ult 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 (1, 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 : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
        If `seed` is None (or `np.random`), the `numpy.random.RandomState`
        singleton is used.
        If `seed` is an int, a new ``RandomState`` instance is used,
        seeded with `seed`.
        If `seed` is already a ``Generator`` or ``RandomState`` instance then
        that instance is used.
        Specify `seed` for repeatable minimizations. 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)

    >>> import numpy as np
    >>> 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)))
    >>> 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]) # random
    >>> ret.fun
    0.000000

    Nz