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""" differential_evolution: The differential evolution global optimization algorithm Added by Andrew Nelson 2014 """
maxiter=1000, popsize=15, tol=0.01, mutation=(0.5, 1), recombination=0.7, seed=None, callback=None, disp=False, polish=True, init='latinhypercube', atol=0): """Finds the global minimum of a multivariate function. Differential Evolution is stochastic in nature (does not use gradient methods) to find the minimium, and can search large areas of candidate space, but often requires larger numbers of function evaluations than conventional gradient based techniques.
The algorithm is due to Storn and Price [1]_.
Parameters ---------- func : callable The objective function to be minimized. Must be in the form ``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array and ``args`` is a tuple of any additional fixed parameters needed to completely specify the function. bounds : sequence Bounds for variables. ``(min, max)`` pairs for each element in ``x``, defining the lower and upper bounds for the optimizing argument of `func`. It is required to have ``len(bounds) == len(x)``. ``len(bounds)`` is used to determine the number of parameters in ``x``. args : tuple, optional Any additional fixed parameters needed to completely specify the objective function. strategy : str, optional The differential evolution strategy to use. Should be one of:
- 'best1bin' - 'best1exp' - 'rand1exp' - 'randtobest1exp' - 'currenttobest1exp' - 'best2exp' - 'rand2exp' - 'randtobest1bin' - 'currenttobest1bin' - 'best2bin' - 'rand2bin' - 'rand1bin'
The default is 'best1bin'. maxiter : int, optional The maximum number of generations over which the entire population is evolved. The maximum number of function evaluations (with no polishing) is: ``(maxiter + 1) * popsize * len(x)`` popsize : int, optional A multiplier for setting the total population size. The population has ``popsize * len(x)`` individuals (unless the initial population is supplied via the `init` keyword). tol : float, optional Relative tolerance for convergence, the solving stops when ``np.std(pop) <= atol + tol * np.abs(np.mean(population_energies))``, where and `atol` and `tol` are the absolute and relative tolerance respectively. mutation : float or tuple(float, float), optional The mutation constant. In the literature this is also known as differential weight, being denoted by F. If specified as a float it should be in the range [0, 2]. If specified as a tuple ``(min, max)`` dithering is employed. Dithering randomly changes the mutation constant on a generation by generation basis. The mutation constant for that generation is taken from ``U[min, max)``. Dithering can help speed convergence significantly. Increasing the mutation constant increases the search radius, but will slow down convergence. recombination : float, optional The recombination constant, should be in the range [0, 1]. In the literature this is also known as the crossover probability, being denoted by CR. Increasing this value allows a larger number of mutants to progress into the next generation, but at the risk of population stability. seed : int or `np.random.RandomState`, optional If `seed` is not specified the `np.RandomState` singleton is used. If `seed` is an int, a new `np.random.RandomState` instance is used, seeded with seed. If `seed` is already a `np.random.RandomState instance`, then that `np.random.RandomState` instance is used. Specify `seed` for repeatable minimizations. disp : bool, optional Display status messages callback : callable, `callback(xk, convergence=val)`, optional A function to follow the progress of the minimization. ``xk`` is the current value of ``x0``. ``val`` represents the fractional value of the population convergence. When ``val`` is greater than one the function halts. If callback returns `True`, then the minimization is halted (any polishing is still carried out). polish : bool, optional If True (default), then `scipy.optimize.minimize` with the `L-BFGS-B` method is used to polish the best population member at the end, which can improve the minimization slightly. init : str or array-like, optional Specify which type of population initialization is performed. Should be one of:
- 'latinhypercube' - 'random' - array specifying the initial population. The array should have shape ``(M, len(x))``, where len(x) is the number of parameters. `init` is clipped to `bounds` before use.
The default is 'latinhypercube'. Latin Hypercube sampling tries to maximize coverage of the available parameter space. 'random' initializes the population randomly - this has the drawback that clustering can occur, preventing the whole of parameter space being covered. Use of an array to specify a population subset could be used, for example, to create a tight bunch of initial guesses in an location where the solution is known to exist, thereby reducing time for convergence. atol : float, optional Absolute tolerance for convergence, the solving stops when ``np.std(pop) <= atol + tol * np.abs(np.mean(population_energies))``, where and `atol` and `tol` are the absolute and relative tolerance respectively.
Returns ------- res : OptimizeResult The optimization result represented as a `OptimizeResult` object. Important attributes are: ``x`` the solution array, ``success`` a Boolean flag indicating if the optimizer exited successfully and ``message`` which describes the cause of the termination. See `OptimizeResult` for a description of other attributes. If `polish` was employed, and a lower minimum was obtained by the polishing, then OptimizeResult also contains the ``jac`` attribute.
Notes ----- Differential evolution is a stochastic population based method that is useful for global optimization problems. At each pass through the population the algorithm mutates each candidate solution by mixing with other candidate solutions to create a trial candidate. There are several strategies [2]_ for creating trial candidates, which suit some problems more than others. The 'best1bin' strategy is a good starting point for many systems. In this strategy two members of the population are randomly chosen. Their difference is used to mutate the best member (the `best` in `best1bin`), :math:`b_0`, so far:
.. math::
b' = b_0 + mutation * (population[rand0] - population[rand1])
A trial vector is then constructed. Starting with a randomly chosen 'i'th parameter the trial is sequentially filled (in modulo) with parameters from `b'` or the original candidate. The choice of whether to use `b'` or the original candidate is made with a binomial distribution (the 'bin' in 'best1bin') - a random number in [0, 1) is generated. If this number is less than the `recombination` constant then the parameter is loaded from `b'`, otherwise it is loaded from the original candidate. The final parameter is always loaded from `b'`. Once the trial candidate is built its fitness is assessed. If the trial is better than the original candidate then it takes its place. If it is also better than the best overall candidate it also replaces that. To improve your chances of finding a global minimum use higher `popsize` values, with higher `mutation` and (dithering), but lower `recombination` values. This has the effect of widening the search radius, but slowing convergence.
.. versionadded:: 0.15.0
Examples -------- Let us consider the problem of minimizing the Rosenbrock function. This function is implemented in `rosen` in `scipy.optimize`.
>>> from scipy.optimize import rosen, differential_evolution >>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)] >>> result = differential_evolution(rosen, bounds) >>> result.x, result.fun (array([1., 1., 1., 1., 1.]), 1.9216496320061384e-19)
Next find the minimum of the Ackley function (http://en.wikipedia.org/wiki/Test_functions_for_optimization).
>>> from scipy.optimize import differential_evolution >>> import numpy as np >>> def ackley(x): ... arg1 = -0.2 * np.sqrt(0.5 * (x[0] ** 2 + x[1] ** 2)) ... arg2 = 0.5 * (np.cos(2. * np.pi * x[0]) + np.cos(2. * np.pi * x[1])) ... return -20. * np.exp(arg1) - np.exp(arg2) + 20. + np.e >>> bounds = [(-5, 5), (-5, 5)] >>> result = differential_evolution(ackley, bounds) >>> result.x, result.fun (array([ 0., 0.]), 4.4408920985006262e-16)
References ---------- .. [1] Storn, R and Price, K, Differential Evolution - a Simple and Efficient Heuristic for Global Optimization over Continuous Spaces, Journal of Global Optimization, 1997, 11, 341 - 359. .. [2] http://www1.icsi.berkeley.edu/~storn/code.html .. [3] http://en.wikipedia.org/wiki/Differential_evolution """
solver = DifferentialEvolutionSolver(func, bounds, args=args, strategy=strategy, maxiter=maxiter, popsize=popsize, tol=tol, mutation=mutation, recombination=recombination, seed=seed, polish=polish, callback=callback, disp=disp, init=init, atol=atol) return solver.solve()
"""This class implements the differential evolution solver
Parameters ---------- func : callable The objective function to be minimized. Must be in the form ``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array and ``args`` is a tuple of any additional fixed parameters needed to completely specify the function. bounds : sequence Bounds for variables. ``(min, max)`` pairs for each element in ``x``, defining the lower and upper bounds for the optimizing argument of `func`. It is required to have ``len(bounds) == len(x)``. ``len(bounds)`` is used to determine the number of parameters in ``x``. args : tuple, optional Any additional fixed parameters needed to completely specify the objective function. strategy : str, optional The differential evolution strategy to use. Should be one of:
- 'best1bin' - 'best1exp' - 'rand1exp' - 'randtobest1exp' - 'currenttobest1exp' - 'best2exp' - 'rand2exp' - 'randtobest1bin' - 'currenttobest1bin' - 'best2bin' - 'rand2bin' - 'rand1bin'
The default is 'best1bin'
maxiter : int, optional The maximum number of generations over which the entire population is evolved. The maximum number of function evaluations (with no polishing) is: ``(maxiter + 1) * popsize * len(x)`` popsize : int, optional A multiplier for setting the total population size. The population has ``popsize * len(x)`` individuals (unless the initial population is supplied via the `init` keyword). tol : float, optional Relative tolerance for convergence, the solving stops when ``np.std(pop) <= atol + tol * np.abs(np.mean(population_energies))``, where and `atol` and `tol` are the absolute and relative tolerance respectively. mutation : float or tuple(float, float), optional The mutation constant. In the literature this is also known as differential weight, being denoted by F. If specified as a float it should be in the range [0, 2]. If specified as a tuple ``(min, max)`` dithering is employed. Dithering randomly changes the mutation constant on a generation by generation basis. The mutation constant for that generation is taken from U[min, max). Dithering can help speed convergence significantly. Increasing the mutation constant increases the search radius, but will slow down convergence. recombination : float, optional The recombination constant, should be in the range [0, 1]. In the literature this is also known as the crossover probability, being denoted by CR. Increasing this value allows a larger number of mutants to progress into the next generation, but at the risk of population stability. seed : int or `np.random.RandomState`, optional If `seed` is not specified the `np.random.RandomState` singleton is used. If `seed` is an int, a new `np.random.RandomState` instance is used, seeded with `seed`. If `seed` is already a `np.random.RandomState` instance, then that `np.random.RandomState` instance is used. Specify `seed` for repeatable minimizations. disp : bool, optional Display status messages callback : callable, `callback(xk, convergence=val)`, optional A function to follow the progress of the minimization. ``xk`` is the current value of ``x0``. ``val`` represents the fractional value of the population convergence. When ``val`` is greater than one the function halts. If callback returns `True`, then the minimization is halted (any polishing is still carried out). polish : bool, optional If True, then `scipy.optimize.minimize` with the `L-BFGS-B` method is used to polish the best population member at the end. This requires a few more function evaluations. maxfun : int, optional Set the maximum number of function evaluations. However, it probably makes more sense to set `maxiter` instead. init : str or array-like, optional Specify which type of population initialization is performed. Should be one of:
- 'latinhypercube' - 'random' - array specifying the initial population. The array should have shape ``(M, len(x))``, where len(x) is the number of parameters. `init` is clipped to `bounds` before use.
The default is 'latinhypercube'. Latin Hypercube sampling tries to maximize coverage of the available parameter space. 'random' initializes the population randomly - this has the drawback that clustering can occur, preventing the whole of parameter space being covered. Use of an array to specify a population could be used, for example, to create a tight bunch of initial guesses in an location where the solution is known to exist, thereby reducing time for convergence. atol : float, optional Absolute tolerance for convergence, the solving stops when ``np.std(pop) <= atol + tol * np.abs(np.mean(population_energies))``, where and `atol` and `tol` are the absolute and relative tolerance respectively. """
# Dispatch of mutation strategy method (binomial or exponential). 'randtobest1bin': '_randtobest1', 'currenttobest1bin': '_currenttobest1', 'best2bin': '_best2', 'rand2bin': '_rand2', 'rand1bin': '_rand1'} 'rand1exp': '_rand1', 'randtobest1exp': '_randtobest1', 'currenttobest1exp': '_currenttobest1', 'best2exp': '_best2', 'rand2exp': '_rand2'}
"'latinhypercube' or 'random', or an array of shape " "(M, N) where N is the number of parameters and M>5")
strategy='best1bin', maxiter=1000, popsize=15, tol=0.01, mutation=(0.5, 1), recombination=0.7, seed=None, maxfun=np.inf, callback=None, disp=False, polish=True, init='latinhypercube', atol=0):
if strategy in self._binomial: self.mutation_func = getattr(self, self._binomial[strategy]) elif strategy in self._exponential: self.mutation_func = getattr(self, self._exponential[strategy]) else: raise ValueError("Please select a valid mutation strategy") self.strategy = strategy
self.callback = callback self.polish = polish
# relative and absolute tolerances for convergence self.tol, self.atol = tol, atol
# Mutation constant should be in [0, 2). If specified as a sequence # then dithering is performed. self.scale = mutation if (not np.all(np.isfinite(mutation)) or np.any(np.array(mutation) >= 2) or np.any(np.array(mutation) < 0)): raise ValueError('The mutation constant must be a float in ' 'U[0, 2), or specified as a tuple(min, max)' ' where min < max and min, max are in U[0, 2).')
self.dither = None if hasattr(mutation, '__iter__') and len(mutation) > 1: self.dither = [mutation[0], mutation[1]] self.dither.sort()
self.cross_over_probability = recombination
self.func = func self.args = args
# convert tuple of lower and upper bounds to limits # [(low_0, high_0), ..., (low_n, high_n] # -> [[low_0, ..., low_n], [high_0, ..., high_n]] self.limits = np.array(bounds, dtype='float').T if (np.size(self.limits, 0) != 2 or not np.all(np.isfinite(self.limits))): raise ValueError('bounds should be a sequence containing ' 'real valued (min, max) pairs for each value' ' in x')
if maxiter is None: # the default used to be None maxiter = 1000 self.maxiter = maxiter if maxfun is None: # the default used to be None maxfun = np.inf self.maxfun = maxfun
# population is scaled to between [0, 1]. # We have to scale between parameter <-> population # save these arguments for _scale_parameter and # _unscale_parameter. This is an optimization self.__scale_arg1 = 0.5 * (self.limits[0] + self.limits[1]) self.__scale_arg2 = np.fabs(self.limits[0] - self.limits[1])
self.parameter_count = np.size(self.limits, 1)
self.random_number_generator = check_random_state(seed)
# default population initialization is a latin hypercube design, but # there are other population initializations possible. # the minimum is 5 because 'best2bin' requires a population that's at # least 5 long self.num_population_members = max(5, popsize * self.parameter_count)
self.population_shape = (self.num_population_members, self.parameter_count)
self._nfev = 0 if isinstance(init, string_types): if init == 'latinhypercube': self.init_population_lhs() elif init == 'random': self.init_population_random() else: raise ValueError(self.__init_error_msg) else: self.init_population_array(init)
self.disp = disp
""" Initializes the population with Latin Hypercube Sampling. Latin Hypercube Sampling ensures that each parameter is uniformly sampled over its range. """ rng = self.random_number_generator
# Each parameter range needs to be sampled uniformly. The scaled # parameter range ([0, 1)) needs to be split into # `self.num_population_members` segments, each of which has the following # size: segsize = 1.0 / self.num_population_members
# Within each segment we sample from a uniform random distribution. # We need to do this sampling for each parameter. samples = (segsize * rng.random_sample(self.population_shape)
# Offset each segment to cover the entire parameter range [0, 1) + np.linspace(0., 1., self.num_population_members, endpoint=False)[:, np.newaxis])
# Create an array for population of candidate solutions. self.population = np.zeros_like(samples)
# Initialize population of candidate solutions by permutation of the # random samples. for j in range(self.parameter_count): order = rng.permutation(range(self.num_population_members)) self.population[:, j] = samples[order, j]
# reset population energies self.population_energies = (np.ones(self.num_population_members) * np.inf)
# reset number of function evaluations counter self._nfev = 0
""" Initialises the population at random. This type of initialization can possess clustering, Latin Hypercube sampling is generally better. """ rng = self.random_number_generator self.population = rng.random_sample(self.population_shape)
# reset population energies self.population_energies = (np.ones(self.num_population_members) * np.inf)
# reset number of function evaluations counter self._nfev = 0
""" Initialises the population with a user specified population.
Parameters ---------- init : np.ndarray Array specifying subset of the initial population. The array should have shape (M, len(x)), where len(x) is the number of parameters. The population is clipped to the lower and upper `bounds`. """ # make sure you're using a float array popn = np.asfarray(init)
if (np.size(popn, 0) < 5 or popn.shape[1] != self.parameter_count or len(popn.shape) != 2): raise ValueError("The population supplied needs to have shape" " (M, len(x)), where M > 4.")
# scale values and clip to bounds, assigning to population self.population = np.clip(self._unscale_parameters(popn), 0, 1)
self.num_population_members = np.size(self.population, 0)
self.population_shape = (self.num_population_members, self.parameter_count)
# reset population energies self.population_energies = (np.ones(self.num_population_members) * np.inf)
# reset number of function evaluations counter self._nfev = 0
def x(self): """ The best solution from the solver
Returns ------- x : ndarray The best solution from the solver. """ return self._scale_parameters(self.population[0])
def convergence(self): """ The standard deviation of the population energies divided by their mean. """ return (np.std(self.population_energies) / np.abs(np.mean(self.population_energies) + _MACHEPS))
""" Runs the DifferentialEvolutionSolver.
Returns ------- res : OptimizeResult The optimization result represented as a ``OptimizeResult`` object. Important attributes are: ``x`` the solution array, ``success`` a Boolean flag indicating if the optimizer exited successfully and ``message`` which describes the cause of the termination. See `OptimizeResult` for a description of other attributes. If `polish` was employed, and a lower minimum was obtained by the polishing, then OptimizeResult also contains the ``jac`` attribute. """ nit, warning_flag = 0, False status_message = _status_message['success']
# The population may have just been initialized (all entries are # np.inf). If it has you have to calculate the initial energies. # Although this is also done in the evolve generator it's possible # that someone can set maxiter=0, at which point we still want the # initial energies to be calculated (the following loop isn't run). if np.all(np.isinf(self.population_energies)): self._calculate_population_energies()
# do the optimisation. for nit in xrange(1, self.maxiter + 1): # evolve the population by a generation try: next(self) except StopIteration: warning_flag = True status_message = _status_message['maxfev'] break
if self.disp: print("differential_evolution step %d: f(x)= %g" % (nit, self.population_energies[0]))
# should the solver terminate? convergence = self.convergence
if (self.callback and self.callback(self._scale_parameters(self.population[0]), convergence=self.tol / convergence) is True):
warning_flag = True status_message = ('callback function requested stop early ' 'by returning True') break
intol = (np.std(self.population_energies) <= self.atol + self.tol * np.abs(np.mean(self.population_energies))) if warning_flag or intol: break
else: status_message = _status_message['maxiter'] warning_flag = True
DE_result = OptimizeResult( x=self.x, fun=self.population_energies[0], nfev=self._nfev, nit=nit, message=status_message, success=(warning_flag is not True))
if self.polish: result = minimize(self.func, np.copy(DE_result.x), method='L-BFGS-B', bounds=self.limits.T, args=self.args)
self._nfev += result.nfev DE_result.nfev = self._nfev
if result.fun < DE_result.fun: DE_result.fun = result.fun DE_result.x = result.x DE_result.jac = result.jac # to keep internal state consistent self.population_energies[0] = result.fun self.population[0] = self._unscale_parameters(result.x)
return DE_result
""" Calculate the energies of all the population members at the same time. Puts the best member in first place. Useful if the population has just been initialised. """ for index, candidate in enumerate(self.population): if self._nfev > self.maxfun: break
parameters = self._scale_parameters(candidate) self.population_energies[index] = self.func(parameters, *self.args) self._nfev += 1
minval = np.argmin(self.population_energies)
# put the lowest energy into the best solution position. lowest_energy = self.population_energies[minval] self.population_energies[minval] = self.population_energies[0] self.population_energies[0] = lowest_energy
self.population[[0, minval], :] = self.population[[minval, 0], :]
return self
""" Evolve the population by a single generation
Returns ------- x : ndarray The best solution from the solver. fun : float Value of objective function obtained from the best solution. """ # the population may have just been initialized (all entries are # np.inf). If it has you have to calculate the initial energies if np.all(np.isinf(self.population_energies)): self._calculate_population_energies()
if self.dither is not None: self.scale = (self.random_number_generator.rand() * (self.dither[1] - self.dither[0]) + self.dither[0])
for candidate in range(self.num_population_members): if self._nfev > self.maxfun: raise StopIteration
# create a trial solution trial = self._mutate(candidate)
# ensuring that it's in the range [0, 1) self._ensure_constraint(trial)
# scale from [0, 1) to the actual parameter value parameters = self._scale_parameters(trial)
# determine the energy of the objective function energy = self.func(parameters, *self.args) self._nfev += 1
# if the energy of the trial candidate is lower than the # original population member then replace it if energy < self.population_energies[candidate]: self.population[candidate] = trial self.population_energies[candidate] = energy
# if the trial candidate also has a lower energy than the # best solution then replace that as well if energy < self.population_energies[0]: self.population_energies[0] = energy self.population[0] = trial
return self.x, self.population_energies[0]
""" Evolve the population by a single generation
Returns ------- x : ndarray The best solution from the solver. fun : float Value of objective function obtained from the best solution. """ # next() is required for compatibility with Python2.7. return self.__next__()
""" scale from a number between 0 and 1 to parameters. """ return self.__scale_arg1 + (trial - 0.5) * self.__scale_arg2
""" scale from parameters to a number between 0 and 1. """ return (parameters - self.__scale_arg1) / self.__scale_arg2 + 0.5
""" make sure the parameters lie between the limits """ for index in np.where((trial < 0) | (trial > 1))[0]: trial[index] = self.random_number_generator.rand()
""" create a trial vector based on a mutation strategy """ trial = np.copy(self.population[candidate])
rng = self.random_number_generator
fill_point = rng.randint(0, self.parameter_count)
if self.strategy in ['currenttobest1exp', 'currenttobest1bin']: bprime = self.mutation_func(candidate, self._select_samples(candidate, 5)) else: bprime = self.mutation_func(self._select_samples(candidate, 5))
if self.strategy in self._binomial: crossovers = rng.rand(self.parameter_count) crossovers = crossovers < self.cross_over_probability # the last one is always from the bprime vector for binomial # If you fill in modulo with a loop you have to set the last one to # true. If you don't use a loop then you can have any random entry # be True. crossovers[fill_point] = True trial = np.where(crossovers, bprime, trial) return trial
elif self.strategy in self._exponential: i = 0 while (i < self.parameter_count and rng.rand() < self.cross_over_probability):
trial[fill_point] = bprime[fill_point] fill_point = (fill_point + 1) % self.parameter_count i += 1
return trial
""" best1bin, best1exp """ r0, r1 = samples[:2] return (self.population[0] + self.scale * (self.population[r0] - self.population[r1]))
""" rand1bin, rand1exp """ r0, r1, r2 = samples[:3] return (self.population[r0] + self.scale * (self.population[r1] - self.population[r2]))
""" randtobest1bin, randtobest1exp """ r0, r1, r2 = samples[:3] bprime = np.copy(self.population[r0]) bprime += self.scale * (self.population[0] - bprime) bprime += self.scale * (self.population[r1] - self.population[r2]) return bprime
""" currenttobest1bin, currenttobest1exp """ r0, r1 = samples[:2] bprime = (self.population[candidate] + self.scale * (self.population[0] - self.population[candidate] + self.population[r0] - self.population[r1])) return bprime
""" best2bin, best2exp """ r0, r1, r2, r3 = samples[:4] bprime = (self.population[0] + self.scale * (self.population[r0] + self.population[r1] - self.population[r2] - self.population[r3]))
return bprime
""" rand2bin, rand2exp """ r0, r1, r2, r3, r4 = samples bprime = (self.population[r0] + self.scale * (self.population[r1] + self.population[r2] - self.population[r3] - self.population[r4]))
return bprime
""" obtain random integers from range(self.num_population_members), without replacement. You can't have the original candidate either. """ idxs = list(range(self.num_population_members)) idxs.remove(candidate) self.random_number_generator.shuffle(idxs) idxs = idxs[:number_samples] return idxs
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