def test_mapwrapper_parallel(): in_arg = np.arange(10.) out_arg = np.sin(in_arg) with MapWrapper(2) as p: out = p(np.sin, in_arg) assert_equal(list(out), out_arg) assert_(p._own_pool is True) assert_(isinstance(p.pool, PWL)) assert_(p._mapfunc is not None) # the context manager should've closed the internal pool # check that it has by asking it to calculate again. with assert_raises(Exception) as excinfo: p(np.sin, in_arg) assert_(excinfo.type is ValueError) # can also set a PoolWrapper up with a map-like callable instance try: p = Pool(2) q = MapWrapper(p.map) assert_(q._own_pool is False) q.close() # closing the PoolWrapper shouldn't close the internal pool # because it didn't create it out = p.map(np.sin, in_arg) assert_equal(list(out), out_arg) finally: p.close()
def test_parallel(self): with MapWrapper(2) as p: out = p(np.sin, self.input) assert_equal(list(out), self.output) assert_(p._own_pool is True) assert_(isinstance(p.pool, PWL)) assert_(p._mapfunc is not None) # the context manager should've closed the internal pool # check that it has by asking it to calculate again. with assert_raises(Exception) as excinfo: p(np.sin, self.input) # on py27 an AssertionError is raised, on >py27 it's a ValueError err_type = excinfo.type assert_((err_type is ValueError) or (err_type is AssertionError)) # can also set a PoolWrapper up with a map-like callable instance try: p = Pool(2) q = MapWrapper(p.map) assert_(q._own_pool is False) q.close() # closing the PoolWrapper shouldn't close the internal pool # because it didn't create it out = p.map(np.sin, self.input) assert_equal(list(out), self.output) finally: p.close()
def test_parallel(self): with MapWrapper(2) as p: out = p(np.sin, self.input) assert_equal(list(out), self.output) assert_(p._own_pool is True) assert_(isinstance(p.pool, PWL)) assert_(p._mapfunc is not None) # the context manager should've closed the internal pool # check that it has by asking it to calculate again. with assert_raises(Exception) as excinfo: p(np.sin, self.input) # on py27 an AssertionError is raised, on >py27 it's a ValueError err_type = excinfo.type assert_((err_type is ValueError) or (err_type is AssertionError)) # can also set a PoolWrapper up with a map-like callable instance try: p = Pool(2) q = MapWrapper(p.map) assert_(q._own_pool is False) q.close() # closing the PoolWrapper shouldn't close the internal pool # because it didn't create it out = p.map(np.sin, self.input) assert_equal(list(out), self.output) finally: p.close()
class EvolutionaryAlgorithmSolver(object): def __init__(self, func, bounds, args = None, crossover = 'arithmetic', mutation = 'uniform', generations = 1000, populationSize = 100, recombinationRate = 0.9, mutationRate = 0.1, seed = None): self.func = _FunctionWrapper(func, args) self.bounds = np.array(bounds) self.seed = check_random_state(seed) self.crossoverOp = crossover self.mutationOp = mutation self.generations = generations self.popsize = populationSize self.recombinationrate = recombinationRate self.mutationrate = mutationRate self._mapwrapper = MapWrapper(4) def tournament(self, parents, fitness): rng = self.seed idx = np.argsort(rng.uniform(size = (self.popsize, self.popsize)))[:,0:4] idx_1 = fitness[idx[:,0]] < fitness[idx[:,1]] idx_2 = fitness[idx[:,2]] < fitness[idx[:,3]] idx_parent_1 = idx[:,1] idx_parent_1[idx_1] = idx[idx_1,0] idx_parent_2 = idx[:,3] idx_parent_2[idx_2] = idx[idx_2,2] return (parents[idx_parent_1,:], parents[idx_parent_2,:]) def arithmetic(self, parents): rng = self.seed size = np.shape(parents[0]) alpha = rng.uniform(size = size) mask = rng.choice([False, True], size = (size[0]), p = [self.recombinationrate, 1-self.recombinationrate]) offspring = np.multiply(alpha, parents[0]) + np.multiply(1-alpha, parents[1]) offspring[offspring > 1] = 1 offspring[offspring < 0] = 0 offspring[mask] = parents[0][mask] return offspring #BLX def blx(self, parents): rng = self.seed size = np.shape(parents[0]) alpha = rng.uniform(size = size) mask = rng.choice([False, True], size = (size[0]), p = [self.recombinationrate, 1-self.recombinationrate]) d = np.absolute(parents[0]-parents[1]) alphaD = np.multiply(alpha, d) q = np.minimum(parents[0],parents[1])-alphaD r=np.maximum(parents[0],parents[1])+alphaD offspring= rng.uniform(low=q,high=r) offspring[offspring > 1] = 1 offspring[offspring < 0] = 0 offspring[mask] = parents[0][mask] return offspring #SBX def sbx(self,parents): rng = self.seed size = np.shape(parents[0]) mask = rng.choice([False, True], size = (size[0]), p = [self.recombinationrate, 1-self.recombinationrate]) u=rng.uniform(size=size) n=10 b=np.power(2*u,1/n) #(2*u)**(1/(n+1)) b[u>.5]=np.power(2*(1-u[u>.5]),1/n) offspring=.5*(parents[0]+parents[1])-.5*np.multiply(b,parents[0]-parents[1]) offspring[offspring > 1] = 1 offspring[offspring < 0] = 0 offspring[mask] = parents[0][mask] return offspring def uniform(self, offspring): rng = self.seed size=np.shape(offspring) mutation = rng.uniform(size = size) mask = rng.choice([True, False], size = size, p = [self.mutationrate, \ 1-self.mutationrate]) offspring[mask] = mutation[mask] return offspring def boundary(self,offspring): rng = self.seed size=np.shape(offspring) mutation =np.round(rng.uniform(size = size)) mask = rng.choice([True, False], size = size, p = [self.mutationrate, \ 1-self.mutationrate]) offspring[mask] = mutation[mask] return offspring def __crossover(self, parents): if self.crossoverOp == 'blx': return self.blx(parents) elif self.crossoverOp == 'sbx': return self.sbx(parents) elif self.crossoverOp == 'arithmetic': return self.arithmetic(parents) else: raise ValueError("Please select a valid crossover strategy") def __mutation(self, offspring): if self.mutationOp == 'uniform': return self.uniform(offspring) elif self.mutationOp == 'boundary': return self.boundary(offspring) else: raise ValueError("Please select a valid mutation strategy") def __reproduce(self, parents): return self.__mutation(self.__crossover(parents)) def __survivalSelection(self, parents, fparents, offspring, foffspring): mergedPop = np.vstack((parents, offspring)) mergedFit = np.hstack((fparents, foffspring)) idx = np.argsort(mergedFit)[0:self.popsize] return mergedPop[idx,:], mergedFit[idx] def __scaleParameters(self, population): scaled = np.multiply((self.bounds[:,1] - self.bounds[:,0]), population) return scaled + self.bounds[:,0] def solve(self): rng = self.seed numVars = np.shape(self.bounds)[0] # Check the bounds if not np.all((self.bounds[:,1] - self.bounds[:,0]) > 0): raise ValueError("Error: lower bound greater than upper bound") pop = rng.uniform(size=(self.popsize,numVars)) fitness = self.func(self.__scaleParameters(pop)) currentGeneration = 0 while currentGeneration < self.generations + 1: parents = self.tournament(pop, fitness) offspring = self.__reproduce(parents) foffspring = self.func(self.__scaleParameters(offspring)) pop, fitness = self.__survivalSelection(pop, fitness, offspring, foffspring) print('The best value in generation '+str(currentGeneration)+ ' is '+str(np.min(fitness))) currentGeneration += 1 idxBest = np.argmin(fitness) return (self.__scaleParameters(pop[idxBest,:]), fitness[idxBest]) def __enter__(self): return self def __exit__(self, *args): self._mapwrapper.close()
class DifferentialEvolutionSolver(object): """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. updating : {'immediate', 'deferred'}, optional If `immediate` the best solution vector is continuously updated within a single generation. This can lead to faster convergence as trial vectors can take advantage of continuous improvements in the best solution. With `deferred` the best solution vector is updated once per generation. Only `deferred` is compatible with parallelization, and the `workers` keyword can over-ride this option. workers : int or map-like callable, optional If `workers` is an int the population is subdivided into `workers` sections and evaluated in parallel (uses `multiprocessing.Pool`). Supply `-1` to use all cores available to the Process. Alternatively supply a map-like callable, such as `multiprocessing.Pool.map` for evaluating the population in parallel. This evaluation is carried out as ``workers(func, iterable)``. This option will override the `updating` keyword to `updating='deferred'` if `workers != 1`. Requires that `func` be pickleable. """ # Dispatch of mutation strategy method (binomial or exponential). _binomial = {'best1bin': '_best1', 'randtobest1bin': '_randtobest1', 'currenttobest1bin': '_currenttobest1', 'best2bin': '_best2', 'rand2bin': '_rand2', 'rand1bin': '_rand1'} _exponential = {'best1exp': '_best1', 'rand1exp': '_rand1', 'randtobest1exp': '_randtobest1', 'currenttobest1exp': '_currenttobest1', 'best2exp': '_best2', 'rand2exp': '_rand2'} __init_error_msg = ("The population initialization method must be one of " "'latinhypercube' or 'random', or an array of shape " "(M, N) where N is the number of parameters and M>5") def __init__(self, func, bounds, args=(), 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, updating='immediate', workers=1): 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 # set the updating / parallelisation options if updating in ['immediate', 'deferred']: self._updating = updating # want to use parallelisation, but updating is immediate if workers != 1 and updating == 'immediate': warnings.warn("differential_evolution: the 'workers' keyword has" " overridden updating='immediate' to" " updating='deferred'", UserWarning) self._updating = 'deferred' # an object with a map method. self._mapwrapper = MapWrapper(workers) # 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 # we create a wrapped function to allow the use of map (and Pool.map # in the future) self.func = _FunctionWrapper(func, args) 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 def init_population_lhs(self): """ 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.full(self.num_population_members, np.inf) # reset number of function evaluations counter self._nfev = 0 def init_population_random(self): """ 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.full(self.num_population_members, np.inf) # reset number of function evaluations counter self._nfev = 0 def init_population_array(self, init): """ 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 @property def x(self): """ The best solution from the solver """ return self._scale_parameters(self.population[0]) @property def convergence(self): """ The standard deviation of the population energies divided by their mean. """ if np.any(np.isinf(self.population_energies)): return np.inf return (np.std(self.population_energies) / np.abs(np.mean(self.population_energies) + _MACHEPS)) def converged(self): """ Return True if the solver has converged. """ return (np.std(self.population_energies) <= self.atol + self.tol * np.abs(np.mean(self.population_energies))) def solve(self): """ 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.population_energies[:] = self._calculate_population_energies( self.population) self._promote_lowest_energy() # 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 if self._nfev > self.maxfun: status_message = _status_message['maxfev'] elif self._nfev == self.maxfun: status_message = ('Maximum number of function evaluations' ' has been reached.') 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 if np.any(np.isinf(self.population_energies)): intol = False else: 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) 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 def _calculate_population_energies(self, population): """ Calculate the energies of all the population members at the same time. Parameters ---------- population : ndarray An array of parameter vectors normalised to [0, 1] using lower and upper limits. Has shape ``(np.size(population, 0), len(x))``. Returns ------- energies : ndarray An array of energies corresponding to each population member. If maxfun will be exceeded during this call, then the number of function evaluations will be reduced and energies will be right-padded with np.inf. Has shape ``(np.size(population, 0),)`` """ num_members = np.size(population, 0) nfevs = min(num_members, self.maxfun - num_members) energies = np.full(num_members, np.inf) parameters_pop = self._scale_parameters(population) try: calc_energies = list(self._mapwrapper(self.func, parameters_pop[0:nfevs])) energies[0:nfevs] = calc_energies except (TypeError, ValueError): # wrong number of arguments for _mapwrapper # or wrong length returned from the mapper raise RuntimeError("The map-like callable must be of the" " form f(func, iterable), returning a sequence" " of numbers the same length as 'iterable'") self._nfev += nfevs return energies def _promote_lowest_energy(self): # promotes the lowest energy to the first entry in the population l = np.argmin(self.population_energies) # put the lowest energy into the best solution position. self.population_energies[[0, l]] = self.population_energies[[l, 0]] self.population[[0, l], :] = self.population[[l, 0], :] def __iter__(self): return self def __enter__(self): return self def __exit__(self, *args): # to make sure resources are closed down self._mapwrapper.close() def __del__(self): # to make sure resources are closed down self._mapwrapper.close() def __next__(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.population_energies[:] = self._calculate_population_energies( self.population) self._promote_lowest_energy() if self.dither is not None: self.scale = (self.random_number_generator.rand() * (self.dither[1] - self.dither[0]) + self.dither[0]) if self._updating == 'immediate': # update best solution immediately 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._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 promote it to the best solution. if energy < self.population_energies[0]: self._promote_lowest_energy() elif self._updating == 'deferred': # update best solution once per generation if self._nfev >= self.maxfun: raise StopIteration # 'deferred' approach, vectorised form. # create trial solutions trial_pop = np.array( [self._mutate(i) for i in range(self.num_population_members)]) # enforce bounds self._ensure_constraint(trial_pop) # determine the energies of the objective function trial_energies = self._calculate_population_energies(trial_pop) # which solutions are improved? loc = trial_energies < self.population_energies self.population = np.where(loc[:, np.newaxis], trial_pop, self.population) self.population_energies = np.where(loc, trial_energies, self.population_energies) # make sure the best solution is updated if updating='deferred'. # put the lowest energy into the best solution position. self._promote_lowest_energy() return self.x, self.population_energies[0] next = __next__ def _scale_parameters(self, trial): """Scale from a number between 0 and 1 to parameters.""" return self.__scale_arg1 + (trial - 0.5) * self.__scale_arg2 def _unscale_parameters(self, parameters): """Scale from parameters to a number between 0 and 1.""" return (parameters - self.__scale_arg1) / self.__scale_arg2 + 0.5 def _ensure_constraint(self, trial): """Make sure the parameters lie between the limits.""" mask = np.where((trial > 1) | (trial < 0)) trial[mask] = self.random_number_generator.rand(mask[0].size) def _mutate(self, candidate): """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 def _best1(self, samples): """best1bin, best1exp""" r0, r1 = samples[:2] return (self.population[0] + self.scale * (self.population[r0] - self.population[r1])) def _rand1(self, samples): """rand1bin, rand1exp""" r0, r1, r2 = samples[:3] return (self.population[r0] + self.scale * (self.population[r1] - self.population[r2])) def _randtobest1(self, samples): """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 def _currenttobest1(self, candidate, samples): """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 def _best2(self, samples): """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 def _rand2(self, samples): """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 def _select_samples(self, candidate, number_samples): """ 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
class DifferentialEvolutionSolver(object): """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. updating : {'immediate', 'deferred'}, optional If `immediate` the best solution vector is continuously updated within a single generation. This can lead to faster convergence as trial vectors can take advantage of continuous improvements in the best solution. With `deferred` the best solution vector is updated once per generation. Only `deferred` is compatible with parallelization, and the `workers` keyword can over-ride this option. workers : int or map-like callable, optional If `workers` is an int the population is subdivided into `workers` sections and evaluated in parallel (uses `multiprocessing.Pool`). Supply `-1` to use all cores available to the Process. Alternatively supply a map-like callable, such as `multiprocessing.Pool.map` for evaluating the population in parallel. This evaluation is carried out as ``workers(func, iterable)``. This option will override the `updating` keyword to `updating='deferred'` if `workers != 1`. Requires that `func` be pickleable. """ # Dispatch of mutation strategy method (binomial or exponential). _binomial = { 'best1bin': '_best1', 'randtobest1bin': '_randtobest1', 'currenttobest1bin': '_currenttobest1', 'best2bin': '_best2', 'rand2bin': '_rand2', 'rand1bin': '_rand1' } _exponential = { 'best1exp': '_best1', 'rand1exp': '_rand1', 'randtobest1exp': '_randtobest1', 'currenttobest1exp': '_currenttobest1', 'best2exp': '_best2', 'rand2exp': '_rand2' } __init_error_msg = ("The population initialization method must be one of " "'latinhypercube' or 'random', or an array of shape " "(M, N) where N is the number of parameters and M>5") def __init__(self, func, bounds, args=(), 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, updating='immediate', workers=1): 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 # set the updating / parallelisation options if updating in ['immediate', 'deferred']: self._updating = updating # want to use parallelisation, but updating is immediate if workers != 1 and updating == 'immediate': warnings.warn( "differential_evolution: the 'workers' keyword has" " overridden updating='immediate' to" " updating='deferred'", UserWarning) self._updating = 'deferred' # an object with a map method. self._mapwrapper = MapWrapper(workers) # 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 # we create a wrapped function to allow the use of map (and Pool.map # in the future) self.func = _FunctionWrapper(func, args) 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 def init_population_lhs(self): """ 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.full(self.num_population_members, np.inf) # reset number of function evaluations counter self._nfev = 0 def init_population_random(self): """ 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.full(self.num_population_members, np.inf) # reset number of function evaluations counter self._nfev = 0 def init_population_array(self, init): """ 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 @property def x(self): """ The best solution from the solver """ return self._scale_parameters(self.population[0]) @property def convergence(self): """ The standard deviation of the population energies divided by their mean. """ if np.any(np.isinf(self.population_energies)): return np.inf return (np.std(self.population_energies) / np.abs(np.mean(self.population_energies) + _MACHEPS)) def converged(self): """ Return True if the solver has converged. """ return (np.std(self.population_energies) <= self.atol + self.tol * np.abs(np.mean(self.population_energies))) def solve(self): """ 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.population_energies[:] = self._calculate_population_energies( self.population) self._promote_lowest_energy() # 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 if self._nfev > self.maxfun: status_message = _status_message['maxfev'] elif self._nfev == self.maxfun: status_message = ('Maximum number of function evaluations' ' has been reached.') 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 if np.any(np.isinf(self.population_energies)): intol = False else: 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) 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 def _calculate_population_energies(self, population): """ Calculate the energies of all the population members at the same time. Parameters ---------- population : ndarray An array of parameter vectors normalised to [0, 1] using lower and upper limits. Has shape ``(np.size(population, 0), len(x))``. Returns ------- energies : ndarray An array of energies corresponding to each population member. If maxfun will be exceeded during this call, then the number of function evaluations will be reduced and energies will be right-padded with np.inf. Has shape ``(np.size(population, 0),)`` """ num_members = np.size(population, 0) nfevs = min(num_members, self.maxfun - num_members) energies = np.full(num_members, np.inf) parameters_pop = self._scale_parameters(population) try: calc_energies = list( self._mapwrapper(self.func, parameters_pop[0:nfevs])) energies[0:nfevs] = calc_energies except (TypeError, ValueError): # wrong number of arguments for _mapwrapper # or wrong length returned from the mapper raise RuntimeError("The map-like callable must be of the" " form f(func, iterable), returning a sequence" " of numbers the same length as 'iterable'") self._nfev += nfevs return energies def _promote_lowest_energy(self): # promotes the lowest energy to the first entry in the population l = np.argmin(self.population_energies) # put the lowest energy into the best solution position. self.population_energies[[0, l]] = self.population_energies[[l, 0]] self.population[[0, l], :] = self.population[[l, 0], :] def __iter__(self): return self def __enter__(self): return self def __exit__(self, *args): # to make sure resources are closed down self._mapwrapper.close() def __del__(self): # to make sure resources are closed down self._mapwrapper.close() def __next__(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.population_energies[:] = self._calculate_population_energies( self.population) self._promote_lowest_energy() if self.dither is not None: self.scale = (self.random_number_generator.rand() * (self.dither[1] - self.dither[0]) + self.dither[0]) if self._updating == 'immediate': # update best solution immediately 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._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 promote it to the best solution. if energy < self.population_energies[0]: self._promote_lowest_energy() elif self._updating == 'deferred': # update best solution once per generation if self._nfev >= self.maxfun: raise StopIteration # 'deferred' approach, vectorised form. # create trial solutions trial_pop = np.array( [self._mutate(i) for i in range(self.num_population_members)]) # enforce bounds self._ensure_constraint(trial_pop) # determine the energies of the objective function trial_energies = self._calculate_population_energies(trial_pop) # which solutions are improved? loc = trial_energies < self.population_energies self.population = np.where(loc[:, np.newaxis], trial_pop, self.population) self.population_energies = np.where(loc, trial_energies, self.population_energies) # make sure the best solution is updated if updating='deferred'. # put the lowest energy into the best solution position. self._promote_lowest_energy() return self.x, self.population_energies[0] next = __next__ def _scale_parameters(self, trial): """Scale from a number between 0 and 1 to parameters.""" return self.__scale_arg1 + (trial - 0.5) * self.__scale_arg2 def _unscale_parameters(self, parameters): """Scale from parameters to a number between 0 and 1.""" return (parameters - self.__scale_arg1) / self.__scale_arg2 + 0.5 def _ensure_constraint(self, trial): """Make sure the parameters lie between the limits.""" mask = np.where((trial > 1) | (trial < 0)) trial[mask] = self.random_number_generator.rand(mask[0].size) def _mutate(self, candidate): """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 def _best1(self, samples): """best1bin, best1exp""" r0, r1 = samples[:2] return (self.population[0] + self.scale * (self.population[r0] - self.population[r1])) def _rand1(self, samples): """rand1bin, rand1exp""" r0, r1, r2 = samples[:3] return (self.population[r0] + self.scale * (self.population[r1] - self.population[r2])) def _randtobest1(self, samples): """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 def _currenttobest1(self, candidate, samples): """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 def _best2(self, samples): """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 def _rand2(self, samples): """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 def _select_samples(self, candidate, number_samples): """ 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