""" 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)