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"""
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
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 <multiprocessing>`).
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 <multiprocessing>`).
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)
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 = 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, 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.any(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.where((trial > 1) | (trial < 0))
trial[mask] = self.random_number_generator.uniform(size=mask[0].shape)
def _mutate(self, candidate):
"""Create a trial vector based on a mutation strategy."""
rng = self.random_number_generator
if callable(self.strategy):
_population = self._scale_parameters(self.population)
trial = np.array(
self.strategy(candidate, _population, rng=rng), dtype=float
)
if trial.shape != (self.parameter_count,):
raise RuntimeError(
"strategy must have signature"
" f(candidate: int, population: np.ndarray, rng=None)"
" returning an array of shape (N,)"
)
return self._unscale_parameters(trial)
trial = np.copy(self.population[candidate])
fill_point = rng.choice(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.uniform(size=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
crossovers = rng.uniform(size=self.parameter_count)
crossovers = crossovers < self.cross_over_probability
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"""
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.
"""
pool = np.arange(self.num_population_members)
self.random_number_generator.shuffle(pool)
idxs = []
while len(idxs) < number_samples and len(pool) > 0:
idx = pool[0]
pool = pool[1:]
if idx != candidate:
idxs.append(idx)
return idxs
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