# Buggy code
def dual_annealing(
    func,
    bounds,
    args=(),
    maxiter=1000,
    local_search_options={},
    initial_temp=5230.0,
    restart_temp_ratio=2.0e-5,
    visit=2.62,
    accept=-5.0,
    maxfun=1e7,
    seed=None,
    no_local_search=False,
    callback=None,
    x0=None,
):
    """
    Find the global minimum of a function using Dual Annealing.

    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, shape (n, 2)
        Bounds for variables.  ``(min, max)`` pairs for each element in ``x``,
        defining bounds for the objective function parameter.
    args : tuple, optional
        Any additional fixed parameters needed to completely specify the
        objective function.
    maxiter : int, optional
        The maximum number of global search iterations. Default value is 1000.
    local_search_options : dict, optional
        Extra keyword arguments to be passed to the local minimizer
        (`minimize`). Some important options could be:
        ``method`` for the minimizer method to use and ``args`` for
        objective function additional arguments.
    initial_temp : float, optional
        The initial temperature, use higher values to facilitates a wider
        search of the energy landscape, allowing dual_annealing to escape
        local minima that it is trapped in. Default value is 5230. Range is
        (0.01, 5.e4].
    restart_temp_ratio : float, optional
        During the annealing process, temperature is decreasing, when it
        reaches ``initial_temp * restart_temp_ratio``, the reannealing process
        is triggered. Default value of the ratio is 2e-5. Range is (0, 1).
    visit : float, optional
        Parameter for visiting distribution. Default value is 2.62. Higher
        values give the visiting distribution a heavier tail, this makes
        the algorithm jump to a more distant region. The value range is (0, 3].
    accept : float, optional
        Parameter for acceptance distribution. It is used to control the
        probability of acceptance. The lower the acceptance parameter, the
        smaller the probability of acceptance. Default value is -5.0 with
        a range (-1e4, -5].
    maxfun : int, optional
        Soft limit for the number of objective function calls. If the
        algorithm is in the middle of a local search, this number will be
        exceeded, the algorithm will stop just after the local search is
        done. Default value is 1e7.
    seed : {int or `~numpy.random.mtrand.RandomState` instance}, optional
        If `seed` is not specified the `~numpy.random.mtrand.RandomState`
        singleton is used.
        If `seed` is an int, a new ``RandomState`` instance is used,
        seeded with `seed`.
        If `seed` is already a ``RandomState`` instance, then that
        instance is used.
        Specify `seed` for repeatable minimizations. The random numbers
        generated with this seed only affect the visiting distribution
        function and new coordinates generation.
    no_local_search : bool, optional
        If `no_local_search` is set to True, a traditional Generalized
        Simulated Annealing will be performed with no local search
        strategy applied.
    callback : callable, optional
        A callback function with signature ``callback(x, f, context)``,
        which will be called for all minima found.
        ``x`` and ``f`` are the coordinates and function value of the
        latest minimum found, and ``context`` has value in [0, 1, 2], with the
        following meaning:

            - 0: minimum detected in the annealing process.
            - 1: detection occurred in the local search process.
            - 2: detection done in the dual annealing process.

        If the callback implementation returns True, the algorithm will stop.
    x0 : ndarray, shape(n,), optional
        Coordinates of a single N-D starting point.

    Returns
    -------
    res : OptimizeResult
        The optimization result represented as a `OptimizeResult` object.
        Important attributes are: ``x`` the solution array, ``fun`` the value
        of the function at the solution, and ``message`` which describes the
        cause of the termination.
        See `OptimizeResult` for a description of other attributes.

    Notes
    -----
    This function implements the Dual Annealing optimization. This stochastic
    approach derived from [3]_ combines the generalization of CSA (Classical
    Simulated Annealing) and FSA (Fast Simulated Annealing) [1]_ [2]_ coupled
    to a strategy for applying a local search on accepted locations [4]_.
    An alternative implementation of this same algorithm is described in [5]_
    and benchmarks are presented in [6]_. This approach introduces an advanced
    method to refine the solution found by the generalized annealing
    process. This algorithm uses a distorted Cauchy-Lorentz visiting
    distribution, with its shape controlled by the parameter :math:`q_{v}`

    .. math::

        g_{q_{v}}(\\Delta x(t)) \\propto \\frac{ \\
        \\left[T_{q_{v}}(t) \\right]^{-\\frac{D}{3-q_{v}}}}{ \\
        \\left[{1+(q_{v}-1)\\frac{(\\Delta x(t))^{2}} { \\
        \\left[T_{q_{v}}(t)\\right]^{\\frac{2}{3-q_{v}}}}}\\right]^{ \\
        \\frac{1}{q_{v}-1}+\\frac{D-1}{2}}}

    Where :math:`t` is the artificial time. This visiting distribution is used
    to generate a trial jump distance :math:`\\Delta x(t)` of variable
    :math:`x(t)` under artificial temperature :math:`T_{q_{v}}(t)`.

    From the starting point, after calling the visiting distribution
    function, the acceptance probability is computed as follows:

    .. math::

        p_{q_{a}} = \\min{\\{1,\\left[1-(1-q_{a}) \\beta \\Delta E \\right]^{ \\
        \\frac{1}{1-q_{a}}}\\}}

    Where :math:`q_{a}` is a acceptance parameter. For :math:`q_{a}<1`, zero
    acceptance probability is assigned to the cases where

    .. math::

        [1-(1-q_{a}) \\beta \\Delta E] < 0

    The artificial temperature :math:`T_{q_{v}}(t)` is decreased according to

    .. math::

        T_{q_{v}}(t) = T_{q_{v}}(1) \\frac{2^{q_{v}-1}-1}{\\left( \\
        1 + t\\right)^{q_{v}-1}-1}

    Where :math:`q_{v}` is the visiting parameter.

    .. versionadded:: 1.2.0

    References
    ----------
    .. [1] Tsallis C. Possible generalization of Boltzmann-Gibbs
        statistics. Journal of Statistical Physics, 52, 479-487 (1998).
    .. [2] Tsallis C, Stariolo DA. Generalized Simulated Annealing.
        Physica A, 233, 395-406 (1996).
    .. [3] Xiang Y, Sun DY, Fan W, Gong XG. Generalized Simulated
        Annealing Algorithm and Its Application to the Thomson Model.
        Physics Letters A, 233, 216-220 (1997).
    .. [4] Xiang Y, Gong XG. Efficiency of Generalized Simulated
        Annealing. Physical Review E, 62, 4473 (2000).
    .. [5] Xiang Y, Gubian S, Suomela B, Hoeng J. Generalized
        Simulated Annealing for Efficient Global Optimization: the GenSA
        Package for R. The R Journal, Volume 5/1 (2013).
    .. [6] Mullen, K. Continuous Global Optimization in R. Journal of
        Statistical Software, 60(6), 1 - 45, (2014). DOI:10.18637/jss.v060.i06

    Examples
    --------
    The following example is a 10-D problem, with many local minima.
    The function involved is called Rastrigin
    (https://en.wikipedia.org/wiki/Rastrigin_function)

    >>> from scipy.optimize import dual_annealing
    >>> func = lambda x: np.sum(x*x - 10*np.cos(2*np.pi*x)) + 10*np.size(x)
    >>> lw = [-5.12] * 10
    >>> up = [5.12] * 10
    >>> ret = dual_annealing(func, bounds=list(zip(lw, up)), seed=1234)
    >>> print("global minimum: xmin = {0}, f(xmin) = {1:.6f}".format(
    ...       ret.x, ret.fun))
    global minimum: xmin = [-4.26437714e-09 -3.91699361e-09 -1.86149218e-09 -3.97165720e-09
     -6.29151648e-09 -6.53145322e-09 -3.93616815e-09 -6.55623025e-09
    -6.05775280e-09 -5.00668935e-09], f(xmin) = 0.000000

    """  # noqa: E501
    if x0 is not None and not len(x0) == len(bounds):
        raise ValueError("Bounds size does not match x0")

    lu = list(zip(*bounds))
    lower = np.array(lu[0])
    upper = np.array(lu[1])
    # Check that restart temperature ratio is correct
    if restart_temp_ratio <= 0.0 or restart_temp_ratio >= 1.0:
        raise ValueError("Restart temperature ratio has to be in range (0, 1)")
    # Checking bounds are valid
    if (
        np.any(np.isinf(lower))
        or np.any(np.isinf(upper))
        or np.any(np.isnan(lower))
        or np.any(np.isnan(upper))
    ):
        raise ValueError("Some bounds values are inf values or nan values")
    # Checking that bounds are consistent
    if not np.all(lower < upper):
        raise ValueError("Bounds are not consistent min < max")
    # Checking that bounds are the same length
    if not len(lower) == len(upper):
        raise ValueError("Bounds do not have the same dimensions")

    # Wrapper for the objective function
    func_wrapper = ObjectiveFunWrapper(func, maxfun, *args)
    # Wrapper fot the minimizer
    minimizer_wrapper = LocalSearchWrapper(bounds, func_wrapper, **local_search_options)
    # Initialization of RandomState for reproducible runs if seed provided
    rand_state = check_random_state(seed)
    # Initialization of the energy state
    energy_state = EnergyState(lower, upper, callback)
    energy_state.reset(func_wrapper, rand_state, x0)
    # Minimum value of annealing temperature reached to perform
    # re-annealing
    temperature_restart = initial_temp * restart_temp_ratio
    # VisitingDistribution instance
    visit_dist = VisitingDistribution(lower, upper, visit, rand_state)
    # Strategy chain instance
    strategy_chain = StrategyChain(
        accept, visit_dist, func_wrapper, minimizer_wrapper, rand_state, energy_state
    )
    need_to_stop = False
    iteration = 0
    message = []
    # OptimizeResult object to be returned
    optimize_res = OptimizeResult()
    optimize_res.success = True
    optimize_res.status = 0

    t1 = np.exp((visit - 1) * np.log(2.0)) - 1.0
    # Run the search loop
    while not need_to_stop:
        for i in range(maxiter):
            # Compute temperature for this step
            s = float(i) + 2.0
            t2 = np.exp((visit - 1) * np.log(s)) - 1.0
            temperature = initial_temp * t1 / t2
            if iteration >= maxiter:
                message.append("Maximum number of iteration reached")
                need_to_stop = True
                break
            # Need a re-annealing process?
            if temperature < temperature_restart:
                energy_state.reset(func_wrapper, rand_state)
                break
            # starting strategy chain
            val = strategy_chain.run(i, temperature)
            if val is not None:
                message.append(val)
                need_to_stop = True
                optimize_res.success = False
                break
            # Possible local search at the end of the strategy chain
            if not no_local_search:
                val = strategy_chain.local_search()
                if val is not None:
                    message.append(val)
                    need_to_stop = True
                    optimize_res.success = False
                    break
            iteration += 1

    # Setting the OptimizeResult values
    optimize_res.x = energy_state.xbest
    optimize_res.fun = energy_state.ebest
    optimize_res.nit = iteration
    optimize_res.nfev = func_wrapper.nfev
    optimize_res.njev = func_wrapper.ngev
    optimize_res.nhev = func_wrapper.nhev
    optimize_res.message = message
    return optimize_res


# Traceback (ValueError)
In [158]: minimize(func, [1.1, 1.2], method='Newton-CG', jac=False)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-158-94c7580e9508> in <module>()
----> 1 minimize(func, [1.1, 1.2], method='Newton-CG', jac=False)

~/miniconda3scipy/lib/python3.5/site-packages/scipy-1.2.0.dev0+6d455f3-py3.5-macosx-10.6-x86_64.egg/scipy/optimize/_minimize.py in minimize(fun, x0, args, method, jac, hess, hessp, bounds, constraints, tol, callback, options)
600     elif meth == 'newton-cg':
601         return _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback,
--> 602                                   **options)
603     elif meth == 'l-bfgs-b':
604         return _minimize_lbfgsb(fun, x0, args, jac, bounds,

~/miniconda3scipy/lib/python3.5/site-packages/scipy-1.2.0.dev0+6d455f3-py3.5-macosx-10.6-x86_64.egg/scipy/optimize/optimize.py in _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback, xtol, eps, maxiter, disp, return_all, **unknown_options)
1494     _check_unknown_options(unknown_options)
1495     if jac is None:
-> 1496         raise ValueError('Jacobian is required for Newton-CG method')
1497     f = fun
1498     fprime = jac

ValueError: Jacobian is required for Newton-CG method

# Fixed code
def dual_annealing(
    func,
    bounds,
    args=(),
    maxiter=1000,
    local_search_options={},
    initial_temp=5230.0,
    restart_temp_ratio=2.0e-5,
    visit=2.62,
    accept=-5.0,
    maxfun=1e7,
    seed=None,
    no_local_search=False,
    callback=None,
    x0=None,
):
    """
    Find the global minimum of a function using Dual Annealing.

    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, shape (n, 2)
        Bounds for variables.  ``(min, max)`` pairs for each element in ``x``,
        defining bounds for the objective function parameter.
    args : tuple, optional
        Any additional fixed parameters needed to completely specify the
        objective function.
    maxiter : int, optional
        The maximum number of global search iterations. Default value is 1000.
    local_search_options : dict, optional
        Extra keyword arguments to be passed to the local minimizer
        (`minimize`). Some important options could be:
        ``method`` for the minimizer method to use and ``args`` for
        objective function additional arguments.
    initial_temp : float, optional
        The initial temperature, use higher values to facilitates a wider
        search of the energy landscape, allowing dual_annealing to escape
        local minima that it is trapped in. Default value is 5230. Range is
        (0.01, 5.e4].
    restart_temp_ratio : float, optional
        During the annealing process, temperature is decreasing, when it
        reaches ``initial_temp * restart_temp_ratio``, the reannealing process
        is triggered. Default value of the ratio is 2e-5. Range is (0, 1).
    visit : float, optional
        Parameter for visiting distribution. Default value is 2.62. Higher
        values give the visiting distribution a heavier tail, this makes
        the algorithm jump to a more distant region. The value range is (0, 3].
    accept : float, optional
        Parameter for acceptance distribution. It is used to control the
        probability of acceptance. The lower the acceptance parameter, the
        smaller the probability of acceptance. Default value is -5.0 with
        a range (-1e4, -5].
    maxfun : int, optional
        Soft limit for the number of objective function calls. If the
        algorithm is in the middle of a local search, this number will be
        exceeded, the algorithm will stop just after the local search is
        done. Default value is 1e7.
    seed : {int or `~numpy.random.mtrand.RandomState` instance}, optional
        If `seed` is not specified the `~numpy.random.mtrand.RandomState`
        singleton is used.
        If `seed` is an int, a new ``RandomState`` instance is used,
        seeded with `seed`.
        If `seed` is already a ``RandomState`` instance, then that
        instance is used.
        Specify `seed` for repeatable minimizations. The random numbers
        generated with this seed only affect the visiting distribution
        function and new coordinates generation.
    no_local_search : bool, optional
        If `no_local_search` is set to True, a traditional Generalized
        Simulated Annealing will be performed with no local search
        strategy applied.
    callback : callable, optional
        A callback function with signature ``callback(x, f, context)``,
        which will be called for all minima found.
        ``x`` and ``f`` are the coordinates and function value of the
        latest minimum found, and ``context`` has value in [0, 1, 2], with the
        following meaning:

            - 0: minimum detected in the annealing process.
            - 1: detection occurred in the local search process.
            - 2: detection done in the dual annealing process.

        If the callback implementation returns True, the algorithm will stop.
    x0 : ndarray, shape(n,), optional
        Coordinates of a single N-D starting point.

    Returns
    -------
    res : OptimizeResult
        The optimization result represented as a `OptimizeResult` object.
        Important attributes are: ``x`` the solution array, ``fun`` the value
        of the function at the solution, and ``message`` which describes the
        cause of the termination.
        See `OptimizeResult` for a description of other attributes.

    Notes
    -----
    This function implements the Dual Annealing optimization. This stochastic
    approach derived from [3]_ combines the generalization of CSA (Classical
    Simulated Annealing) and FSA (Fast Simulated Annealing) [1]_ [2]_ coupled
    to a strategy for applying a local search on accepted locations [4]_.
    An alternative implementation of this same algorithm is described in [5]_
    and benchmarks are presented in [6]_. This approach introduces an advanced
    method to refine the solution found by the generalized annealing
    process. This algorithm uses a distorted Cauchy-Lorentz visiting
    distribution, with its shape controlled by the parameter :math:`q_{v}`

    .. math::

        g_{q_{v}}(\\Delta x(t)) \\propto \\frac{ \\
        \\left[T_{q_{v}}(t) \\right]^{-\\frac{D}{3-q_{v}}}}{ \\
        \\left[{1+(q_{v}-1)\\frac{(\\Delta x(t))^{2}} { \\
        \\left[T_{q_{v}}(t)\\right]^{\\frac{2}{3-q_{v}}}}}\\right]^{ \\
        \\frac{1}{q_{v}-1}+\\frac{D-1}{2}}}

    Where :math:`t` is the artificial time. This visiting distribution is used
    to generate a trial jump distance :math:`\\Delta x(t)` of variable
    :math:`x(t)` under artificial temperature :math:`T_{q_{v}}(t)`.

    From the starting point, after calling the visiting distribution
    function, the acceptance probability is computed as follows:

    .. math::

        p_{q_{a}} = \\min{\\{1,\\left[1-(1-q_{a}) \\beta \\Delta E \\right]^{ \\
        \\frac{1}{1-q_{a}}}\\}}

    Where :math:`q_{a}` is a acceptance parameter. For :math:`q_{a}<1`, zero
    acceptance probability is assigned to the cases where

    .. math::

        [1-(1-q_{a}) \\beta \\Delta E] < 0

    The artificial temperature :math:`T_{q_{v}}(t)` is decreased according to

    .. math::

        T_{q_{v}}(t) = T_{q_{v}}(1) \\frac{2^{q_{v}-1}-1}{\\left( \\
        1 + t\\right)^{q_{v}-1}-1}

    Where :math:`q_{v}` is the visiting parameter.

    .. versionadded:: 1.2.0

    References
    ----------
    .. [1] Tsallis C. Possible generalization of Boltzmann-Gibbs
        statistics. Journal of Statistical Physics, 52, 479-487 (1998).
    .. [2] Tsallis C, Stariolo DA. Generalized Simulated Annealing.
        Physica A, 233, 395-406 (1996).
    .. [3] Xiang Y, Sun DY, Fan W, Gong XG. Generalized Simulated
        Annealing Algorithm and Its Application to the Thomson Model.
        Physics Letters A, 233, 216-220 (1997).
    .. [4] Xiang Y, Gong XG. Efficiency of Generalized Simulated
        Annealing. Physical Review E, 62, 4473 (2000).
    .. [5] Xiang Y, Gubian S, Suomela B, Hoeng J. Generalized
        Simulated Annealing for Efficient Global Optimization: the GenSA
        Package for R. The R Journal, Volume 5/1 (2013).
    .. [6] Mullen, K. Continuous Global Optimization in R. Journal of
        Statistical Software, 60(6), 1 - 45, (2014). DOI:10.18637/jss.v060.i06

    Examples
    --------
    The following example is a 10-D problem, with many local minima.
    The function involved is called Rastrigin
    (https://en.wikipedia.org/wiki/Rastrigin_function)

    >>> from scipy.optimize import dual_annealing
    >>> func = lambda x: np.sum(x*x - 10*np.cos(2*np.pi*x)) + 10*np.size(x)
    >>> lw = [-5.12] * 10
    >>> up = [5.12] * 10
    >>> ret = dual_annealing(func, bounds=list(zip(lw, up)), seed=1234)
    >>> ret.x
    array([-4.26437714e-09, -3.91699361e-09, -1.86149218e-09, -3.97165720e-09,
           -6.29151648e-09, -6.53145322e-09, -3.93616815e-09, -6.55623025e-09,
           -6.05775280e-09, -5.00668935e-09]) # may vary
    >>> ret.fun
    0.000000

    """  # noqa: E501
    if x0 is not None and not len(x0) == len(bounds):
        raise ValueError("Bounds size does not match x0")

    lu = list(zip(*bounds))
    lower = np.array(lu[0])
    upper = np.array(lu[1])
    # Check that restart temperature ratio is correct
    if restart_temp_ratio <= 0.0 or restart_temp_ratio >= 1.0:
        raise ValueError("Restart temperature ratio has to be in range (0, 1)")
    # Checking bounds are valid
    if (
        np.any(np.isinf(lower))
        or np.any(np.isinf(upper))
        or np.any(np.isnan(lower))
        or np.any(np.isnan(upper))
    ):
        raise ValueError("Some bounds values are inf values or nan values")
    # Checking that bounds are consistent
    if not np.all(lower < upper):
        raise ValueError("Bounds are not consistent min < max")
    # Checking that bounds are the same length
    if not len(lower) == len(upper):
        raise ValueError("Bounds do not have the same dimensions")

    # Wrapper for the objective function
    func_wrapper = ObjectiveFunWrapper(func, maxfun, *args)
    # Wrapper fot the minimizer
    minimizer_wrapper = LocalSearchWrapper(bounds, func_wrapper, **local_search_options)
    # Initialization of RandomState for reproducible runs if seed provided
    rand_state = check_random_state(seed)
    # Initialization of the energy state
    energy_state = EnergyState(lower, upper, callback)
    energy_state.reset(func_wrapper, rand_state, x0)
    # Minimum value of annealing temperature reached to perform
    # re-annealing
    temperature_restart = initial_temp * restart_temp_ratio
    # VisitingDistribution instance
    visit_dist = VisitingDistribution(lower, upper, visit, rand_state)
    # Strategy chain instance
    strategy_chain = StrategyChain(
        accept, visit_dist, func_wrapper, minimizer_wrapper, rand_state, energy_state
    )
    need_to_stop = False
    iteration = 0
    message = []
    # OptimizeResult object to be returned
    optimize_res = OptimizeResult()
    optimize_res.success = True
    optimize_res.status = 0

    t1 = np.exp((visit - 1) * np.log(2.0)) - 1.0
    # Run the search loop
    while not need_to_stop:
        for i in range(maxiter):
            # Compute temperature for this step
            s = float(i) + 2.0
            t2 = np.exp((visit - 1) * np.log(s)) - 1.0
            temperature = initial_temp * t1 / t2
            if iteration >= maxiter:
                message.append("Maximum number of iteration reached")
                need_to_stop = True
                break
            # Need a re-annealing process?
            if temperature < temperature_restart:
                energy_state.reset(func_wrapper, rand_state)
                break
            # starting strategy chain
            val = strategy_chain.run(i, temperature)
            if val is not None:
                message.append(val)
                need_to_stop = True
                optimize_res.success = False
                break
            # Possible local search at the end of the strategy chain
            if not no_local_search:
                val = strategy_chain.local_search()
                if val is not None:
                    message.append(val)
                    need_to_stop = True
                    optimize_res.success = False
                    break
            iteration += 1

    # Setting the OptimizeResult values
    optimize_res.x = energy_state.xbest
    optimize_res.fun = energy_state.ebest
    optimize_res.nit = iteration
    optimize_res.nfev = func_wrapper.nfev
    optimize_res.njev = func_wrapper.ngev
    optimize_res.nhev = func_wrapper.nhev
    optimize_res.message = message
    return optimize_res