""" differential_evolution: The differential evolution global optimization algorithm Added by Andrew Nelson 2014 """ import warnings import numpy as np from scipy.optimize import OptimizeResult, minimize from scipy.optimize._optimize import _status_message, _wrap_callback from scipy._lib._util import (check_random_state, MapWrapper, _FunctionWrapper, rng_integers) from scipy.optimize._constraints import (Bounds, new_bounds_to_old, NonlinearConstraint, LinearConstraint) from scipy.sparse import issparse __all__ = ['differential_evolution'] _MACHEPS = np.finfo(np.float64).eps def differential_evolution(func, bounds, args=(), strategy='best1bin', 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, updating='immediate', workers=1, constraints=(), x0=None, *, integrality=None, vectorized=False): """Finds the global minimum of a multivariate function. The differential evolution method [1]_ is stochastic in nature. It does not use gradient methods to find the minimum, 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 [2]_. 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. The number of parameters, N, is equal to ``len(x)``. bounds : sequence or `Bounds` Bounds for variables. There are two ways to specify the bounds: 1. Instance of `Bounds` class. 2. ``(min, max)`` pairs for each element in ``x``, defining the finite lower and upper bounds for the optimizing argument of `func`. The total number of bounds is used to determine the number of parameters, N. If there are parameters whose bounds are equal the total number of free parameters is ``N - N_equal``. args : tuple, optional Any additional fixed parameters needed to completely specify the objective function. strategy : {str, callable}, optional The differential evolution strategy to use. Should be one of: - 'best1bin' - 'best1exp' - 'rand1bin' - 'rand1exp' - 'rand2bin' - 'rand2exp' - 'randtobest1bin' - 'randtobest1exp' - 'currenttobest1bin' - 'currenttobest1exp' - 'best2exp' - 'best2bin' The default is 'best1bin'. Strategies that may be implemented are outlined in 'Notes'. Alternatively the differential evolution strategy can be customized by providing a callable that constructs a trial vector. The callable must have the form ``strategy(candidate: int, population: np.ndarray, rng=None)``, where ``candidate`` is an integer specifying which entry of the population is being evolved, ``population`` is an array of shape ``(S, N)`` containing all the population members (where S is the total population size), and ``rng`` is the random number generator being used within the solver. ``candidate`` will be in the range ``[0, S)``. ``strategy`` must return a trial vector with shape `(N,)`. The fitness of this trial vector is compared against the fitness of ``population[candidate]``. .. versionchanged:: 1.12.0 Customization of evolution strategy via a callable. 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 * (N - N_equal)`` popsize : int, optional A multiplier for setting the total population size. The population has ``popsize * (N - N_equal)`` individuals. This keyword is overridden if an initial population is supplied via the `init` keyword. When using ``init='sobol'`` the population size is calculated as the next power of 2 after ``popsize * (N - N_equal)``. 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 : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional If `seed` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `seed` is an int, a new ``RandomState`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Specify `seed` for repeatable minimizations. disp : bool, optional Prints the evaluated `func` at every iteration. callback : callable, optional A callable called after each iteration. Has the signature: ``callback(intermediate_result: OptimizeResult)`` where ``intermediate_result`` is a keyword parameter containing an `OptimizeResult` with attributes ``x`` and ``fun``, the best solution found so far and the objective function. Note that the name of the parameter must be ``intermediate_result`` for the callback to be passed an `OptimizeResult`. The callback also supports a signature like: ``callback(x, convergence: float=val)`` ``val`` represents the fractional value of the population convergence. When ``val`` is greater than ``1.0``, the function halts. Introspection is used to determine which of the signatures is invoked. Global minimization will halt if the callback raises ``StopIteration`` or returns ``True``; any polishing is still carried out. .. versionchanged:: 1.12.0 callback accepts the ``intermediate_result`` keyword. 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. If a constrained problem is being studied then the `trust-constr` method is used instead. For large problems with many constraints, polishing can take a long time due to the Jacobian computations. init : str or array-like, optional Specify which type of population initialization is performed. Should be one of: - 'latinhypercube' - 'sobol' - 'halton' - 'random' - array specifying the initial population. The array should have shape ``(S, N)``, where S is the total population size and N 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. 'sobol' and 'halton' are superior alternatives and maximize even more the parameter space. 'sobol' will enforce an initial population size which is calculated as the next power of 2 after ``popsize * (N - N_equal)``. 'halton' has no requirements but is a bit less efficient. See `scipy.stats.qmc` for more details. '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 [4]_. 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 or vectorization, and the `workers` and `vectorized` keywords can over-ride this option. .. versionadded:: 1.2.0 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 available CPU cores. 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``. This option overrides the `vectorized` keyword if ``workers != 1``. Requires that `func` be pickleable. .. versionadded:: 1.2.0 constraints : {NonLinearConstraint, LinearConstraint, Bounds} Constraints on the solver, over and above those applied by the `bounds` kwd. Uses the approach by Lampinen [5]_. .. versionadded:: 1.4.0 x0 : None or array-like, optional Provides an initial guess to the minimization. Once the population has been initialized this vector replaces the first (best) member. This replacement is done even if `init` is given an initial population. ``x0.shape == (N,)``. .. versionadded:: 1.7.0 integrality : 1-D array, optional For each decision variable, a boolean value indicating whether the decision variable is constrained to integer values. The array is broadcast to ``(N,)``. If any decision variables are constrained to be integral, they will not be changed during polishing. Only integer values lying between the lower and upper bounds are used. If there are no integer values lying between the bounds then a `ValueError` is raised. .. versionadded:: 1.9.0 vectorized : bool, optional If ``vectorized is True``, `func` is sent an `x` array with ``x.shape == (N, S)``, and is expected to return an array of shape ``(S,)``, where `S` is the number of solution vectors to be calculated. If constraints are applied, each of the functions used to construct a `Constraint` object should accept an `x` array with ``x.shape == (N, S)``, and return an array of shape ``(M, S)``, where `M` is the number of constraint components. This option is an alternative to the parallelization offered by `workers`, and may help in optimization speed by reducing interpreter overhead from multiple function calls. This keyword is ignored if ``workers != 1``. This option will override the `updating` keyword to ``updating='deferred'``. See the notes section for further discussion on when to use ``'vectorized'``, and when to use ``'workers'``. .. versionadded:: 1.9.0 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, ``message`` which describes the cause of the termination, ``population`` the solution vectors present in the population, and ``population_energies`` the value of the objective function for each entry in ``population``. 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. If the eventual solution does not satisfy the applied constraints ``success`` will be `False`. 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 [3]_ 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:`x_0`, so far: .. math:: b' = x_0 + mutation * (x_{r_0} - x_{r_1}) A trial vector is then constructed. Starting with a randomly chosen ith 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. The other strategies available are outlined in Qiang and Mitchell (2014) [3]_. .. math:: rand1* : b' = x_{r_0} + mutation*(x_{r_1} - x_{r_2}) rand2* : b' = x_{r_0} + mutation*(x_{r_1} + x_{r_2} - x_{r_3} - x_{r_4}) best1* : b' = x_0 + mutation*(x_{r_0} - x_{r_1}) best2* : b' = x_0 + mutation*(x_{r_0} + x_{r_1} - x_{r_2} - x_{r_3}) currenttobest1* : b' = x_i + mutation*(x_0 - x_i + x_{r_0} - x_{r_1}) randtobest1* : b' = x_{r_0} + mutation*(x_0 - x_{r_0} + x_{r_1} - x_{r_2}) where the integers :math:`r_0, r_1, r_2, r_3, r_4` are chosen randomly from the interval [0, NP) with `NP` being the total population size and the original candidate having index `i`. The user can fully customize the generation of the trial candidates by supplying a callable to ``strategy``. 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. By default the best solution vector is updated continuously within a single iteration (``updating='immediate'``). This is a modification [4]_ of the original differential evolution algorithm which can lead to faster convergence as trial vectors can immediately benefit from improved solutions. To use the original Storn and Price behaviour, updating the best solution once per iteration, set ``updating='deferred'``. The ``'deferred'`` approach is compatible with both parallelization and vectorization (``'workers'`` and ``'vectorized'`` keywords). These may improve minimization speed by using computer resources more efficiently. The ``'workers'`` distribute calculations over multiple processors. By default the Python `multiprocessing` module is used, but other approaches are also possible, such as the Message Passing Interface (MPI) used on clusters [6]_ [7]_. The overhead from these approaches (creating new Processes, etc) may be significant, meaning that computational speed doesn't necessarily scale with the number of processors used. Parallelization is best suited to computationally expensive objective functions. If the objective function is less expensive, then ``'vectorized'`` may aid by only calling the objective function once per iteration, rather than multiple times for all the population members; the interpreter overhead is reduced. .. versionadded:: 0.15.0 References ---------- .. [1] Differential evolution, Wikipedia, http://en.wikipedia.org/wiki/Differential_evolution .. [2] 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. .. [3] Qiang, J., Mitchell, C., A Unified Differential Evolution Algorithm for Global Optimization, 2014, https://www.osti.gov/servlets/purl/1163659 .. [4] Wormington, M., Panaccione, C., Matney, K. M., Bowen, D. K., - Characterization of structures from X-ray scattering data using genetic algorithms, Phil. Trans. R. Soc. Lond. A, 1999, 357, 2827-2848 .. [5] Lampinen, J., A constraint handling approach for the differential evolution algorithm. Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No. 02TH8600). Vol. 2. IEEE, 2002. .. [6] https://mpi4py.readthedocs.io/en/stable/ .. [7] https://schwimmbad.readthedocs.io/en/latest/ Examples -------- Let us consider the problem of minimizing the Rosenbrock function. This function is implemented in `rosen` in `scipy.optimize`. >>> import numpy as np >>> 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) Now repeat, but with parallelization. >>> result = differential_evolution(rosen, bounds, updating='deferred', ... workers=2) >>> result.x, result.fun (array([1., 1., 1., 1., 1.]), 1.9216496320061384e-19) Let's do a constrained minimization. >>> from scipy.optimize import LinearConstraint, Bounds We add the constraint that the sum of ``x[0]`` and ``x[1]`` must be less than or equal to 1.9. This is a linear constraint, which may be written ``A @ x <= 1.9``, where ``A = array([[1, 1]])``. This can be encoded as a `LinearConstraint` instance: >>> lc = LinearConstraint([[1, 1]], -np.inf, 1.9) Specify limits using a `Bounds` object. >>> bounds = Bounds([0., 0.], [2., 2.]) >>> result = differential_evolution(rosen, bounds, constraints=lc, ... seed=1) >>> result.x, result.fun (array([0.96632622, 0.93367155]), 0.0011352416852625719) Next find the minimum of the Ackley function (https://en.wikipedia.org/wiki/Test_functions_for_optimization). >>> 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, seed=1) >>> result.x, result.fun (array([0., 0.]), 4.440892098500626e-16) The Ackley function is written in a vectorized manner, so the ``'vectorized'`` keyword can be employed. Note the reduced number of function evaluations. >>> result = differential_evolution( ... ackley, bounds, vectorized=True, updating='deferred', seed=1 ... ) >>> result.x, result.fun (array([0., 0.]), 4.440892098500626e-16) The following custom strategy function mimics 'best1bin': >>> def custom_strategy_fn(candidate, population, rng=None): ... parameter_count = population.shape(-1) ... mutation, recombination = 0.7, 0.9 ... trial = np.copy(population[candidate]) ... fill_point = rng.choice(parameter_count) ... ... pool = np.arange(len(population)) ... rng.shuffle(pool) ... ... # two unique random numbers that aren't the same, and ... # aren't equal to candidate. ... idxs = [] ... while len(idxs) < 2 and len(pool) > 0: ... idx = pool[0] ... pool = pool[1:] ... if idx != candidate: ... idxs.append(idx) ... ... r0, r1 = idxs[:2] ... ... bprime = (population[0] + mutation * ... (population[r0] - population[r1])) ... ... crossovers = rng.uniform(size=parameter_count) ... crossovers = crossovers < recombination ... crossovers[fill_point] = True ... trial = np.where(crossovers, bprime, trial) ... return trial """ # using a context manager means that any created Pool objects are # cleared up. with 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, updating=updating, workers=workers, constraints=constraints, x0=x0, integrality=integrality, vectorized=vectorized) as solver: ret = solver.solve() return ret class DifferentialEvolutionSolver: """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. The number of parameters, N, is equal to ``len(x)``. bounds : sequence or `Bounds` Bounds for variables. There are two ways to specify the bounds: 1. Instance of `Bounds` class. 2. ``(min, max)`` pairs for each element in ``x``, defining the finite lower and upper bounds for the optimizing argument of `func`. The total number of bounds is used to determine the number of parameters, N. If there are parameters whose bounds are equal the total number of free parameters is ``N - N_equal``. args : tuple, optional Any additional fixed parameters needed to completely specify the objective function. strategy : {str, callable}, optional The differential evolution strategy to use. Should be one of: - 'best1bin' - 'best1exp' - 'rand1bin' - 'rand1exp' - 'rand2bin' - 'rand2exp' - 'randtobest1bin' - 'randtobest1exp' - 'currenttobest1bin' - 'currenttobest1exp' - 'best2exp' - 'best2bin' The default is 'best1bin'. Strategies that may be implemented are outlined in 'Notes'. Alternatively the differential evolution strategy can be customized by providing a callable that constructs a trial vector. The callable must have the form ``strategy(candidate: int, population: np.ndarray, rng=None)``, where ``candidate`` is an integer specifying which entry of the population is being evolved, ``population`` is an array of shape ``(S, N)`` containing all the population members (where S is the total population size), and ``rng`` is the random number generator being used within the solver. ``candidate`` will be in the range ``[0, S)``. ``strategy`` must return a trial vector with shape `(N,)`. The fitness of this trial vector is compared against the fitness of ``population[candidate]``. 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 * (N - N_equal)`` popsize : int, optional A multiplier for setting the total population size. The population has ``popsize * (N - N_equal)`` individuals. This keyword is overridden if an initial population is supplied via the `init` keyword. When using ``init='sobol'`` the population size is calculated as the next power of 2 after ``popsize * (N - N_equal)``. 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 : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional If `seed` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `seed` is an int, a new ``RandomState`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Specify `seed` for repeatable minimizations. disp : bool, optional Prints the evaluated `func` at every iteration. callback : callable, optional A callable called after each iteration. Has the signature: ``callback(intermediate_result: OptimizeResult)`` where ``intermediate_result`` is a keyword parameter containing an `OptimizeResult` with attributes ``x`` and ``fun``, the best solution found so far and the objective function. Note that the name of the parameter must be ``intermediate_result`` for the callback to be passed an `OptimizeResult`. The callback also supports a signature like: ``callback(x, convergence: float=val)`` ``val`` represents the fractional value of the population convergence. When ``val`` is greater than ``1.0``, the function halts. Introspection is used to determine which of the signatures is invoked. Global minimization will halt if the callback raises ``StopIteration`` or returns ``True``; any polishing is still carried out. .. versionchanged:: 1.12.0 callback accepts the ``intermediate_result`` keyword. 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. If a constrained problem is being studied then the `trust-constr` method is used instead. For large problems with many constraints, polishing can take a long time due to the Jacobian computations. 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' - 'sobol' - 'halton' - 'random' - array specifying the initial population. The array should have shape ``(S, N)``, where S is the total population size and N 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. 'sobol' and 'halton' are superior alternatives and maximize even more the parameter space. 'sobol' will enforce an initial population size which is calculated as the next power of 2 after ``popsize * (N - N_equal)``. 'halton' has no requirements but is a bit less efficient. See `scipy.stats.qmc` for more details. '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 [4]_. 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 or vectorization, and the `workers` and `vectorized` keywords 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. constraints : {NonLinearConstraint, LinearConstraint, Bounds} Constraints on the solver, over and above those applied by the `bounds` kwd. Uses the approach by Lampinen. x0 : None or array-like, optional Provides an initial guess to the minimization. Once the population has been initialized this vector replaces the first (best) member. This replacement is done even if `init` is given an initial population. ``x0.shape == (N,)``. integrality : 1-D array, optional For each decision variable, a boolean value indicating whether the decision variable is constrained to integer values. The array is broadcast to ``(N,)``. If any decision variables are constrained to be integral, they will not be changed during polishing. Only integer values lying between the lower and upper bounds are used. If there are no integer values lying between the bounds then a `ValueError` is raised. vectorized : bool, optional If ``vectorized is True``, `func` is sent an `x` array with ``x.shape == (N, S)``, and is expected to return an array of shape ``(S,)``, where `S` is the number of solution vectors to be calculated. If constraints are applied, each of the functions used to construct a `Constraint` object should accept an `x` array with ``x.shape == (N, S)``, and return an array of shape ``(M, S)``, where `M` is the number of constraint components. This option is an alternative to the parallelization offered by `workers`, and may help in optimization speed. This keyword is ignored if ``workers != 1``. This option will override the `updating` keyword to ``updating='deferred'``. """ # 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 " "(S, N) where N is the number of parameters and S>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, constraints=(), x0=None, *, integrality=None, vectorized=False): if callable(strategy): # a callable strategy is going to be stored in self.strategy anyway pass elif 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 = _wrap_callback(callback, "differential_evolution") self.polish = polish # set the updating / parallelisation options if updating in ['immediate', 'deferred']: self._updating = updating self.vectorized = vectorized # 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, stacklevel=2) self._updating = 'deferred' if vectorized and workers != 1: warnings.warn("differential_evolution: the 'workers' keyword" " overrides the 'vectorized' keyword", stacklevel=2) self.vectorized = vectorized = False if vectorized and updating == 'immediate': warnings.warn("differential_evolution: the 'vectorized' keyword" " has overridden updating='immediate' to updating" "='deferred'", UserWarning, stacklevel=2) self._updating = 'deferred' # an object with a map method. if vectorized: def maplike_for_vectorized_func(func, x): # send an array (N, S) to the user func, # expect to receive (S,). Transposition is required because # internally the population is held as (S, N) return np.atleast_1d(func(x.T)) workers = maplike_for_vectorized_func 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]] if isinstance(bounds, Bounds): self.limits = np.array(new_bounds_to_old(bounds.lb, bounds.ub, len(bounds.lb)), dtype=float).T else: 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 finite ' '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]) with np.errstate(divide='ignore'): # if lb == ub then the following line will be 1/0, which is why # we ignore the divide by zero warning. The result from 1/0 is # inf, so replace those values by 0. self.__recip_scale_arg2 = 1 / self.__scale_arg2 self.__recip_scale_arg2[~np.isfinite(self.__recip_scale_arg2)] = 0 self.parameter_count = np.size(self.limits, 1) self.random_number_generator = check_random_state(seed) # Which parameters are going to be integers? if np.any(integrality): # # user has provided a truth value for integer constraints integrality = np.broadcast_to( integrality, self.parameter_count ) integrality = np.asarray(integrality, bool) # For integrality parameters change the limits to only allow # integer values lying between the limits. lb, ub = np.copy(self.limits) lb = np.ceil(lb) ub = np.floor(ub) if not (lb[integrality] <= ub[integrality]).all(): # there's a parameter that doesn't have an integer value # lying between the limits raise ValueError("One of the integrality constraints does not" " have any possible integer values between" " the lower/upper bounds.") nlb = np.nextafter(lb[integrality] - 0.5, np.inf) nub = np.nextafter(ub[integrality] + 0.5, -np.inf) self.integrality = integrality self.limits[0, self.integrality] = nlb self.limits[1, self.integrality] = nub else: self.integrality = False # check for equal bounds eb = self.limits[0] == self.limits[1] eb_count = np.count_nonzero(eb) # 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 # 202301 - reduced population size to account for parameters with # equal bounds. If there are no varying parameters set N to at least 1 self.num_population_members = max( 5, popsize * max(1, self.parameter_count - eb_count) ) self.population_shape = (self.num_population_members, self.parameter_count) self._nfev = 0 # check first str otherwise will fail to compare str with array if isinstance(init, str): if init == 'latinhypercube': self.init_population_lhs() elif init == 'sobol': # must be Ns = 2**m for Sobol' n_s = int(2 ** np.ceil(np.log2(self.num_population_members))) self.num_population_members = n_s self.population_shape = (self.num_population_members, self.parameter_count) self.init_population_qmc(qmc_engine='sobol') elif init == 'halton': self.init_population_qmc(qmc_engine='halton') elif init == 'random': self.init_population_random() else: raise ValueError(self.__init_error_msg) else: self.init_population_array(init) if x0 is not None: # scale to within unit interval and # ensure parameters are within bounds. x0_scaled = self._unscale_parameters(np.asarray(x0)) if ((x0_scaled > 1.0) | (x0_scaled < 0.0)).any(): raise ValueError( "Some entries in x0 lay outside the specified bounds" ) self.population[0] = x0_scaled # infrastructure for constraints self.constraints = constraints self._wrapped_constraints = [] if hasattr(constraints, '__len__'): # sequence of constraints, this will also deal with default # keyword parameter for c in constraints: self._wrapped_constraints.append( _ConstraintWrapper(c, self.x) ) else: self._wrapped_constraints = [ _ConstraintWrapper(constraints, self.x) ] self.total_constraints = np.sum( [c.num_constr for c in self._wrapped_constraints] ) self.constraint_violation = np.zeros((self.num_population_members, 1)) self.feasible = np.ones(self.num_population_members, bool) # an array to shuffle when selecting candidates. Create it here # rather than repeatedly creating it in _select_samples. self._random_population_index = np.arange(self.num_population_members) 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.uniform(size=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_qmc(self, qmc_engine): """Initializes the population with a QMC method. QMC methods ensures that each parameter is uniformly sampled over its range. Parameters ---------- qmc_engine : str The QMC method to use for initialization. Can be one of ``latinhypercube``, ``sobol`` or ``halton``. """ from scipy.stats import qmc rng = self.random_number_generator # Create an array for population of candidate solutions. if qmc_engine == 'latinhypercube': sampler = qmc.LatinHypercube(d=self.parameter_count, seed=rng) elif qmc_engine == 'sobol': sampler = qmc.Sobol(d=self.parameter_count, seed=rng) elif qmc_engine == 'halton': sampler = qmc.Halton(d=self.parameter_count, seed=rng) else: raise ValueError(self.__init_error_msg) self.population = sampler.random(n=self.num_population_members) # 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): """ Initializes 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.uniform(size=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): """ Initializes the population with a user specified population. Parameters ---------- init : np.ndarray Array specifying subset of the initial population. The array should have shape (S, N), where N 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.asarray(init, dtype=np.float64) 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" " (S, len(x)), where S > 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.full(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. """ if np.any(np.isinf(self.population_energies)): return False 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, ``message`` which describes the cause of the termination, ``population`` the solution vectors present in the population, and ``population_energies`` the value of the objective function for each entry in ``population``. 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. If the eventual solution does not satisfy the applied constraints ``success`` will be `False`. """ 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.feasible, self.constraint_violation = ( self._calculate_population_feasibilities(self.population)) # only work out population energies for feasible solutions self.population_energies[self.feasible] = ( self._calculate_population_energies( self.population[self.feasible])) self._promote_lowest_energy() # do the optimization. for nit in range(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(f"differential_evolution step {nit}: f(x)=" f" {self.population_energies[0]}" ) if self.callback: c = self.tol / (self.convergence + _MACHEPS) res = self._result(nit=nit, message="in progress") res.convergence = c try: warning_flag = bool(self.callback(res)) except StopIteration: warning_flag = True if warning_flag: status_message = 'callback function requested stop early' # should the solver terminate? if warning_flag or self.converged(): break else: status_message = _status_message['maxiter'] warning_flag = True DE_result = self._result( nit=nit, message=status_message, warning_flag=warning_flag ) if self.polish and not np.all(self.integrality): # can't polish if all the parameters are integers if np.any(self.integrality): # set the lower/upper bounds equal so that any integrality # constraints work. limits, integrality = self.limits, self.integrality limits[0, integrality] = DE_result.x[integrality] limits[1, integrality] = DE_result.x[integrality] polish_method = 'L-BFGS-B' if self._wrapped_constraints: polish_method = 'trust-constr' constr_violation = self._constraint_violation_fn(DE_result.x) if np.any(constr_violation > 0.): warnings.warn("differential evolution didn't find a " "solution satisfying the constraints, " "attempting to polish from the least " "infeasible solution", UserWarning, stacklevel=2) if self.disp: print(f"Polishing solution with '{polish_method}'") result = minimize(self.func, np.copy(DE_result.x), method=polish_method, bounds=self.limits.T, constraints=self.constraints) self._nfev += result.nfev DE_result.nfev = self._nfev # Polishing solution is only accepted if there is an improvement in # cost function, the polishing was successful and the solution lies # within the bounds. if (result.fun < DE_result.fun and result.success and np.all(result.x <= self.limits[1]) and np.all(self.limits[0] <= result.x)): 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) if self._wrapped_constraints: DE_result.constr = [c.violation(DE_result.x) for c in self._wrapped_constraints] DE_result.constr_violation = np.max( np.concatenate(DE_result.constr)) DE_result.maxcv = DE_result.constr_violation if DE_result.maxcv > 0: # if the result is infeasible then success must be False DE_result.success = False DE_result.message = ("The solution does not satisfy the " f"constraints, MAXCV = {DE_result.maxcv}") return DE_result def _result(self, **kwds): # form an intermediate OptimizeResult nit = kwds.get('nit', None) message = kwds.get('message', None) warning_flag = kwds.get('warning_flag', False) result = OptimizeResult( x=self.x, fun=self.population_energies[0], nfev=self._nfev, nit=nit, message=message, success=(warning_flag is not True), population=self._scale_parameters(self.population), population_energies=self.population_energies ) if self._wrapped_constraints: result.constr = [c.violation(result.x) for c in self._wrapped_constraints] result.constr_violation = np.max(np.concatenate(result.constr)) result.maxcv = result.constr_violation if result.maxcv > 0: result.success = False return result def _calculate_population_energies(self, population): """ Calculate the energies of a population. Parameters ---------- population : ndarray An array of parameter vectors normalised to [0, 1] using lower and upper limits. Has shape ``(np.size(population, 0), N)``. 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) # S is the number of function evals left to stay under the # maxfun budget S = min(num_members, self.maxfun - self._nfev) energies = np.full(num_members, np.inf) parameters_pop = self._scale_parameters(population) try: calc_energies = list( self._mapwrapper(self.func, parameters_pop[0:S]) ) calc_energies = np.squeeze(calc_energies) except (TypeError, ValueError) as e: # 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'" ) from e if calc_energies.size != S: if self.vectorized: raise RuntimeError("The vectorized function must return an" " array of shape (S,) when given an array" " of shape (len(x), S)") raise RuntimeError("func(x, *args) must return a scalar value") energies[0:S] = calc_energies if self.vectorized: self._nfev += 1 else: self._nfev += S return energies def _promote_lowest_energy(self): # swaps 'best solution' into first population entry idx = np.arange(self.num_population_members) feasible_solutions = idx[self.feasible] if feasible_solutions.size: # find the best feasible solution idx_t = np.argmin(self.population_energies[feasible_solutions]) l = feasible_solutions[idx_t] else: # no solution was feasible, use 'best' infeasible solution, which # will violate constraints the least l = np.argmin(np.sum(self.constraint_violation, axis=1)) self.population_energies[[0, l]] = self.population_energies[[l, 0]] self.population[[0, l], :] = self.population[[l, 0], :] self.feasible[[0, l]] = self.feasible[[l, 0]] self.constraint_violation[[0, l], :] = ( self.constraint_violation[[l, 0], :]) def _constraint_violation_fn(self, x): """ Calculates total constraint violation for all the constraints, for a set of solutions. Parameters ---------- x : ndarray Solution vector(s). Has shape (S, N), or (N,), where S is the number of solutions to investigate and N is the number of parameters. Returns ------- cv : ndarray Total violation of constraints. Has shape ``(S, M)``, where M is the total number of constraint components (which is not necessarily equal to len(self._wrapped_constraints)). """ # how many solution vectors you're calculating constraint violations # for S = np.size(x) // self.parameter_count _out = np.zeros((S, self.total_constraints)) offset = 0 for con in self._wrapped_constraints: # the input/output of the (vectorized) constraint function is # {(N, S), (N,)} --> (M, S) # The input to _constraint_violation_fn is (S, N) or (N,), so # transpose to pass it to the constraint. The output is transposed # from (M, S) to (S, M) for further use. c = con.violation(x.T).T # The shape of c should be (M,), (1, M), or (S, M). Check for # those shapes, as an incorrect shape indicates that the # user constraint function didn't return the right thing, and # the reshape operation will fail. Intercept the wrong shape # to give a reasonable error message. I'm not sure what failure # modes an inventive user will come up with. if c.shape[-1] != con.num_constr or (S > 1 and c.shape[0] != S): raise RuntimeError("An array returned from a Constraint has" " the wrong shape. If `vectorized is False`" " the Constraint should return an array of" " shape (M,). If `vectorized is True` then" " the Constraint must return an array of" " shape (M, S), where S is the number of" " solution vectors and M is the number of" " constraint components in a given" " Constraint object.") # the violation function may return a 1D array, but is it a # sequence of constraints for one solution (S=1, M>=1), or the # value of a single constraint for a sequence of solutions # (S>=1, M=1) c = np.reshape(c, (S, con.num_constr)) _out[:, offset:offset + con.num_constr] = c offset += con.num_constr return _out def _calculate_population_feasibilities(self, population): """ Calculate the feasibilities of a population. Parameters ---------- population : ndarray An array of parameter vectors normalised to [0, 1] using lower and upper limits. Has shape ``(np.size(population, 0), N)``. Returns ------- feasible, constraint_violation : ndarray, ndarray Boolean array of feasibility for each population member, and an array of the constraint violation for each population member. constraint_violation has shape ``(np.size(population, 0), M)``, where M is the number of constraints. """ num_members = np.size(population, 0) if not self._wrapped_constraints: # shortcut for no constraints return np.ones(num_members, bool), np.zeros((num_members, 1)) # (S, N) parameters_pop = self._scale_parameters(population) if self.vectorized: # (S, M) constraint_violation = np.array( self._constraint_violation_fn(parameters_pop) ) else: # (S, 1, M) constraint_violation = np.array([self._constraint_violation_fn(x) for x in parameters_pop]) # if you use the list comprehension in the line above it will # create an array of shape (S, 1, M), because each iteration # generates an array of (1, M). In comparison the vectorized # version returns (S, M). It's therefore necessary to remove axis 1 constraint_violation = constraint_violation[:, 0] feasible = ~(np.sum(constraint_violation, axis=1) > 0) return feasible, constraint_violation def __iter__(self): return self def __enter__(self): return self def __exit__(self, *args): return self._mapwrapper.__exit__(*args) def _accept_trial(self, energy_trial, feasible_trial, cv_trial, energy_orig, feasible_orig, cv_orig): """ Trial is accepted if: * it satisfies all constraints and provides a lower or equal objective function value, while both the compared solutions are feasible - or - * it is feasible while the original solution is infeasible, - or - * it is infeasible, but provides a lower or equal constraint violation for all constraint functions. This test corresponds to section III of Lampinen [1]_. Parameters ---------- energy_trial : float Energy of the trial solution feasible_trial : float Feasibility of trial solution cv_trial : array-like Excess constraint violation for the trial solution energy_orig : float Energy of the original solution feasible_orig : float Feasibility of original solution cv_orig : array-like Excess constraint violation for the original solution Returns ------- accepted : bool """ if feasible_orig and feasible_trial: return energy_trial <= energy_orig elif feasible_trial and not feasible_orig: return True elif not feasible_trial and (cv_trial <= cv_orig).all(): # cv_trial < cv_orig would imply that both trial and orig are not # feasible return True return False 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.feasible, self.constraint_violation = ( self._calculate_population_feasibilities(self.population)) # only need to work out population energies for those that are # feasible self.population_energies[self.feasible] = ( self._calculate_population_energies( self.population[self.feasible])) self._promote_lowest_energy() if self.dither is not None: self.scale = self.random_number_generator.uniform(self.dither[0], self.dither[1]) 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 if self._wrapped_constraints: cv = self._constraint_violation_fn(parameters) feasible = False energy = np.inf if not np.sum(cv) > 0: # solution is feasible feasible = True energy = self.func(parameters) self._nfev += 1 else: feasible = True cv = np.atleast_2d([0.]) energy = self.func(parameters) self._nfev += 1 # compare trial and population member if self._accept_trial(energy, feasible, cv, self.population_energies[candidate], self.feasible[candidate], self.constraint_violation[candidate]): self.population[candidate] = trial self.population_energies[candidate] = np.squeeze(energy) self.feasible[candidate] = feasible self.constraint_violation[candidate] = cv # if the trial candidate is also better than the best # solution then promote it. if self._accept_trial(energy, feasible, cv, self.population_energies[0], self.feasible[0], self.constraint_violation[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 = self._mutate_many( np.arange(self.num_population_members) ) # enforce bounds self._ensure_constraint(trial_pop) # determine the energies of the objective function, but only for # feasible trials feasible, cv = self._calculate_population_feasibilities(trial_pop) trial_energies = np.full(self.num_population_members, np.inf) # only calculate for feasible entries trial_energies[feasible] = self._calculate_population_energies( trial_pop[feasible]) # which solutions are 'improved'? loc = [self._accept_trial(*val) for val in zip(trial_energies, feasible, cv, self.population_energies, self.feasible, self.constraint_violation)] loc = np.array(loc) self.population = np.where(loc[:, np.newaxis], trial_pop, self.population) self.population_energies = np.where(loc, trial_energies, self.population_energies) self.feasible = np.where(loc, feasible, self.feasible) self.constraint_violation = np.where(loc[:, np.newaxis], cv, self.constraint_violation) # 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] def _scale_parameters(self, trial): """Scale from a number between 0 and 1 to parameters.""" # trial either has shape (N, ) or (L, N), where L is the number of # solutions being scaled scaled = self.__scale_arg1 + (trial - 0.5) * self.__scale_arg2 if np.count_nonzero(self.integrality): i = np.broadcast_to(self.integrality, scaled.shape) scaled[i] = np.round(scaled[i]) return scaled def _unscale_parameters(self, parameters): """Scale from parameters to a number between 0 and 1.""" return (parameters - self.__scale_arg1) * self.__recip_scale_arg2 + 0.5 def _ensure_constraint(self, trial): """Make sure the parameters lie between the limits.""" mask = np.bitwise_or(trial > 1, trial < 0) if oob := np.count_nonzero(mask): trial[mask] = self.random_number_generator.uniform(size=oob) def _mutate_custom(self, candidate): rng = self.random_number_generator msg = ( "strategy must have signature" " f(candidate: int, population: np.ndarray, rng=None) returning an" " array of shape (N,)" ) _population = self._scale_parameters(self.population) if not len(np.shape(candidate)): # single entry in population trial = self.strategy(candidate, _population, rng=rng) if trial.shape != (self.parameter_count,): raise RuntimeError(msg) else: S = candidate.shape[0] trial = np.array( [self.strategy(c, _population, rng=rng) for c in candidate], dtype=float ) if trial.shape != (S, self.parameter_count): raise RuntimeError(msg) return self._unscale_parameters(trial) def _mutate_many(self, candidates): """Create trial vectors based on a mutation strategy.""" rng = self.random_number_generator S = len(candidates) if callable(self.strategy): return self._mutate_custom(candidates) trial = np.copy(self.population[candidates]) samples = np.array([self._select_samples(c, 5) for c in candidates]) if self.strategy in ['currenttobest1exp', 'currenttobest1bin']: bprime = self.mutation_func(candidates, samples) else: bprime = self.mutation_func(samples) fill_point = rng_integers(rng, self.parameter_count, size=S) crossovers = rng.uniform(size=(S, self.parameter_count)) crossovers = crossovers < self.cross_over_probability if self.strategy in self._binomial: # 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. i = np.arange(S) crossovers[i, fill_point[i]] = True trial = np.where(crossovers, bprime, trial) return trial elif self.strategy in self._exponential: crossovers[..., 0] = True for j in range(S): i = 0 init_fill = fill_point[j] while (i < self.parameter_count and crossovers[j, i]): trial[j, init_fill] = bprime[j, init_fill] init_fill = (init_fill + 1) % self.parameter_count i += 1 return trial def _mutate(self, candidate): """Create a trial vector based on a mutation strategy.""" rng = self.random_number_generator if callable(self.strategy): return self._mutate_custom(candidate) fill_point = rng_integers(rng, self.parameter_count) samples = self._select_samples(candidate, 5) trial = np.copy(self.population[candidate]) if self.strategy in ['currenttobest1exp', 'currenttobest1bin']: bprime = self.mutation_func(candidate, samples) else: bprime = self.mutation_func(samples) crossovers = rng.uniform(size=self.parameter_count) crossovers = crossovers < self.cross_over_probability if self.strategy in self._binomial: # 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 crossovers[0] = True while i < self.parameter_count and crossovers[i]: trial[fill_point] = bprime[fill_point] fill_point = (fill_point + 1) % self.parameter_count i += 1 return trial def _best1(self, samples): """best1bin, best1exp""" # samples.shape == (S, 5) # or # samples.shape(5,) r0, r1 = samples[..., :2].T return (self.population[0] + self.scale * (self.population[r0] - self.population[r1])) def _rand1(self, samples): """rand1bin, rand1exp""" r0, r1, r2 = samples[..., :3].T return (self.population[r0] + self.scale * (self.population[r1] - self.population[r2])) def _randtobest1(self, samples): """randtobest1bin, randtobest1exp""" r0, r1, r2 = samples[..., :3].T 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].T 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].T 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[..., :5].T 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. """ self.random_number_generator.shuffle(self._random_population_index) idxs = self._random_population_index[:number_samples + 1] return idxs[idxs != candidate][:number_samples] class _ConstraintWrapper: """Object to wrap/evaluate user defined constraints. Very similar in practice to `PreparedConstraint`, except that no evaluation of jac/hess is performed (explicit or implicit). If created successfully, it will contain the attributes listed below. Parameters ---------- constraint : {`NonlinearConstraint`, `LinearConstraint`, `Bounds`} Constraint to check and prepare. x0 : array_like Initial vector of independent variables, shape (N,) Attributes ---------- fun : callable Function defining the constraint wrapped by one of the convenience classes. bounds : 2-tuple Contains lower and upper bounds for the constraints --- lb and ub. These are converted to ndarray and have a size equal to the number of the constraints. Notes ----- _ConstraintWrapper.fun and _ConstraintWrapper.violation can get sent arrays of shape (N, S) or (N,), where S is the number of vectors of shape (N,) to consider constraints for. """ def __init__(self, constraint, x0): self.constraint = constraint if isinstance(constraint, NonlinearConstraint): def fun(x): x = np.asarray(x) return np.atleast_1d(constraint.fun(x)) elif isinstance(constraint, LinearConstraint): def fun(x): if issparse(constraint.A): A = constraint.A else: A = np.atleast_2d(constraint.A) res = A.dot(x) # x either has shape (N, S) or (N) # (M, N) x (N, S) --> (M, S) # (M, N) x (N,) --> (M,) # However, if (M, N) is a matrix then: # (M, N) * (N,) --> (M, 1), we need this to be (M,) if x.ndim == 1 and res.ndim == 2: # deal with case that constraint.A is an np.matrix # see gh20041 res = np.asarray(res)[:, 0] return res elif isinstance(constraint, Bounds): def fun(x): return np.asarray(x) else: raise ValueError("`constraint` of an unknown type is passed.") self.fun = fun lb = np.asarray(constraint.lb, dtype=float) ub = np.asarray(constraint.ub, dtype=float) x0 = np.asarray(x0) # find out the number of constraints f0 = fun(x0) self.num_constr = m = f0.size self.parameter_count = x0.size if lb.ndim == 0: lb = np.resize(lb, m) if ub.ndim == 0: ub = np.resize(ub, m) self.bounds = (lb, ub) def __call__(self, x): return np.atleast_1d(self.fun(x)) def violation(self, x): """How much the constraint is exceeded by. Parameters ---------- x : array-like Vector of independent variables, (N, S), where N is number of parameters and S is the number of solutions to be investigated. Returns ------- excess : array-like How much the constraint is exceeded by, for each of the constraints specified by `_ConstraintWrapper.fun`. Has shape (M, S) where M is the number of constraint components. """ # expect ev to have shape (num_constr, S) or (num_constr,) ev = self.fun(np.asarray(x)) try: excess_lb = np.maximum(self.bounds[0] - ev.T, 0) excess_ub = np.maximum(ev.T - self.bounds[1], 0) except ValueError as e: raise RuntimeError("An array returned from a Constraint has" " the wrong shape. If `vectorized is False`" " the Constraint should return an array of" " shape (M,). If `vectorized is True` then" " the Constraint must return an array of" " shape (M, S), where S is the number of" " solution vectors and M is the number of" " constraint components in a given" " Constraint object.") from e v = (excess_lb + excess_ub).T return v