# Source code for deeprobust.image.optimizer

    """
    This module include the following optimizer:
    1. differential_evolution:
    The differential evolution global optimization algorithm
    https://github.com/scipy/scipy/blob/70e61dee181de23fdd8d893eaa9491100e2218d7/scipy/optimize/_differentialevolution.py

    modified by:
    https://github.com/DebangLi/one-pixel-attack-pytorch/blob/master/differential_evolution.py

    2. Basic Adam Optimizer

    """

    from __future__ import division, print_function, absolute_import
    import numpy as np
    from scipy.optimize import OptimizeResult, minimize
    from scipy.optimize.optimize import _status_message
    from scipy._lib._util import check_random_state
    import warnings

    __all__ = ['differential_evolution', 'AdamOptimizer']

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

    def differential_evolution(func, bounds, args=(), strategy='best1bin',
                               maxiter=1000, popsize=15, tol=0.01,
                               mutation=(0.5, 1), recombination=0.7, seed=None,
                               callback=None, disp=False, polish=True,
                               init='latinhypercube', atol=0):
        """Finds the global minimum of a multivariate function.
        Differential Evolution is stochastic in nature (does not use gradient
        methods) to find the minimium, 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 [1]_.
        Parameters
        ----------
        func : callable
            The objective function to be minimized.  Must be in the form
            ``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array
            and ``args`` is a  tuple of any additional fixed parameters needed to
            completely specify the function.
        bounds : sequence
            Bounds for variables.  ``(min, max)`` pairs for each element in ``x``,
            defining the lower and upper bounds for the optimizing argument of
            `func`. It is required to have ``len(bounds) == len(x)``.
            ``len(bounds)`` is used to determine the number of parameters in ``x``.
        args : tuple, optional
            Any additional fixed parameters needed to
            completely specify the objective function.
        strategy : str, optional
            The differential evolution strategy to use. Should be one of:
                - 'best1bin'
                - 'best1exp'
                - 'rand1exp'
                - 'randtobest1exp'
                - 'currenttobest1exp'
                - 'best2exp'
                - 'rand2exp'
                - 'randtobest1bin'
                - 'currenttobest1bin'
                - 'best2bin'
                - 'rand2bin'
                - 'rand1bin'
            The default is 'best1bin'.
        maxiter : int, optional
            The maximum number of generations over which the entire population is
            evolved. The maximum number of function evaluations (with no polishing)
            is: ``(maxiter + 1) * popsize * len(x)``
        popsize : int, optional
            A multiplier for setting the total population size.  The population has
            ``popsize * len(x)`` individuals (unless the initial population is
            supplied via the `init` keyword).
        tol : float, optional
            Relative tolerance for convergence, the solving stops when
            ``np.std(pop) <= atol + tol * np.abs(np.mean(population_energies))``,
            where and `atol` and `tol` are the absolute and relative tolerance
            respectively.
        mutation : float or tuple(float, float), optional
            The mutation constant. In the literature this is also known as
            differential weight, being denoted by F.
            If specified as a float it should be in the range [0, 2].
            If specified as a tuple ``(min, max)`` dithering is employed. Dithering
            randomly changes the mutation constant on a generation by generation
            basis. The mutation constant for that generation is taken from
            ``U[min, max)``. Dithering can help speed convergence significantly.
            Increasing the mutation constant increases the search radius, but will
            slow down convergence.
        recombination : float, optional
            The recombination constant, should be in the range [0, 1]. In the
            literature this is also known as the crossover probability, being
            denoted by CR. Increasing this value allows a larger number of mutants
            to progress into the next generation, but at the risk of population
            stability.
        seed : int or `np.random.RandomState`, optional
            If `seed` is not specified the `np.RandomState` singleton is used.
            If `seed` is an int, a new `np.random.RandomState` instance is used,
            seeded with seed.
            If `seed` is already a `np.random.RandomState instance`, then that
            `np.random.RandomState` instance is used.
            Specify `seed` for repeatable minimizations.
        disp : bool, optional
            Display status messages
        callback : callable, `callback(xk, convergence=val)`, optional
            A function to follow the progress of the minimization. ``xk`` is
            the current value of ``x0``. ``val`` represents the fractional
            value of the population convergence.  When ``val`` is greater than one
            the function halts. If callback returns `True`, then the minimization
            is halted (any polishing is still carried out).
        polish : bool, optional
            If True (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.
        init : str or array-like, optional
            Specify which type of population initialization is performed. Should be
            one of:
                - 'latinhypercube'
                - 'random'
                - array specifying the initial population. The array should have
                  shape ``(M, len(x))``, where len(x) is the number of parameters.
                  `init` is clipped to `bounds` before use.
            The default is 'latinhypercube'. Latin Hypercube sampling tries to
            maximize coverage of the available parameter space. 'random'
            initializes the population randomly - this has the drawback that
            clustering can occur, preventing the whole of parameter space being
            covered. Use of an array to specify a population subset 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.
        Returns
        -------
        res : OptimizeResult
            The optimization result represented as a `OptimizeResult` object.
            Important attributes are: ``x`` the solution array, ``success`` a
            Boolean flag indicating if the optimizer exited successfully and
            ``message`` which describes the cause of the termination. See
            `OptimizeResult` for a description of other attributes.  If `polish`
            was employed, and a lower minimum was obtained by the polishing, then
            OptimizeResult also contains the ``jac`` attribute.
        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 [2]_ 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:`b_0`,
        so far:
        .. math::
            b' = b_0 + mutation * (population[rand0] - population[rand1])
        A trial vector is then constructed. Starting with a randomly chosen 'i'th
        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.
        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.
        .. versionadded:: 0.15.0

        References
        ----------
        .. [1] 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.
        .. [2] http://www1.icsi.berkeley.edu/~storn/code.html
        .. [3] http://en.wikipedia.org/wiki/Differential_evolution
        """

        solver = DifferentialEvolutionSolver(func, bounds, args=args,
                                             strategy=strategy, maxiter=maxiter,
                                             popsize=popsize, tol=tol,
                                             mutation=mutation,
                                             recombination=recombination,
                                             seed=seed, polish=polish,
                                             callback=callback,
                                             disp=disp, init=init, atol=atol)
        return solver.solve()

    class DifferentialEvolutionSolver(object):

        """This class implements the differential evolution solver
        Parameters
        ----------
        func : callable
            The objective function to be minimized.  Must be in the form
            ``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array
            and ``args`` is a  tuple of any additional fixed parameters needed to
            completely specify the function.
        bounds : sequence
            Bounds for variables.  ``(min, max)`` pairs for each element in ``x``,
            defining the lower and upper bounds for the optimizing argument of
            `func`. It is required to have ``len(bounds) == len(x)``.
            ``len(bounds)`` is used to determine the number of parameters in ``x``.
        args : tuple, optional
            Any additional fixed parameters needed to
            completely specify the objective function.
        strategy : str, optional
            The differential evolution strategy to use. Should be one of:
                - 'best1bin'
                - 'best1exp'
                - 'rand1exp'
                - 'randtobest1exp'
                - 'currenttobest1exp'
                - 'best2exp'
                - 'rand2exp'
                - 'randtobest1bin'
                - 'currenttobest1bin'
                - 'best2bin'
                - 'rand2bin'
                - 'rand1bin'
            The default is 'best1bin'
        maxiter : int, optional
            The maximum number of generations over which the entire population is
            evolved. The maximum number of function evaluations (with no polishing)
            is: ``(maxiter + 1) * popsize * len(x)``
        popsize : int, optional
            A multiplier for setting the total population size.  The population has
            ``popsize * len(x)`` individuals (unless the initial population is
            supplied via the `init` keyword).
        tol : float, optional
            Relative tolerance for convergence, the solving stops when
            ``np.std(pop) <= atol + tol * np.abs(np.mean(population_energies))``,
            where and `atol` and `tol` are the absolute and relative tolerance
            respectively.
        mutation : float or tuple(float, float), optional
            The mutation constant. In the literature this is also known as
            differential weight, being denoted by F.
            If specified as a float it should be in the range [0, 2].
            If specified as a tuple ``(min, max)`` dithering is employed. Dithering
            randomly changes the mutation constant on a generation by generation
            basis. The mutation constant for that generation is taken from
            U[min, max). Dithering can help speed convergence significantly.
            Increasing the mutation constant increases the search radius, but will
            slow down convergence.
        recombination : float, optional
            The recombination constant, should be in the range [0, 1]. In the
            literature this is also known as the crossover probability, being
            denoted by CR. Increasing this value allows a larger number of mutants
            to progress into the next generation, but at the risk of population
            stability.
        seed : int or `np.random.RandomState`, optional
            If `seed` is not specified the `np.random.RandomState` singleton is
            used.
            If `seed` is an int, a new `np.random.RandomState` instance is used,
            seeded with `seed`.
            If `seed` is already a `np.random.RandomState` instance, then that
            `np.random.RandomState` instance is used.
            Specify `seed` for repeatable minimizations.
        disp : bool, optional
            Display status messages
        callback : callable, `callback(xk, convergence=val)`, optional
            A function to follow the progress of the minimization. ``xk`` is
            the current value of ``x0``. ``val`` represents the fractional
            value of the population convergence.  When ``val`` is greater than one
            the function halts. If callback returns `True`, then the minimization
            is halted (any polishing is still carried out).
        polish : bool, optional
            If True, then `scipy.optimize.minimize` with the `L-BFGS-B` method
            is used to polish the best population member at the end. This requires
            a few more function evaluations.
        maxfun : int, optional
            Set the maximum number of function evaluations. However, it probably
            makes more sense to set `maxiter` instead.
        init : str or array-like, optional
            Specify which type of population initialization is performed. Should be
            one of:
                - 'latinhypercube'
                - 'random'
                - array specifying the initial population. The array should have
                  shape ``(M, len(x))``, where len(x) is the number of parameters.
                  `init` is clipped to `bounds` before use.
            The default is 'latinhypercube'. Latin Hypercube sampling tries to
            maximize coverage of the available parameter space. 'random'
            initializes the population randomly - this has the drawback that
            clustering can occur, preventing the whole of parameter space being
            covered. Use of an array to specify a population could be used, for
            example, to create a tight bunch of initial guesses in an location
            where the solution is known to exist, thereby reducing time for
            convergence.
        atol : float, optional
            Absolute tolerance for convergence, the solving stops when
            ``np.std(pop) <= atol + tol * np.abs(np.mean(population_energies))``,
            where and `atol` and `tol` are the absolute and relative tolerance
            respectively.
        """

        # Dispatch of mutation strategy method (binomial or exponential).
        _binomial = {'best1bin': '_best1',
                     'randtobest1bin': '_randtobest1',
                     'currenttobest1bin': '_currenttobest1',
                     'best2bin': '_best2',
                     'rand2bin': '_rand2',
                     'rand1bin': '_rand1'}
        _exponential = {'best1exp': '_best1',
                        'rand1exp': '_rand1',
                        'randtobest1exp': '_randtobest1',
                        'currenttobest1exp': '_currenttobest1',
                        'best2exp': '_best2',
                        'rand2exp': '_rand2'}

        __init_error_msg = ("The population initialization method must be one of "
                            "'latinhypercube' or 'random', or an array of shape "
                            "(M, N) where N is the number of parameters and M>5")

        def __init__(self, func, bounds, args=(),
                     strategy='best1bin', maxiter=1000, popsize=15,
                     tol=0.01, mutation=(0.5, 1), recombination=0.7, seed=None,
                     maxfun=np.inf, callback=None, disp=False, polish=True,
                     init='latinhypercube', atol=0):

            if strategy in self._binomial:
                self.mutation_func = getattr(self, self._binomial[strategy])
            elif strategy in self._exponential:
                self.mutation_func = getattr(self, self._exponential[strategy])
            else:
                raise ValueError("Please select a valid mutation strategy")
            self.strategy = strategy

            self.callback = callback
            self.polish = polish

            # 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

            self.func = func
            self.args = args

            # convert tuple of lower and upper bounds to limits
            # [(low_0, high_0), ..., (low_n, high_n]
            #     -> [[low_0, ..., low_n], [high_0, ..., high_n]]
            self.limits = np.array(bounds, dtype='float').T
            if (np.size(self.limits, 0) != 2 or not
                    np.all(np.isfinite(self.limits))):
                raise ValueError('bounds should be a sequence containing '
                                 'real valued (min, max) pairs for each value'
                                 ' in x')

            if maxiter is None:  # the default used to be None
                maxiter = 1000
            self.maxiter = maxiter
            if maxfun is None:  # the default used to be None
                maxfun = np.inf
            self.maxfun = maxfun

            # population is scaled to between [0, 1].
            # We have to scale between parameter <-> population
            # save these arguments for _scale_parameter and
            # _unscale_parameter. This is an optimization
            self.__scale_arg1 = 0.5 * (self.limits[0] + self.limits[1])
            self.__scale_arg2 = np.fabs(self.limits[0] - self.limits[1])

            self.parameter_count = np.size(self.limits, 1)

            self.random_number_generator = check_random_state(seed)

            # default population initialization is a latin hypercube design, but
            # there are other population initializations possible.
            # the minimum is 5 because 'best2bin' requires a population that's at
            # least 5 long
            self.num_population_members = max(5, popsize * self.parameter_count)

            self.population_shape = (self.num_population_members,
                                     self.parameter_count)

            self._nfev = 0
            if isinstance(init, str):
                if init == 'latinhypercube':
                    self.init_population_lhs()
                elif init == 'random':
                    self.init_population_random()
                else:
                    raise ValueError(self.__init_error_msg)
            else:
                self.init_population_array(init)

            self.disp = disp

        def init_population_lhs(self):
            """
            Initializes the population with Latin Hypercube Sampling.
            Latin Hypercube Sampling ensures that each parameter is uniformly
            sampled over its range.
            """
            rng = self.random_number_generator

            # Each parameter range needs to be sampled uniformly. The scaled
            # parameter range ([0, 1)) needs to be split into
            # `self.num_population_members` segments, each of which has the following
            # size:
            segsize = 1.0 / self.num_population_members

            # Within each segment we sample from a uniform random distribution.
            # We need to do this sampling for each parameter.
            samples = (segsize * rng.random_sample(self.population_shape)

            # Offset each segment to cover the entire parameter range [0, 1)
                       + np.linspace(0., 1., self.num_population_members,
                                     endpoint=False)[:, np.newaxis])

            # Create an array for population of candidate solutions.
            self.population = np.zeros_like(samples)

            # Initialize population of candidate solutions by permutation of the
            # random samples.
            for j in range(self.parameter_count):
                order = rng.permutation(range(self.num_population_members))
                self.population[:, j] = samples[order, j]

            # reset population energies
            self.population_energies = (np.ones(self.num_population_members) *
                                        np.inf)

            # reset number of function evaluations counter
            self._nfev = 0

        def init_population_random(self):
            """
            Initialises the population at random.  This type of initialization
            can possess clustering, Latin Hypercube sampling is generally better.
            """
            rng = self.random_number_generator
            self.population = rng.random_sample(self.population_shape)

            # reset population energies
            self.population_energies = (np.ones(self.num_population_members) *
                                        np.inf)

            # reset number of function evaluations counter
            self._nfev = 0

        def init_population_array(self, init):
            """
            Initialises the population with a user specified population.
            Parameters
            ----------
            init : np.ndarray
                Array specifying subset of the initial population. The array should
                have shape (M, len(x)), where len(x) is the number of parameters.
                The population is clipped to the lower and upper `bounds`.
            """
            # make sure you're using a float array
            popn = np.asfarray(init)

            if (np.size(popn, 0) < 5 or
                    popn.shape[1] != self.parameter_count or
                    len(popn.shape) != 2):
                raise ValueError("The population supplied needs to have shape"
                                 " (M, len(x)), where M > 4.")

            # scale values and clip to bounds, assigning to population
            self.population = np.clip(self._unscale_parameters(popn), 0, 1)

            self.num_population_members = np.size(self.population, 0)

            self.population_shape = (self.num_population_members,
                                     self.parameter_count)

            # reset population energies
            self.population_energies = (np.ones(self.num_population_members) *
                                        np.inf)

            # reset number of function evaluations counter
            self._nfev = 0

        @property
        def x(self):
            """
            The best solution from the solver
            Returns
            -------
            x : ndarray
                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.
            """
            return (np.std(self.population_energies) /
                    np.abs(np.mean(self.population_energies) + _MACHEPS))

        def solve(self):
            """
            Runs the DifferentialEvolutionSolver.
            Returns
            -------
            res : OptimizeResult
                The optimization result represented as a ``OptimizeResult`` object.
                Important attributes are: ``x`` the solution array, ``success`` a
                Boolean flag indicating if the optimizer exited successfully and
                ``message`` which describes the cause of the termination. See
                `OptimizeResult` for a description of other attributes.  If `polish`
                was employed, and a lower minimum was obtained by the polishing,
                then OptimizeResult also contains the ``jac`` attribute.
            """
            nit, warning_flag = 0, False
            status_message = _status_message['success']

            # The population may have just been initialized (all entries are
            # np.inf). If it has you have to calculate the initial energies.
            # Although this is also done in the evolve generator it's possible
            # that someone can set maxiter=0, at which point we still want the
            # initial energies to be calculated (the following loop isn't run).
            if np.all(np.isinf(self.population_energies)):
                self._calculate_population_energies()

            # do the optimisation.
            for nit in range(1, self.maxiter + 1):
                # evolve the population by a generation
                try:
                    next(self)
                except StopIteration:
                    warning_flag = True
                    status_message = _status_message['maxfev']
                    break

                if self.disp:
                    print("differential_evolution step %d: f(x)= %g"
                          % (nit,
                             self.population_energies[0]))

                # should the solver terminate?
                convergence = self.convergence

                if (self.callback and
                        self.callback(self._scale_parameters(self.population[0]),
                                      convergence=self.tol / convergence) is True):

                    warning_flag = True
                    status_message = ('callback function requested stop early '
                                      'by returning True')
                    break

                intol = (np.std(self.population_energies) <=
                         self.atol +
                         self.tol * np.abs(np.mean(self.population_energies)))
                if warning_flag or intol:
                    break

            else:
                status_message = _status_message['maxiter']
                warning_flag = True

            DE_result = OptimizeResult(
                x=self.x,
                fun=self.population_energies[0],
                nfev=self._nfev,
                nit=nit,
                message=status_message,
                success=(warning_flag is not True))

            if self.polish:
                result = minimize(self.func,
                                  np.copy(DE_result.x),
                                  method='L-BFGS-B',
                                  bounds=self.limits.T,
                                  args=self.args)

                self._nfev += result.nfev
                DE_result.nfev = self._nfev

                if result.fun < DE_result.fun:
                    DE_result.fun = result.fun
                    DE_result.x = result.x
                    DE_result.jac = result.jac
                    # to keep internal state consistent
                    self.population_energies[0] = result.fun
                    self.population[0] = self._unscale_parameters(result.x)

            return DE_result

        def _calculate_population_energies(self):
            """
            Calculate the energies of all the population members at the same time.
            Puts the best member in first place. Useful if the population has just
            been initialised.
            """

            ##############
            ## CHANGES: self.func operates on the entire parameters array
            ##############
            itersize = max(0, min(len(self.population), self.maxfun - self._nfev + 1))
            candidates = self.population[:itersize]
            parameters = np.array([self._scale_parameters(c) for c in candidates]) # TODO: vectorize
            energies = self.func(parameters, *self.args)
            self.population_energies = energies
            self._nfev += itersize

            # for index, candidate in enumerate(self.population):
            #     if self._nfev > self.maxfun:
            #         break

            #     parameters = self._scale_parameters(candidate)
            #     self.population_energies[index] = self.func(parameters,
            #                                                 *self.args)
            #     self._nfev += 1

            ##############
            ##############

            minval = np.argmin(self.population_energies)

            # put the lowest energy into the best solution position.
            lowest_energy = self.population_energies[minval]
            self.population_energies[minval] = self.population_energies[0]
            self.population_energies[0] = lowest_energy

            self.population[[0, minval], :] = self.population[[minval, 0], :]

        def __iter__(self):
            return self

        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._calculate_population_energies()

            if self.dither is not None:
                self.scale = (self.random_number_generator.rand()
                              * (self.dither[1] - self.dither[0]) + self.dither[0])

            ##############
            ## CHANGES: self.func operates on the entire parameters array
            ##############

            itersize = max(0, min(self.num_population_members, self.maxfun - self._nfev + 1))
            trials = np.array([self._mutate(c) for c in range(itersize)]) # TODO: vectorize
            for trial in trials: self._ensure_constraint(trial)
            parameters = np.array([self._scale_parameters(trial) for trial in trials])
            energies = self.func(parameters, *self.args)
            self._nfev += itersize

            for candidate,(energy,trial) in enumerate(zip(energies, trials)):
                # if the energy of the trial candidate is lower than the
                # original population member then replace it
                if energy < self.population_energies[candidate]:
                    self.population[candidate] = trial
                    self.population_energies[candidate] = energy

                    # if the trial candidate also has a lower energy than the
                    # best solution then replace that as well
                    if energy < self.population_energies[0]:
                        self.population_energies[0] = energy
                        self.population[0] = trial

            # for candidate in range(self.num_population_members):
            #     if self._nfev > self.maxfun:
            #         raise StopIteration

            #     # create a trial solution
            #     trial = self._mutate(candidate)

            #     # ensuring that it's in the range [0, 1)
            #     self._ensure_constraint(trial)

            #     # scale from [0, 1) to the actual parameter value
            #     parameters = self._scale_parameters(trial)

            #     # determine the energy of the objective function
            #     energy = self.func(parameters, *self.args)
            #     self._nfev += 1

            #     # if the energy of the trial candidate is lower than the
            #     # original population member then replace it
            #     if energy < self.population_energies[candidate]:
            #         self.population[candidate] = trial
            #         self.population_energies[candidate] = energy

            #         # if the trial candidate also has a lower energy than the
            #         # best solution then replace that as well
            #         if energy < self.population_energies[0]:
            #             self.population_energies[0] = energy
            #             self.population[0] = trial

            ##############
            ##############

            return self.x, self.population_energies[0]

        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.
            """
            # next() is required for compatibility with Python2.7.
            return self.__next__()

        def _scale_parameters(self, trial):
            """
            scale from a number between 0 and 1 to parameters.
            """
            return self.__scale_arg1 + (trial - 0.5) * self.__scale_arg2

        def _unscale_parameters(self, parameters):
            """
            scale from parameters to a number between 0 and 1.
            """
            return (parameters - self.__scale_arg1) / self.__scale_arg2 + 0.5

        def _ensure_constraint(self, trial):
            """
            make sure the parameters lie between the limits
            """
            for index in np.where((trial < 0) | (trial > 1))[0]:
                trial[index] = self.random_number_generator.rand()

        def _mutate(self, candidate):
            """
            create a trial vector based on a mutation strategy
            """
            trial = np.copy(self.population[candidate])

            rng = self.random_number_generator

            fill_point = rng.randint(0, self.parameter_count)

            if self.strategy in ['currenttobest1exp', 'currenttobest1bin']:
                bprime = self.mutation_func(candidate,
                                            self._select_samples(candidate, 5))
            else:
                bprime = self.mutation_func(self._select_samples(candidate, 5))

            if self.strategy in self._binomial:
                crossovers = rng.rand(self.parameter_count)
                crossovers = crossovers < self.cross_over_probability
                # the last one is always from the bprime vector for binomial
                # If you fill in modulo with a loop you have to set the last one to
                # true. If you don't use a loop then you can have any random entry
                # be True.
                crossovers[fill_point] = True
                trial = np.where(crossovers, bprime, trial)
                return trial

            elif self.strategy in self._exponential:
                i = 0
                while (i < self.parameter_count and
                       rng.rand() < self.cross_over_probability):

                    trial[fill_point] = bprime[fill_point]
                    fill_point = (fill_point + 1) % self.parameter_count
                    i += 1

                return trial

        def _best1(self, samples):
            """
            best1bin, best1exp
            """
            r0, r1 = samples[:2]
            return (self.population[0] + self.scale *
                    (self.population[r0] - self.population[r1]))

        def _rand1(self, samples):
            """
            rand1bin, rand1exp
            """
            r0, r1, r2 = samples[:3]
            return (self.population[r0] + self.scale *
                    (self.population[r1] - self.population[r2]))

        def _randtobest1(self, samples):
            """
            randtobest1bin, randtobest1exp
            """
            r0, r1, r2 = samples[:3]
            bprime = np.copy(self.population[r0])
            bprime += self.scale * (self.population[0] - bprime)
            bprime += self.scale * (self.population[r1] -
                                    self.population[r2])
            return bprime

        def _currenttobest1(self, candidate, samples):
            """
            currenttobest1bin, currenttobest1exp
            """
            r0, r1 = samples[:2]
            bprime = (self.population[candidate] + self.scale *
                      (self.population[0] - self.population[candidate] +
                       self.population[r0] - self.population[r1]))
            return bprime

        def _best2(self, samples):
            """
            best2bin, best2exp
            """
            r0, r1, r2, r3 = samples[:4]
            bprime = (self.population[0] + self.scale *
                      (self.population[r0] + self.population[r1] -
                       self.population[r2] - self.population[r3]))

            return bprime

        def _rand2(self, samples):
            """
            rand2bin, rand2exp
            """
            r0, r1, r2, r3, r4 = samples
            bprime = (self.population[r0] + self.scale *
                      (self.population[r1] + self.population[r2] -
                       self.population[r3] - self.population[r4]))

            return bprime

        def _select_samples(self, candidate, number_samples):
            """
            obtain random integers from range(self.num_population_members),
            without replacement.  You can't have the original candidate either.
            """
            idxs = list(range(self.num_population_members))
            idxs.remove(candidate)
            self.random_number_generator.shuffle(idxs)
            idxs = idxs[:number_samples]
            return idxs

    class AdamOptimizer:
        """Basic Adam optimizer implementation that can minimize w.r.t.
        a single variable.
        Parameters
        ----------
        shape : tuple
            shape of the variable w.r.t. which the loss should be minimized
        """
        #TODO Add reference or rewrite the function.
        def __init__(self, shape):
            self.m = np.zeros(shape)
            self.v = np.zeros(shape)
            self.t = 0

        def __call__(self, gradient, learning_rate, beta1=0.9, beta2=0.999, epsilon=1e-8):
            """Updates internal parameters of the optimizer and returns
            the change that should be applied to the variable.
            Parameters
            ----------
            gradient : `np.ndarray`
                the gradient of the loss w.r.t. to the variable
            learning_rate: float
                the learning rate in the current iteration
            beta1: float
                decay rate for calculating the exponentially
                decaying average of past gradients
            beta2: float
                decay rate for calculating the exponentially
                decaying average of past squared gradients
            epsilon: float
                small value to avoid division by zero
            """

            self.t += 1

            self.m = beta1 * self.m + (1 - beta1) * gradient
            self.v = beta2 * self.v + (1 - beta2) * gradient ** 2

            bias_correction_1 = 1 - beta1 ** self.t
            bias_correction_2 = 1 - beta2 ** self.t

            m_hat = self.m / bias_correction_1
            v_hat = self.v / bias_correction_2

            return -learning_rate * m_hat / (np.sqrt(v_hat) + epsilon)