image-match/python/py/Lib/site-packages/scipy/optimize/_differentialevolution.py

"""
differential_evolution: The differential evolution global optimization algorithm
Added by Andrew Nelson 2014
"""
from functools import partial
import warnings

import numpy as np

from scipy.optimize import OptimizeResult, minimize
from scipy.optimize._constraints import (Bounds, new_bounds_to_old,
                                         NonlinearConstraint, LinearConstraint)
from scipy.optimize._optimize import _status_message, _wrap_callback
from scipy._lib._util import (check_random_state, MapWrapper, _FunctionWrapper,
                              rng_integers, _transition_to_rng)
from scipy._lib._sparse import issparse


__all__ = ['differential_evolution']


_MACHEPS = np.finfo(np.float64).eps


@_transition_to_rng("seed", position_num=9)
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):
    r"""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 : `numpy.random.Generator`, optional
        Pseudorandom number generator state. When `rng` is None, a new
        `numpy.random.Generator` is created using entropy from the
        operating system. Types other than `numpy.random.Generator` are
        passed to `numpy.random.default_rng` to instantiate a ``Generator``.
    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, callable}, 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.
        Alternatively supply a callable that has a `minimize`-like signature,
        ``polish_func(func, x0, **kwds)`` and returns an `OptimizeResult`. This
        allows the user to have fine control over how the polishing occurs.
        `bounds` and `constraints` will be present in ``kwds``. Extra keywords
        could be supplied to `polish_func` using `functools.partial`. It is the
        user's responsibility to ensure that the polishing function obeys
        bounds, any constraints (including integrality constraints), and that
        appropriate attributes are set in the `OptimizeResult`, such as ``fun``,
        ``x``, ``nfev``, ``jac``.

        .. 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.

        .. versionchanged:: 1.17.0
            A callable obeying the `minimize` signature can be supplied to
            polish the best population member.

    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(population_energies) <= 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. A randomly
    selected parameter is always loaded from ``b'``. For binomial crossover,
    this is a single random parameter. For exponential crossover, this is the
    starting point of a consecutive sequence of parameters 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 final polishing step can be customised by providing a callable that
    mimics the `minimize` interface. The user is responsible for ensuring that
    the minimizer obeys any bounds and constraints.

    >>> from functools import partial
    >>> from scipy.optimize import minimize
    >>> # supply extra parameters to the polishing function using partial
    >>> polish_func = partial(minimize, method="SLSQP")
    >>> result = differential_evolution(
    ...     ackley, bounds, vectorized=True, updating='deferred', rng=1,
    ...     polish=polish_func
    ... )
    >>> 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

    """# noqa: E501

    # 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,
                                     rng=rng, 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(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 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

        ..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 (only specify
            one of them). After the interim period using the `seed` keyword will emit
            warnings. The behavior of the `seed` and `rng` keywords is outlined below.

        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.

        If this argument is passed by position or `seed` is passed by keyword, the
        behavior is:

        - 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`/`rng` 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, callable}, 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.
        Alternatively supply a callable that has a `minimize`-like signature,
        ``polish_func(func, x0, **kwds)`` and returns an `OptimizeResult`. This
        allows the user to have fine control over how the polishing occurs.
        `bounds` and `constraints` will be present in ``kwds``. Extra keywords
        could be supplied to `polish_func` using `functools.partial`. It is the
        user's responsibility to ensure that the polishing function obeys
        bounds, any constraints (including integrality constraints), and that
        appropriate attributes are set in the `OptimizeResult`, such as ``fun``,
        ```x``, ``nfev``, ``jac``.
    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'``.
    """ # noqa: E501

    # 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'}
    __combined = _binomial | _exponential

    __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, rng=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 not in self.__combined:
            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.original_func = func
        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(rng)

        # 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

    @property
    def mutation_func(self):
        return getattr(self, self.__combined[self.strategy])

    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)

            pf = self.polish
            _f = self.original_func
            if not callable(pf):
                pf = partial(minimize, method=polish_method)
                def _f(x):
                    return list(self._mapwrapper(self.func, np.atleast_2d(x)))[0]

                if self.disp:
                    print(f"Polishing solution with '{polish_method}'")

            result = pf(
                _f,
                np.copy(DE_result.x),
                bounds=Bounds(lb=self.limits[0], ub=self.limits[1]),
                constraints=self.constraints
            )

            if not isinstance(result, OptimizeResult):
                raise ValueError(
                    "The result from a user defined polishing function "
                     "should return an OptimizeResult."
                )

            self._nfev += result.get("nfev", 0)
            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.get("jac", None)
                # 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:
            # A randomly selected parameter is always from the bprime vector for
            # binomial crossover. The fill_point ensures at least one parameter
            # comes from bprime, preventing the possibility of no mutation
            # influence in the trial vector.
            i = np.arange(S)
            crossovers[i, fill_point[i]] = True
            trial = np.where(crossovers, bprime, trial)
            return trial

        elif self.strategy in self._exponential:
            # For exponential crossover, fill_point determines the starting index
            # for a consecutive sequence of parameters from bprime. The sequence
            # continues until a crossover probability check fails. The starting
            # index is always from the bprime vector ensuring at least one
            # parameter comes from bprime.
            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:
            # A randomly selected parameter is always from the bprime vector for
            # binomial crossover. The fill_point ensures at least one parameter
            # comes from bprime, preventing the possibility of no mutation
            # influence in the trial vector.
            crossovers[fill_point] = True
            trial = np.where(crossovers, bprime, trial)
            return trial

        elif self.strategy in self._exponential:
            # For exponential crossover, fill_point determines the starting index
            # for a consecutive sequence of parameters from bprime. The sequence
            # continues until a crossover probability check fails. The starting
            # index is always from the bprime vector ensuring at least one
            # parameter comes from bprime.
            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