f_new`, `x_new`, `f_old` and
|
||
|
|
`x_old`. These tests will be used to judge whether or not to accept
|
||
|
|
the step. The acceptable return values are True, False, or ``"force
|
||
|
|
accept"``. If any of the tests return False then the step is rejected.
|
||
|
|
If ``"force accept"``, then this will override any other tests in
|
||
|
|
order to accept the step. This can be used, for example, to forcefully
|
||
|
|
escape from a local minimum that ``basinhopping`` is trapped in.
|
||
|
|
disp : bool, optional
|
||
|
|
Display status messages.
|
||
|
|
|
||
|
|
"""
|
||
|
|
def __init__(self, x0, minimizer, step_taking, accept_tests, disp=False):
|
||
|
|
self.x = np.copy(x0)
|
||
|
|
self.minimizer = minimizer
|
||
|
|
self.step_taking = step_taking
|
||
|
|
self.accept_tests = accept_tests
|
||
|
|
self.disp = disp
|
||
|
|
|
||
|
|
self.nstep = 0
|
||
|
|
|
||
|
|
# initialize return object
|
||
|
|
self.res = scipy.optimize.OptimizeResult()
|
||
|
|
self.res.minimization_failures = 0
|
||
|
|
|
||
|
|
# do initial minimization
|
||
|
|
minres = minimizer(self.x)
|
||
|
|
if not minres.success:
|
||
|
|
self.res.minimization_failures += 1
|
||
|
|
if self.disp:
|
||
|
|
print("warning: basinhopping: local minimization failure")
|
||
|
|
self.x = np.copy(minres.x)
|
||
|
|
self.energy = minres.fun
|
||
|
|
self.incumbent_minres = minres # best minimize result found so far
|
||
|
|
if self.disp:
|
||
|
|
print("basinhopping step %d: f %g" % (self.nstep, self.energy))
|
||
|
|
|
||
|
|
# initialize storage class
|
||
|
|
self.storage = Storage(minres)
|
||
|
|
|
||
|
|
if hasattr(minres, "nfev"):
|
||
|
|
self.res.nfev = minres.nfev
|
||
|
|
if hasattr(minres, "njev"):
|
||
|
|
self.res.njev = minres.njev
|
||
|
|
if hasattr(minres, "nhev"):
|
||
|
|
self.res.nhev = minres.nhev
|
||
|
|
|
||
|
|
def _monte_carlo_step(self):
|
||
|
|
"""Do one Monte Carlo iteration
|
||
|
|
|
||
|
|
Randomly displace the coordinates, minimize, and decide whether
|
||
|
|
or not to accept the new coordinates.
|
||
|
|
"""
|
||
|
|
# Take a random step. Make a copy of x because the step_taking
|
||
|
|
# algorithm might change x in place
|
||
|
|
x_after_step = np.copy(self.x)
|
||
|
|
x_after_step = self.step_taking(x_after_step)
|
||
|
|
|
||
|
|
# do a local minimization
|
||
|
|
minres = self.minimizer(x_after_step)
|
||
|
|
x_after_quench = minres.x
|
||
|
|
energy_after_quench = minres.fun
|
||
|
|
if not minres.success:
|
||
|
|
self.res.minimization_failures += 1
|
||
|
|
if self.disp:
|
||
|
|
print("warning: basinhopping: local minimization failure")
|
||
|
|
if hasattr(minres, "nfev"):
|
||
|
|
self.res.nfev += minres.nfev
|
||
|
|
if hasattr(minres, "njev"):
|
||
|
|
self.res.njev += minres.njev
|
||
|
|
if hasattr(minres, "nhev"):
|
||
|
|
self.res.nhev += minres.nhev
|
||
|
|
|
||
|
|
# accept the move based on self.accept_tests. If any test is False,
|
||
|
|
# then reject the step. If any test returns the special string
|
||
|
|
# 'force accept', then accept the step regardless. This can be used
|
||
|
|
# to forcefully escape from a local minimum if normal basin hopping
|
||
|
|
# steps are not sufficient.
|
||
|
|
accept = True
|
||
|
|
for test in self.accept_tests:
|
||
|
|
if inspect.signature(test) == _new_accept_test_signature:
|
||
|
|
testres = test(res_new=minres, res_old=self.incumbent_minres)
|
||
|
|
else:
|
||
|
|
testres = test(f_new=energy_after_quench, x_new=x_after_quench,
|
||
|
|
f_old=self.energy, x_old=self.x)
|
||
|
|
|
||
|
|
if testres == 'force accept':
|
||
|
|
accept = True
|
||
|
|
break
|
||
|
|
elif testres is None:
|
||
|
|
raise ValueError("accept_tests must return True, False, or "
|
||
|
|
"'force accept'")
|
||
|
|
elif not testres:
|
||
|
|
accept = False
|
||
|
|
|
||
|
|
# Report the result of the acceptance test to the take step class.
|
||
|
|
# This is for adaptive step taking
|
||
|
|
if hasattr(self.step_taking, "report"):
|
||
|
|
self.step_taking.report(accept, f_new=energy_after_quench,
|
||
|
|
x_new=x_after_quench, f_old=self.energy,
|
||
|
|
x_old=self.x)
|
||
|
|
|
||
|
|
return accept, minres
|
||
|
|
|
||
|
|
def one_cycle(self):
|
||
|
|
"""Do one cycle of the basinhopping algorithm
|
||
|
|
"""
|
||
|
|
self.nstep += 1
|
||
|
|
new_global_min = False
|
||
|
|
|
||
|
|
accept, minres = self._monte_carlo_step()
|
||
|
|
|
||
|
|
if accept:
|
||
|
|
self.energy = minres.fun
|
||
|
|
self.x = np.copy(minres.x)
|
||
|
|
self.incumbent_minres = minres # best minimize result found so far
|
||
|
|
new_global_min = self.storage.update(minres)
|
||
|
|
|
||
|
|
# print some information
|
||
|
|
if self.disp:
|
||
|
|
self.print_report(minres.fun, accept)
|
||
|
|
if new_global_min:
|
||
|
|
print("found new global minimum on step %d with function"
|
||
|
|
" value %g" % (self.nstep, self.energy))
|
||
|
|
|
||
|
|
# save some variables as BasinHoppingRunner attributes
|
||
|
|
self.xtrial = minres.x
|
||
|
|
self.energy_trial = minres.fun
|
||
|
|
self.accept = accept
|
||
|
|
|
||
|
|
return new_global_min
|
||
|
|
|
||
|
|
def print_report(self, energy_trial, accept):
|
||
|
|
"""print a status update"""
|
||
|
|
minres = self.storage.get_lowest()
|
||
|
|
print("basinhopping step %d: f %g trial_f %g accepted %d "
|
||
|
|
" lowest_f %g" % (self.nstep, self.energy, energy_trial,
|
||
|
|
accept, minres.fun))
|
||
|
|
|
||
|
|
|
||
|
|
class AdaptiveStepsize:
|
||
|
|
"""
|
||
|
|
Class to implement adaptive stepsize.
|
||
|
|
|
||
|
|
This class wraps the step taking class and modifies the stepsize to
|
||
|
|
ensure the true acceptance rate is as close as possible to the target.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
takestep : callable
|
||
|
|
The step taking routine. Must contain modifiable attribute
|
||
|
|
takestep.stepsize
|
||
|
|
accept_rate : float, optional
|
||
|
|
The target step acceptance rate
|
||
|
|
interval : int, optional
|
||
|
|
Interval for how often to update the stepsize
|
||
|
|
factor : float, optional
|
||
|
|
The step size is multiplied or divided by this factor upon each
|
||
|
|
update.
|
||
|
|
verbose : bool, optional
|
||
|
|
Print information about each update
|
||
|
|
|
||
|
|
"""
|
||
|
|
def __init__(self, takestep, accept_rate=0.5, interval=50, factor=0.9,
|
||
|
|
verbose=True):
|
||
|
|
self.takestep = takestep
|
||
|
|
self.target_accept_rate = accept_rate
|
||
|
|
self.interval = interval
|
||
|
|
self.factor = factor
|
||
|
|
self.verbose = verbose
|
||
|
|
|
||
|
|
self.nstep = 0
|
||
|
|
self.nstep_tot = 0
|
||
|
|
self.naccept = 0
|
||
|
|
|
||
|
|
def __call__(self, x):
|
||
|
|
return self.take_step(x)
|
||
|
|
|
||
|
|
def _adjust_step_size(self):
|
||
|
|
old_stepsize = self.takestep.stepsize
|
||
|
|
accept_rate = float(self.naccept) / self.nstep
|
||
|
|
if accept_rate > self.target_accept_rate:
|
||
|
|
# We're accepting too many steps. This generally means we're
|
||
|
|
# trapped in a basin. Take bigger steps.
|
||
|
|
self.takestep.stepsize /= self.factor
|
||
|
|
else:
|
||
|
|
# We're not accepting enough steps. Take smaller steps.
|
||
|
|
self.takestep.stepsize *= self.factor
|
||
|
|
if self.verbose:
|
||
|
|
print("adaptive stepsize: acceptance rate {:f} target {:f} new "
|
||
|
|
"stepsize {:g} old stepsize {:g}".format(accept_rate,
|
||
|
|
self.target_accept_rate, self.takestep.stepsize,
|
||
|
|
old_stepsize))
|
||
|
|
|
||
|
|
def take_step(self, x):
|
||
|
|
self.nstep += 1
|
||
|
|
self.nstep_tot += 1
|
||
|
|
if self.nstep % self.interval == 0:
|
||
|
|
self._adjust_step_size()
|
||
|
|
return self.takestep(x)
|
||
|
|
|
||
|
|
def report(self, accept, **kwargs):
|
||
|
|
"called by basinhopping to report the result of the step"
|
||
|
|
if accept:
|
||
|
|
self.naccept += 1
|
||
|
|
|
||
|
|
|
||
|
|
class RandomDisplacement:
|
||
|
|
"""Add a random displacement of maximum size `stepsize` to each coordinate.
|
||
|
|
|
||
|
|
Calling this updates `x` in-place.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
stepsize : float, optional
|
||
|
|
Maximum stepsize in any dimension
|
||
|
|
random_gen : {None, int, `numpy.random.Generator`,
|
||
|
|
`numpy.random.RandomState`}, optional
|
||
|
|
|
||
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
||
|
|
singleton is used.
|
||
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `seed`.
|
||
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance then
|
||
|
|
that instance is used.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, stepsize=0.5, random_gen=None):
|
||
|
|
self.stepsize = stepsize
|
||
|
|
self.random_gen = check_random_state(random_gen)
|
||
|
|
|
||
|
|
def __call__(self, x):
|
||
|
|
x += self.random_gen.uniform(-self.stepsize, self.stepsize,
|
||
|
|
np.shape(x))
|
||
|
|
return x
|
||
|
|
|
||
|
|
|
||
|
|
class MinimizerWrapper:
|
||
|
|
"""
|
||
|
|
wrap a minimizer function as a minimizer class
|
||
|
|
"""
|
||
|
|
def __init__(self, minimizer, func=None, **kwargs):
|
||
|
|
self.minimizer = minimizer
|
||
|
|
self.func = func
|
||
|
|
self.kwargs = kwargs
|
||
|
|
|
||
|
|
def __call__(self, x0):
|
||
|
|
if self.func is None:
|
||
|
|
return self.minimizer(x0, **self.kwargs)
|
||
|
|
else:
|
||
|
|
return self.minimizer(self.func, x0, **self.kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
class Metropolis:
|
||
|
|
"""Metropolis acceptance criterion.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
T : float
|
||
|
|
The "temperature" parameter for the accept or reject criterion.
|
||
|
|
random_gen : {None, int, `numpy.random.Generator`,
|
||
|
|
`numpy.random.RandomState`}, optional
|
||
|
|
|
||
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
||
|
|
singleton is used.
|
||
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `seed`.
|
||
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance then
|
||
|
|
that instance is used.
|
||
|
|
Random number generator used for acceptance test.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, T, random_gen=None):
|
||
|
|
# Avoid ZeroDivisionError since "MBH can be regarded as a special case
|
||
|
|
# of the BH framework with the Metropolis criterion, where temperature
|
||
|
|
# T = 0." (Reject all steps that increase energy.)
|
||
|
|
self.beta = 1.0 / T if T != 0 else float('inf')
|
||
|
|
self.random_gen = check_random_state(random_gen)
|
||
|
|
|
||
|
|
def accept_reject(self, res_new, res_old):
|
||
|
|
"""
|
||
|
|
Assuming the local search underlying res_new was successful:
|
||
|
|
If new energy is lower than old, it will always be accepted.
|
||
|
|
If new is higher than old, there is a chance it will be accepted,
|
||
|
|
less likely for larger differences.
|
||
|
|
"""
|
||
|
|
with np.errstate(invalid='ignore'):
|
||
|
|
# The energy values being fed to Metropolis are 1-length arrays, and if
|
||
|
|
# they are equal, their difference is 0, which gets multiplied by beta,
|
||
|
|
# which is inf, and array([0]) * float('inf') causes
|
||
|
|
#
|
||
|
|
# RuntimeWarning: invalid value encountered in multiply
|
||
|
|
#
|
||
|
|
# Ignore this warning so when the algorithm is on a flat plane, it always
|
||
|
|
# accepts the step, to try to move off the plane.
|
||
|
|
prod = -(res_new.fun - res_old.fun) * self.beta
|
||
|
|
w = math.exp(min(0, prod))
|
||
|
|
|
||
|
|
rand = self.random_gen.uniform()
|
||
|
|
return w >= rand and (res_new.success or not res_old.success)
|
||
|
|
|
||
|
|
def __call__(self, *, res_new, res_old):
|
||
|
|
"""
|
||
|
|
f_new and f_old are mandatory in kwargs
|
||
|
|
"""
|
||
|
|
return bool(self.accept_reject(res_new, res_old))
|
||
|
|
|
||
|
|
|
||
|
|
def basinhopping(func, x0, niter=100, T=1.0, stepsize=0.5,
|
||
|
|
minimizer_kwargs=None, take_step=None, accept_test=None,
|
||
|
|
callback=None, interval=50, disp=False, niter_success=None,
|
||
|
|
seed=None, *, target_accept_rate=0.5, stepwise_factor=0.9):
|
||
|
|
"""Find the global minimum of a function using the basin-hopping algorithm.
|
||
|
|
|
||
|
|
Basin-hopping is a two-phase method that combines a global stepping
|
||
|
|
algorithm with local minimization at each step. Designed to mimic
|
||
|
|
the natural process of energy minimization of clusters of atoms, it works
|
||
|
|
well for similar problems with "funnel-like, but rugged" energy landscapes
|
||
|
|
[5]_.
|
||
|
|
|
||
|
|
As the step-taking, step acceptance, and minimization methods are all
|
||
|
|
customizable, this function can also be used to implement other two-phase
|
||
|
|
methods.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable ``f(x, *args)``
|
||
|
|
Function to be optimized. ``args`` can be passed as an optional item
|
||
|
|
in the dict `minimizer_kwargs`
|
||
|
|
x0 : array_like
|
||
|
|
Initial guess.
|
||
|
|
niter : integer, optional
|
||
|
|
The number of basin-hopping iterations. There will be a total of
|
||
|
|
``niter + 1`` runs of the local minimizer.
|
||
|
|
T : float, optional
|
||
|
|
The "temperature" parameter for the acceptance or rejection criterion.
|
||
|
|
Higher "temperatures" mean that larger jumps in function value will be
|
||
|
|
accepted. For best results `T` should be comparable to the
|
||
|
|
separation (in function value) between local minima.
|
||
|
|
stepsize : float, optional
|
||
|
|
Maximum step size for use in the random displacement.
|
||
|
|
minimizer_kwargs : dict, optional
|
||
|
|
Extra keyword arguments to be passed to the local minimizer
|
||
|
|
`scipy.optimize.minimize` Some important options could be:
|
||
|
|
|
||
|
|
method : str
|
||
|
|
The minimization method (e.g. ``"L-BFGS-B"``)
|
||
|
|
args : tuple
|
||
|
|
Extra arguments passed to the objective function (`func`) and
|
||
|
|
its derivatives (Jacobian, Hessian).
|
||
|
|
|
||
|
|
take_step : callable ``take_step(x)``, optional
|
||
|
|
Replace the default step-taking routine with this routine. The default
|
||
|
|
step-taking routine is a random displacement of the coordinates, but
|
||
|
|
other step-taking algorithms may be better for some systems.
|
||
|
|
`take_step` can optionally have the attribute ``take_step.stepsize``.
|
||
|
|
If this attribute exists, then `basinhopping` will adjust
|
||
|
|
``take_step.stepsize`` in order to try to optimize the global minimum
|
||
|
|
search.
|
||
|
|
accept_test : callable, ``accept_test(f_new=f_new, x_new=x_new, f_old=fold, x_old=x_old)``, optional
|
||
|
|
Define a test which will be used to judge whether to accept the
|
||
|
|
step. This will be used in addition to the Metropolis test based on
|
||
|
|
"temperature" `T`. The acceptable return values are True,
|
||
|
|
False, or ``"force accept"``. If any of the tests return False
|
||
|
|
then the step is rejected. If the latter, then this will override any
|
||
|
|
other tests in order to accept the step. This can be used, for example,
|
||
|
|
to forcefully escape from a local minimum that `basinhopping` is
|
||
|
|
trapped in.
|
||
|
|
callback : callable, ``callback(x, f, accept)``, optional
|
||
|
|
A callback function which will be called for all minima found. ``x``
|
||
|
|
and ``f`` are the coordinates and function value of the trial minimum,
|
||
|
|
and ``accept`` is whether that minimum was accepted. This can
|
||
|
|
be used, for example, to save the lowest N minima found. Also,
|
||
|
|
`callback` can be used to specify a user defined stop criterion by
|
||
|
|
optionally returning True to stop the `basinhopping` routine.
|
||
|
|
interval : integer, optional
|
||
|
|
interval for how often to update the `stepsize`
|
||
|
|
disp : bool, optional
|
||
|
|
Set to True to print status messages
|
||
|
|
niter_success : integer, optional
|
||
|
|
Stop the run if the global minimum candidate remains the same for this
|
||
|
|
number of iterations.
|
||
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
||
|
|
|
||
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
||
|
|
singleton is used.
|
||
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `seed`.
|
||
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance then
|
||
|
|
that instance is used.
|
||
|
|
Specify `seed` for repeatable minimizations. The random numbers
|
||
|
|
generated with this seed only affect the default Metropolis
|
||
|
|
`accept_test` and the default `take_step`. If you supply your own
|
||
|
|
`take_step` and `accept_test`, and these functions use random
|
||
|
|
number generation, then those functions are responsible for the state
|
||
|
|
of their random number generator.
|
||
|
|
target_accept_rate : float, optional
|
||
|
|
The target acceptance rate that is used to adjust the `stepsize`.
|
||
|
|
If the current acceptance rate is greater than the target,
|
||
|
|
then the `stepsize` is increased. Otherwise, it is decreased.
|
||
|
|
Range is (0, 1). Default is 0.5.
|
||
|
|
|
||
|
|
.. versionadded:: 1.8.0
|
||
|
|
|
||
|
|
stepwise_factor : float, optional
|
||
|
|
The `stepsize` is multiplied or divided by this stepwise factor upon
|
||
|
|
each update. Range is (0, 1). Default is 0.9.
|
||
|
|
|
||
|
|
.. versionadded:: 1.8.0
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : OptimizeResult
|
||
|
|
The optimization result represented as a `OptimizeResult` object.
|
||
|
|
Important attributes are: ``x`` the solution array, ``fun`` the value
|
||
|
|
of the function at the solution, and ``message`` which describes the
|
||
|
|
cause of the termination. The ``OptimizeResult`` object returned by the
|
||
|
|
selected minimizer at the lowest minimum is also contained within this
|
||
|
|
object and can be accessed through the ``lowest_optimization_result``
|
||
|
|
attribute. See `OptimizeResult` for a description of other attributes.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
minimize :
|
||
|
|
The local minimization function called once for each basinhopping step.
|
||
|
|
`minimizer_kwargs` is passed to this routine.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Basin-hopping is a stochastic algorithm which attempts to find the global
|
||
|
|
minimum of a smooth scalar function of one or more variables [1]_ [2]_ [3]_
|
||
|
|
[4]_. The algorithm in its current form was described by David Wales and
|
||
|
|
Jonathan Doye [2]_ http://www-wales.ch.cam.ac.uk/.
|
||
|
|
|
||
|
|
The algorithm is iterative with each cycle composed of the following
|
||
|
|
features
|
||
|
|
|
||
|
|
1) random perturbation of the coordinates
|
||
|
|
|
||
|
|
2) local minimization
|
||
|
|
|
||
|
|
3) accept or reject the new coordinates based on the minimized function
|
||
|
|
value
|
||
|
|
|
||
|
|
The acceptance test used here is the Metropolis criterion of standard Monte
|
||
|
|
Carlo algorithms, although there are many other possibilities [3]_.
|
||
|
|
|
||
|
|
This global minimization method has been shown to be extremely efficient
|
||
|
|
for a wide variety of problems in physics and chemistry. It is
|
||
|
|
particularly useful when the function has many minima separated by large
|
||
|
|
barriers. See the `Cambridge Cluster Database
|
||
|
|
<https://www-wales.ch.cam.ac.uk/CCD.html>`_ for databases of molecular
|
||
|
|
systems that have been optimized primarily using basin-hopping. This
|
||
|
|
database includes minimization problems exceeding 300 degrees of freedom.
|
||
|
|
|
||
|
|
See the free software program `GMIN <https://www-wales.ch.cam.ac.uk/GMIN>`_
|
||
|
|
for a Fortran implementation of basin-hopping. This implementation has many
|
||
|
|
variations of the procedure described above, including more
|
||
|
|
advanced step taking algorithms and alternate acceptance criterion.
|
||
|
|
|
||
|
|
For stochastic global optimization there is no way to determine if the true
|
||
|
|
global minimum has actually been found. Instead, as a consistency check,
|
||
|
|
the algorithm can be run from a number of different random starting points
|
||
|
|
to ensure the lowest minimum found in each example has converged to the
|
||
|
|
global minimum. For this reason, `basinhopping` will by default simply
|
||
|
|
run for the number of iterations `niter` and return the lowest minimum
|
||
|
|
found. It is left to the user to ensure that this is in fact the global
|
||
|
|
minimum.
|
||
|
|
|
||
|
|
Choosing `stepsize`: This is a crucial parameter in `basinhopping` and
|
||
|
|
depends on the problem being solved. The step is chosen uniformly in the
|
||
|
|
region from x0-stepsize to x0+stepsize, in each dimension. Ideally, it
|
||
|
|
should be comparable to the typical separation (in argument values) between
|
||
|
|
local minima of the function being optimized. `basinhopping` will, by
|
||
|
|
default, adjust `stepsize` to find an optimal value, but this may take
|
||
|
|
many iterations. You will get quicker results if you set a sensible
|
||
|
|
initial value for ``stepsize``.
|
||
|
|
|
||
|
|
Choosing `T`: The parameter `T` is the "temperature" used in the
|
||
|
|
Metropolis criterion. Basinhopping steps are always accepted if
|
||
|
|
``func(xnew) < func(xold)``. Otherwise, they are accepted with
|
||
|
|
probability::
|
||
|
|
|
||
|
|
exp( -(func(xnew) - func(xold)) / T )
|
||
|
|
|
||
|
|
So, for best results, `T` should to be comparable to the typical
|
||
|
|
difference (in function values) between local minima. (The height of
|
||
|
|
"walls" between local minima is irrelevant.)
|
||
|
|
|
||
|
|
If `T` is 0, the algorithm becomes Monotonic Basin-Hopping, in which all
|
||
|
|
steps that increase energy are rejected.
|
||
|
|
|
||
|
|
.. versionadded:: 0.12.0
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Wales, David J. 2003, Energy Landscapes, Cambridge University Press,
|
||
|
|
Cambridge, UK.
|
||
|
|
.. [2] Wales, D J, and Doye J P K, Global Optimization by Basin-Hopping and
|
||
|
|
the Lowest Energy Structures of Lennard-Jones Clusters Containing up to
|
||
|
|
110 Atoms. Journal of Physical Chemistry A, 1997, 101, 5111.
|
||
|
|
.. [3] Li, Z. and Scheraga, H. A., Monte Carlo-minimization approach to the
|
||
|
|
multiple-minima problem in protein folding, Proc. Natl. Acad. Sci. USA,
|
||
|
|
1987, 84, 6611.
|
||
|
|
.. [4] Wales, D. J. and Scheraga, H. A., Global optimization of clusters,
|
||
|
|
crystals, and biomolecules, Science, 1999, 285, 1368.
|
||
|
|
.. [5] Olson, B., Hashmi, I., Molloy, K., and Shehu1, A., Basin Hopping as
|
||
|
|
a General and Versatile Optimization Framework for the Characterization
|
||
|
|
of Biological Macromolecules, Advances in Artificial Intelligence,
|
||
|
|
Volume 2012 (2012), Article ID 674832, :doi:`10.1155/2012/674832`
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
The following example is a 1-D minimization problem, with many
|
||
|
|
local minima superimposed on a parabola.
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from scipy.optimize import basinhopping
|
||
|
|
>>> func = lambda x: np.cos(14.5 * x - 0.3) + (x + 0.2) * x
|
||
|
|
>>> x0 = [1.]
|
||
|
|
|
||
|
|
Basinhopping, internally, uses a local minimization algorithm. We will use
|
||
|
|
the parameter `minimizer_kwargs` to tell basinhopping which algorithm to
|
||
|
|
use and how to set up that minimizer. This parameter will be passed to
|
||
|
|
`scipy.optimize.minimize`.
|
||
|
|
|
||
|
|
>>> minimizer_kwargs = {"method": "BFGS"}
|
||
|
|
>>> ret = basinhopping(func, x0, minimizer_kwargs=minimizer_kwargs,
|
||
|
|
... niter=200)
|
||
|
|
>>> print("global minimum: x = %.4f, f(x) = %.4f" % (ret.x, ret.fun))
|
||
|
|
global minimum: x = -0.1951, f(x) = -1.0009
|
||
|
|
|
||
|
|
Next consider a 2-D minimization problem. Also, this time, we
|
||
|
|
will use gradient information to significantly speed up the search.
|
||
|
|
|
||
|
|
>>> def func2d(x):
|
||
|
|
... f = np.cos(14.5 * x[0] - 0.3) + (x[1] + 0.2) * x[1] + (x[0] +
|
||
|
|
... 0.2) * x[0]
|
||
|
|
... df = np.zeros(2)
|
||
|
|
... df[0] = -14.5 * np.sin(14.5 * x[0] - 0.3) + 2. * x[0] + 0.2
|
||
|
|
... df[1] = 2. * x[1] + 0.2
|
||
|
|
... return f, df
|
||
|
|
|
||
|
|
We'll also use a different local minimization algorithm. Also, we must tell
|
||
|
|
the minimizer that our function returns both energy and gradient (Jacobian).
|
||
|
|
|
||
|
|
>>> minimizer_kwargs = {"method":"L-BFGS-B", "jac":True}
|
||
|
|
>>> x0 = [1.0, 1.0]
|
||
|
|
>>> ret = basinhopping(func2d, x0, minimizer_kwargs=minimizer_kwargs,
|
||
|
|
... niter=200)
|
||
|
|
>>> print("global minimum: x = [%.4f, %.4f], f(x) = %.4f" % (ret.x[0],
|
||
|
|
... ret.x[1],
|
||
|
|
... ret.fun))
|
||
|
|
global minimum: x = [-0.1951, -0.1000], f(x) = -1.0109
|
||
|
|
|
||
|
|
Here is an example using a custom step-taking routine. Imagine you want
|
||
|
|
the first coordinate to take larger steps than the rest of the coordinates.
|
||
|
|
This can be implemented like so:
|
||
|
|
|
||
|
|
>>> class MyTakeStep:
|
||
|
|
... def __init__(self, stepsize=0.5):
|
||
|
|
... self.stepsize = stepsize
|
||
|
|
... self.rng = np.random.default_rng()
|
||
|
|
... def __call__(self, x):
|
||
|
|
... s = self.stepsize
|
||
|
|
... x[0] += self.rng.uniform(-2.*s, 2.*s)
|
||
|
|
... x[1:] += self.rng.uniform(-s, s, x[1:].shape)
|
||
|
|
... return x
|
||
|
|
|
||
|
|
Since ``MyTakeStep.stepsize`` exists basinhopping will adjust the magnitude
|
||
|
|
of `stepsize` to optimize the search. We'll use the same 2-D function as
|
||
|
|
before
|
||
|
|
|
||
|
|
>>> mytakestep = MyTakeStep()
|
||
|
|
>>> ret = basinhopping(func2d, x0, minimizer_kwargs=minimizer_kwargs,
|
||
|
|
... niter=200, take_step=mytakestep)
|
||
|
|
>>> print("global minimum: x = [%.4f, %.4f], f(x) = %.4f" % (ret.x[0],
|
||
|
|
... ret.x[1],
|
||
|
|
... ret.fun))
|
||
|
|
global minimum: x = [-0.1951, -0.1000], f(x) = -1.0109
|
||
|
|
|
||
|
|
Now, let's do an example using a custom callback function which prints the
|
||
|
|
value of every minimum found
|
||
|
|
|
||
|
|
>>> def print_fun(x, f, accepted):
|
||
|
|
... print("at minimum %.4f accepted %d" % (f, int(accepted)))
|
||
|
|
|
||
|
|
We'll run it for only 10 basinhopping steps this time.
|
||
|
|
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> ret = basinhopping(func2d, x0, minimizer_kwargs=minimizer_kwargs,
|
||
|
|
... niter=10, callback=print_fun, seed=rng)
|
||
|
|
at minimum 0.4159 accepted 1
|
||
|
|
at minimum -0.4317 accepted 1
|
||
|
|
at minimum -1.0109 accepted 1
|
||
|
|
at minimum -0.9073 accepted 1
|
||
|
|
at minimum -0.4317 accepted 0
|
||
|
|
at minimum -0.1021 accepted 1
|
||
|
|
at minimum -0.7425 accepted 1
|
||
|
|
at minimum -0.9073 accepted 1
|
||
|
|
at minimum -0.4317 accepted 0
|
||
|
|
at minimum -0.7425 accepted 1
|
||
|
|
at minimum -0.9073 accepted 1
|
||
|
|
|
||
|
|
The minimum at -1.0109 is actually the global minimum, found already on the
|
||
|
|
8th iteration.
|
||
|
|
|
||
|
|
""" # numpy/numpydoc#87 # noqa: E501
|
||
|
|
if target_accept_rate <= 0. or target_accept_rate >= 1.:
|
||
|
|
raise ValueError('target_accept_rate has to be in range (0, 1)')
|
||
|
|
if stepwise_factor <= 0. or stepwise_factor >= 1.:
|
||
|
|
raise ValueError('stepwise_factor has to be in range (0, 1)')
|
||
|
|
|
||
|
|
x0 = np.array(x0)
|
||
|
|
|
||
|
|
# set up the np.random generator
|
||
|
|
rng = check_random_state(seed)
|
||
|
|
|
||
|
|
# set up minimizer
|
||
|
|
if minimizer_kwargs is None:
|
||
|
|
minimizer_kwargs = dict()
|
||
|
|
wrapped_minimizer = MinimizerWrapper(scipy.optimize.minimize, func,
|
||
|
|
**minimizer_kwargs)
|
||
|
|
|
||
|
|
# set up step-taking algorithm
|
||
|
|
if take_step is not None:
|
||
|
|
if not callable(take_step):
|
||
|
|
raise TypeError("take_step must be callable")
|
||
|
|
# if take_step.stepsize exists then use AdaptiveStepsize to control
|
||
|
|
# take_step.stepsize
|
||
|
|
if hasattr(take_step, "stepsize"):
|
||
|
|
take_step_wrapped = AdaptiveStepsize(
|
||
|
|
take_step, interval=interval,
|
||
|
|
accept_rate=target_accept_rate,
|
||
|
|
factor=stepwise_factor,
|
||
|
|
verbose=disp)
|
||
|
|
else:
|
||
|
|
take_step_wrapped = take_step
|
||
|
|
else:
|
||
|
|
# use default
|
||
|
|
displace = RandomDisplacement(stepsize=stepsize, random_gen=rng)
|
||
|
|
take_step_wrapped = AdaptiveStepsize(displace, interval=interval,
|
||
|
|
accept_rate=target_accept_rate,
|
||
|
|
factor=stepwise_factor,
|
||
|
|
verbose=disp)
|
||
|
|
|
||
|
|
# set up accept tests
|
||
|
|
accept_tests = []
|
||
|
|
if accept_test is not None:
|
||
|
|
if not callable(accept_test):
|
||
|
|
raise TypeError("accept_test must be callable")
|
||
|
|
accept_tests = [accept_test]
|
||
|
|
|
||
|
|
# use default
|
||
|
|
metropolis = Metropolis(T, random_gen=rng)
|
||
|
|
accept_tests.append(metropolis)
|
||
|
|
|
||
|
|
if niter_success is None:
|
||
|
|
niter_success = niter + 2
|
||
|
|
|
||
|
|
bh = BasinHoppingRunner(x0, wrapped_minimizer, take_step_wrapped,
|
||
|
|
accept_tests, disp=disp)
|
||
|
|
|
||
|
|
# The wrapped minimizer is called once during construction of
|
||
|
|
# BasinHoppingRunner, so run the callback
|
||
|
|
if callable(callback):
|
||
|
|
callback(bh.storage.minres.x, bh.storage.minres.fun, True)
|
||
|
|
|
||
|
|
# start main iteration loop
|
||
|
|
count, i = 0, 0
|
||
|
|
message = ["requested number of basinhopping iterations completed"
|
||
|
|
" successfully"]
|
||
|
|
for i in range(niter):
|
||
|
|
new_global_min = bh.one_cycle()
|
||
|
|
|
||
|
|
if callable(callback):
|
||
|
|
# should we pass a copy of x?
|
||
|
|
val = callback(bh.xtrial, bh.energy_trial, bh.accept)
|
||
|
|
if val is not None:
|
||
|
|
if val:
|
||
|
|
message = ["callback function requested stop early by"
|
||
|
|
"returning True"]
|
||
|
|
break
|
||
|
|
|
||
|
|
count += 1
|
||
|
|
if new_global_min:
|
||
|
|
count = 0
|
||
|
|
elif count > niter_success:
|
||
|
|
message = ["success condition satisfied"]
|
||
|
|
break
|
||
|
|
|
||
|
|
# prepare return object
|
||
|
|
res = bh.res
|
||
|
|
res.lowest_optimization_result = bh.storage.get_lowest()
|
||
|
|
res.x = np.copy(res.lowest_optimization_result.x)
|
||
|
|
res.fun = res.lowest_optimization_result.fun
|
||
|
|
res.message = message
|
||
|
|
res.nit = i + 1
|
||
|
|
res.success = res.lowest_optimization_result.success
|
||
|
|
return res
|