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def differential_evolution(func, bounds, args=(), strategy='best1bin', maxiter=1000, popsize=15, tol=0.01, mutation=(0.5, 1), recombination=0.7, rng=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(population_energies) <= 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 :math:`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.
rng : {None, int, `numpy.random.Generator`}, optional
If `rng` is passed by keyword, types other than `numpy.random.Generator` are
passed to `numpy.random.default_rng` to instantiate a ``Generator``.
If `rng` is already a ``Generator`` instance, then the provided instance is
used. Specify `rng` for repeatable function behavior.
If this argument is passed by position or `seed` is passed by keyword,
legacy behavior for the argument `seed` applies:
- If `seed` is None (or `numpy.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.
.. versionchanged:: 1.15.0
As part of the `SPEC-007 <https://scientific-python.org/specs/spec-0007/>`_
transition from use of `numpy.random.RandomState` to
`numpy.random.Generator`, this keyword was changed from `seed` to `rng`.
For an interim period, both keywords will continue to work, although only one
may be specified at a time. After the interim period, function calls using the
`seed` keyword will emit warnings. The behavior of both `seed` and
`rng` are outlined above, but only the `rng` keyword should be used in new code.
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.
.. versionchanged:: 1.15.0
If `workers` is specified then the map-like callable that wraps
`func` is supplied to `minimize` instead of it using `func`
directly. This allows the caller to control how and where the
invocations actually run.
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 + F \cdot (x_{r_0} - x_{r_1})
where :math:`F` is the `mutation` parameter.
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]_.
- ``rand1`` : :math:`b' = x_{r_0} + F \cdot (x_{r_1} - x_{r_2})`
- ``rand2`` : :math:`b' = x_{r_0} + F \cdot (x_{r_1} + x_{r_2} - x_{r_3} - x_{r_4})`
- ``best1`` : :math:`b' = x_0 + F \cdot (x_{r_0} - x_{r_1})`
- ``best2`` : :math:`b' = x_0 + F \cdot (x_{r_0} + x_{r_1} - x_{r_2} - x_{r_3})`
- ``currenttobest1`` : :math:`b' = x_i + F \cdot (x_0 - x_i + x_{r_0} - x_{r_1})`
- ``randtobest1`` : :math:`b' = x_{r_0} + F \cdot (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,
... rng=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, rng=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', rng=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
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