"""
basinhopping: The basinhopping global optimization algorithm
"""
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy import cos, sin
import scipy.optimize
import collections
__all__ = ['basinhopping']
class Storage(object):
"""
Class used to store the lowest energy structure
"""
def __init__(self, x, f):
self._add(x, f)
def _add(self, x, f):
self.x = np.copy(x)
self.f = f
def update(self, x, f):
if f < self.f:
self._add(x, f)
return True
else:
return False
def get_lowest(self):
return self.x, self.f
class BasinHoppingRunner(object):
"""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
To 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 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.
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
if self.disp:
print("basinhopping step %d: f %g" % (self.nstep, self.energy))
# initialize storage class
self.storage = Storage(self.x, self.energy)
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,
# than reject the step. If any test returns the special value, the
# string 'force accept', accept the step regardless. This can be used
# to forcefully escape from a local minima if normal basin hopping
# steps are not sufficient.
accept = True
for test in self.accept_tests:
testres = test(f_new=energy_after_quench, x_new=x_after_quench,
f_old=self.energy, x_old=self.x)
if isinstance(testres, bool):
if not testres:
accept = False
elif isinstance(testres, str):
if testres == "force accept":
accept = True
break
else:
raise ValueError("accept test must return bool or string "
"'force accept'. Type is", type(testres))
else:
raise ValueError("accept test must return bool or string "
"'force accept'. Type is", type(testres))
# 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 x_after_quench, energy_after_quench, accept
def one_cycle(self):
"""Do one cycle of the basinhopping algorithm
"""
self.nstep += 1
new_global_min = False
xtrial, energy_trial, accept = self._monte_carlo_step()
if accept:
self.energy = energy_trial
self.x = np.copy(xtrial)
new_global_min = self.storage.update(self.x, self.energy)
# print some information
if self.disp:
self.print_report(energy_trial, 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 = xtrial
self.energy_trial = energy_trial
self.accept = accept
return new_global_min
def print_report(self, energy_trial, accept):
"""print a status update"""
xlowest, energy_lowest = 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, energy_lowest))
class AdaptiveStepsize(object):
"""
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" % (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(object):
"""
Add a random displacement of maximum size, stepsize, to the coordinates
update x inplace
"""
def __init__(self, stepsize=0.5):
self.stepsize = stepsize
def __call__(self, x):
x += np.random.uniform(-self.stepsize, self.stepsize, np.shape(x))
return x
class MinimizerWrapper(object):
"""
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(object):
"""
Metropolis acceptance criterion
"""
def __init__(self, T):
self.beta = 1.0 / T
def accept_reject(self, energy_new, energy_old):
w = min(1.0, np.exp(-(energy_new - energy_old) * self.beta))
rand = np.random.rand()
return w >= rand
def __call__(self, **kwargs):
"""
f_new and f_old are mandatory in kwargs
"""
return bool(self.accept_reject(kwargs["f_new"],
kwargs["f_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):
"""
Find the global minimum of a function using the basin-hopping algorithm
.. versionadded:: 0.12.0
Parameters
----------
func : callable ``f(x, *args)``
Function to be optimized. ``args`` can be passed as an optional item
in the dict ``minimizer_kwargs``
x0 : ndarray
Initial guess.
niter : integer, optional
The number of basin hopping iterations
T : float, optional
The "temperature" parameter for the accept or reject 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
initial step size for use in the random displacement.
minimizer_kwargs : dict, optional
Extra keyword arguments to be passed to the 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 or not 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 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 minimum found. ``x``
and ``f`` are the coordinates and function value of the trial minima,
and ``accept`` is whether or not that minima 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.
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. 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
http://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 (http://www-wales.ch.cam.ac.uk/GMIN) for
a Fortran implementation of basin-hopping. This implementation has many
different 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. Ideally it should be comparable to
the typical separation 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 value for ``stepsize``.
Choosing ``T``: The parameter ``T`` is the temperature used in the
metropolis criterion. Basinhopping steps are accepted with probability
``1`` if ``func(xnew) < func(xold)``, or otherwise 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.
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.
Examples
--------
The following example is a one-dimensional minimization problem, with many
local minima superimposed on a parabola.
>>> func = lambda x: 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(x0) = %.4f" % (ret.x, ret.fun))
global minimum: x = -0.1951, f(x0) = -1.0009
Next consider a two-dimensional minimization problem. Also, this time we
will use gradient information to significantly speed up the search.
>>> def func2d(x):
... f = 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 * 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(x0) = %.4f" % (ret.x[0],
... ret.x[1],
... ret.fun))
global minimum: x = [-0.1951, -0.1000], f(x0) = -1.0109
Here is an example using a custom step taking routine. Imagine you want
the first coordinate to take larger steps then the rest of the coordinates.
This can be implemented like so:
>>> class MyTakeStep(object):
... def __init__(self, stepsize=0.5):
... self.stepsize = stepsize
... def __call__(self, x):
... s = self.stepsize
... x[0] += np.random.uniform(-2.*s, 2.*s)
... x[1:] += np.random.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(x0) = %.4f" % (ret.x[0],
... ret.x[1],
... ret.fun))
global minimum: x = [-0.1951, -0.1000], f(x0) = -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 minima %.4f accepted %d" % (f, int(accepted)))
We'll run it for only 10 basinhopping steps this time.
>>> np.random.seed(1)
>>> ret = basinhopping(func2d, x0, minimizer_kwargs=minimizer_kwargs,
... niter=10, callback=print_fun)
at minima 0.4159 accepted 1
at minima -0.9073 accepted 1
at minima -0.1021 accepted 1
at minima -0.1021 accepted 1
at minima 0.9102 accepted 1
at minima 0.9102 accepted 1
at minima 2.2945 accepted 0
at minima -0.1021 accepted 1
at minima -1.0109 accepted 1
at minima -1.0109 accepted 1
The minima at -1.0109 is actually the global minimum, found already on the
8th iteration.
Now let's implement bounds on the problem using a custom ``accept_test``:
>>> class MyBounds(object):
... def __init__(self, xmax=[1.1,1.1], xmin=[-1.1,-1.1] ):
... self.xmax = np.array(xmax)
... self.xmin = np.array(xmin)
... def __call__(self, **kwargs):
... x = kwargs["x_new"]
... tmax = bool(np.all(x <= self.xmax))
... tmin = bool(np.all(x >= self.xmin))
... return tmax and tmin
>>> mybounds = MyBounds()
>>> ret = basinhopping(func2d, x0, minimizer_kwargs=minimizer_kwargs,
... niter=10, accept_test=mybounds)
"""
x0 = np.array(x0)
# 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 isinstance(take_step, collections.Callable):
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,
verbose=disp)
else:
take_step_wrapped = take_step
else:
# use default
displace = RandomDisplacement(stepsize=stepsize)
take_step_wrapped = AdaptiveStepsize(displace, interval=interval,
verbose=disp)
# set up accept tests
if accept_test is not None:
if not isinstance(accept_test, collections.Callable):
raise TypeError("accept_test must be callable")
accept_tests = [accept_test]
else:
accept_tests = []
# use default
metropolis = Metropolis(T)
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)
# start main iteration loop
count = 0
message = ["requested number of basinhopping iterations completed"
" successfully"]
for i in range(niter):
new_global_min = bh.one_cycle()
if isinstance(callback, collections.Callable):
# 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
lowest = bh.storage.get_lowest()
res = bh.res
res.x = np.copy(lowest[0])
res.fun = lowest[1]
res.message = message
res.nit = i + 1
return res
def _test_func2d_nograd(x):
f = (cos(14.5 * x[0] - 0.3) + (x[1] + 0.2) * x[1] + (x[0] + 0.2) * x[0]
+ 1.010876184442655)
return f
def _test_func2d(x):
f = (cos(14.5 * x[0] - 0.3) + (x[0] + 0.2) * x[0] + cos(14.5 * x[1] -
0.3) + (x[1] + 0.2) * x[1] + x[0] * x[1] + 1.963879482144252)
df = np.zeros(2)
df[0] = -14.5 * sin(14.5 * x[0] - 0.3) + 2. * x[0] + 0.2 + x[1]
df[1] = -14.5 * sin(14.5 * x[1] - 0.3) + 2. * x[1] + 0.2 + x[0]
return f, df
if __name__ == "__main__":
print("\n\nminimize a 2d function without gradient")
# minimum expected at ~[-0.195, -0.1]
kwargs = {"method": "L-BFGS-B"}
x0 = np.array([1.0, 1.])
scipy.optimize.minimize(_test_func2d_nograd, x0, **kwargs)
ret = basinhopping(_test_func2d_nograd, x0, minimizer_kwargs=kwargs,
niter=200, disp=False)
print("minimum expected at func([-0.195, -0.1]) = 0.0")
print(ret)
print("\n\ntry a harder 2d problem")
kwargs = {"method": "L-BFGS-B", "jac": True}
x0 = np.array([1.0, 1.0])
ret = basinhopping(_test_func2d, x0, minimizer_kwargs=kwargs, niter=200,
disp=False)
print("minimum expected at ~, func([-0.19415263, -0.19415263]) = 0")
print(ret)