_linesearch.py

"""
Functions
---------
.. autosummary::
   :toctree: generated/

    line_search_armijo
    line_search_wolfe1
    line_search_wolfe2
    scalar_search_wolfe1
    scalar_search_wolfe2

"""
from warnings import warn

from scipy.optimize import _minpack2 as minpack2
import numpy as np

__all__ = ['LineSearchWarning', 'line_search_wolfe1', 'line_search_wolfe2',
           'scalar_search_wolfe1', 'scalar_search_wolfe2',
           'line_search_armijo']

class LineSearchWarning(RuntimeWarning):
    pass


#------------------------------------------------------------------------------
# Minpack's Wolfe line and scalar searches
#------------------------------------------------------------------------------

def line_search_wolfe1(f, fprime, xk, pk, gfk=None,
                       old_fval=None, old_old_fval=None,
                       args=(), c1=1e-4, c2=0.9, amax=50, amin=1e-8,
                       xtol=1e-14):
    """
    As `scalar_search_wolfe1` but do a line search to direction `pk`

    Parameters
    ----------
    f : callable
        Function `f(x)`
    fprime : callable
        Gradient of `f`
    xk : array_like
        Current point
    pk : array_like
        Search direction

    gfk : array_like, optional
        Gradient of `f` at point `xk`
    old_fval : float, optional
        Value of `f` at point `xk`
    old_old_fval : float, optional
        Value of `f` at point preceding `xk`

    The rest of the parameters are the same as for `scalar_search_wolfe1`.

    Returns
    -------
    stp, f_count, g_count, fval, old_fval
        As in `line_search_wolfe1`
    gval : array
        Gradient of `f` at the final point

    """
    if gfk is None:
        gfk = fprime(xk, *args)

    gval = [gfk]
    gc = [0]
    fc = [0]

    def phi(s):
        fc[0] += 1
        return f(xk + s*pk, *args)

    def derphi(s):
        gval[0] = fprime(xk + s*pk, *args)
        gc[0] += 1
        return np.dot(gval[0], pk)

    derphi0 = np.dot(gfk, pk)

    stp, fval, old_fval = scalar_search_wolfe1(
            phi, derphi, old_fval, old_old_fval, derphi0,
            c1=c1, c2=c2, amax=amax, amin=amin, xtol=xtol)

    return stp, fc[0], gc[0], fval, old_fval, gval[0]


def scalar_search_wolfe1(phi, derphi, phi0=None, old_phi0=None, derphi0=None,
                         c1=1e-4, c2=0.9,
                         amax=50, amin=1e-8, xtol=1e-14):
    """
    Scalar function search for alpha that satisfies strong Wolfe conditions

    alpha > 0 is assumed to be a descent direction.

    Parameters
    ----------
    phi : callable phi(alpha)
        Function at point `alpha`
    derphi : callable phi'(alpha)
        Objective function derivative. Returns a scalar.
    phi0 : float, optional
        Value of phi at 0
    old_phi0 : float, optional
        Value of phi at previous point
    derphi0 : float, optional
        Value derphi at 0
    c1 : float, optional
        Parameter for Armijo condition rule.
    c2 : float, optional
        Parameter for curvature condition rule.
    amax, amin : float, optional
        Maximum and minimum step size
    xtol : float, optional
        Relative tolerance for an acceptable step.

    Returns
    -------
    alpha : float
        Step size, or None if no suitable step was found
    phi : float
        Value of `phi` at the new point `alpha`
    phi0 : float
        Value of `phi` at `alpha=0`

    Notes
    -----
    Uses routine DCSRCH from MINPACK.

    """

    if phi0 is None:
        phi0 = phi(0.)
    if derphi0 is None:
        derphi0 = derphi(0.)

    if old_phi0 is not None and derphi0 != 0:
        alpha1 = min(1.0, 1.01*2*(phi0 - old_phi0)/derphi0)
        if alpha1 < 0:
            alpha1 = 1.0
    else:
        alpha1 = 1.0

    phi1 = phi0
    derphi1 = derphi0
    isave = np.zeros((2,), np.intc)
    dsave = np.zeros((13,), float)
    task = b'START'

    maxiter = 100
    for i in range(maxiter):
        stp, phi1, derphi1, task = minpack2.dcsrch(alpha1, phi1, derphi1,
                                                   c1, c2, xtol, task,
                                                   amin, amax, isave, dsave)
        if task[:2] == b'FG':
            alpha1 = stp
            phi1 = phi(stp)
            derphi1 = derphi(stp)
        else:
            break
    else:
        # maxiter reached, the line search did not converge
        stp = None

    if task[:5] == b'ERROR' or task[:4] == b'WARN':
        stp = None  # failed

    return stp, phi1, phi0


line_search = line_search_wolfe1


#------------------------------------------------------------------------------
# Pure-Python Wolfe line and scalar searches
#------------------------------------------------------------------------------

def line_search_wolfe2(f, myfprime, xk, pk, gfk=None, old_fval=None,
                       old_old_fval=None, args=(), c1=1e-4, c2=0.9, amax=None,
                       extra_condition=None, maxiter=10):
    """Find alpha that satisfies strong Wolfe conditions.

    Parameters
    ----------
    f : callable f(x,*args)
        Objective function.
    myfprime : callable f'(x,*args)
        Objective function gradient.
    xk : ndarray
        Starting point.
    pk : ndarray
        Search direction.
    gfk : ndarray, optional
        Gradient value for x=xk (xk being the current parameter
        estimate). Will be recomputed if omitted.
    old_fval : float, optional
        Function value for x=xk. Will be recomputed if omitted.
    old_old_fval : float, optional
        Function value for the point preceding x=xk.
    args : tuple, optional
        Additional arguments passed to objective function.
    c1 : float, optional
        Parameter for Armijo condition rule.
    c2 : float, optional
        Parameter for curvature condition rule.
    amax : float, optional
        Maximum step size
    extra_condition : callable, optional
        A callable of the form ``extra_condition(alpha, x, f, g)``
        returning a boolean. Arguments are the proposed step ``alpha``
        and the corresponding ``x``, ``f`` and ``g`` values. The line search
        accepts the value of ``alpha`` only if this
        callable returns ``True``. If the callable returns ``False``
        for the step length, the algorithm will continue with
        new iterates. The callable is only called for iterates
        satisfying the strong Wolfe conditions.
    maxiter : int, optional
        Maximum number of iterations to perform.

    Returns
    -------
    alpha : float or None
        Alpha for which ``x_new = x0 + alpha * pk``,
        or None if the line search algorithm did not converge.
    fc : int
        Number of function evaluations made.
    gc : int
        Number of gradient evaluations made.
    new_fval : float or None
        New function value ``f(x_new)=f(x0+alpha*pk)``,
        or None if the line search algorithm did not converge.
    old_fval : float
        Old function value ``f(x0)``.
    new_slope : float or None
        The local slope along the search direction at the
        new value ``<myfprime(x_new), pk>``,
        or None if the line search algorithm did not converge.


    Notes
    -----
    Uses the line search algorithm to enforce strong Wolfe
    conditions. See Wright and Nocedal, 'Numerical Optimization',
    1999, pp. 59-61.

    Examples
    --------
    >>> from scipy.optimize import line_search

    A objective function and its gradient are defined.

    >>> def obj_func(x):
    ...     return (x[0])**2+(x[1])**2
    >>> def obj_grad(x):
    ...     return [2*x[0], 2*x[1]]

    We can find alpha that satisfies strong Wolfe conditions.

    >>> start_point = np.array([1.8, 1.7])
    >>> search_gradient = np.array([-1.0, -1.0])
    >>> line_search(obj_func, obj_grad, start_point, search_gradient)
    (1.0, 2, 1, 1.1300000000000001, 6.13, [1.6, 1.4])

    """
    fc = [0]
    gc = [0]
    gval = [None]
    gval_alpha = [None]

    def phi(alpha):
        fc[0] += 1
        return f(xk + alpha * pk, *args)

    fprime = myfprime

    def derphi(alpha):
        gc[0] += 1
        gval[0] = fprime(xk + alpha * pk, *args)  # store for later use
        gval_alpha[0] = alpha
        return np.dot(gval[0], pk)

    if gfk is None:
        gfk = fprime(xk, *args)
    derphi0 = np.dot(gfk, pk)

    if extra_condition is not None:
        # Add the current gradient as argument, to avoid needless
        # re-evaluation
        def extra_condition2(alpha, phi):
            if gval_alpha[0] != alpha:
                derphi(alpha)
            x = xk + alpha * pk
            return extra_condition(alpha, x, phi, gval[0])
    else:
        extra_condition2 = None

    alpha_star, phi_star, old_fval, derphi_star = scalar_search_wolfe2(
            phi, derphi, old_fval, old_old_fval, derphi0, c1, c2, amax,
            extra_condition2, maxiter=maxiter)

    if derphi_star is None:
        warn('The line search algorithm did not converge', LineSearchWarning)
    else:
        # derphi_star is a number (derphi) -- so use the most recently
        # calculated gradient used in computing it derphi = gfk*pk
        # this is the gradient at the next step no need to compute it
        # again in the outer loop.
        derphi_star = gval[0]

    return alpha_star, fc[0], gc[0], phi_star, old_fval, derphi_star


def scalar_search_wolfe2(phi, derphi, phi0=None,
                         old_phi0=None, derphi0=None,
                         c1=1e-4, c2=0.9, amax=None,
                         extra_condition=None, maxiter=10):
    """Find alpha that satisfies strong Wolfe conditions.

    alpha > 0 is assumed to be a descent direction.

    Parameters
    ----------
    phi : callable phi(alpha)
        Objective scalar function.
    derphi : callable phi'(alpha)
        Objective function derivative. Returns a scalar.
    phi0 : float, optional
        Value of phi at 0.
    old_phi0 : float, optional
        Value of phi at previous point.
    derphi0 : float, optional
        Value of derphi at 0
    c1 : float, optional
        Parameter for Armijo condition rule.
    c2 : float, optional
        Parameter for curvature condition rule.
    amax : float, optional
        Maximum step size.
    extra_condition : callable, optional
        A callable of the form ``extra_condition(alpha, phi_value)``
        returning a boolean. The line search accepts the value
        of ``alpha`` only if this callable returns ``True``.
        If the callable returns ``False`` for the step length,
        the algorithm will continue with new iterates.
        The callable is only called for iterates satisfying
        the strong Wolfe conditions.
    maxiter : int, optional
        Maximum number of iterations to perform.

    Returns
    -------
    alpha_star : float or None
        Best alpha, or None if the line search algorithm did not converge.
    phi_star : float
        phi at alpha_star.
    phi0 : float
        phi at 0.
    derphi_star : float or None
        derphi at alpha_star, or None if the line search algorithm
        did not converge.

    Notes
    -----
    Uses the line search algorithm to enforce strong Wolfe
    conditions. See Wright and Nocedal, 'Numerical Optimization',
    1999, pp. 59-61.

    """

    if phi0 is None:
        phi0 = phi(0.)

    if derphi0 is None:
        derphi0 = derphi(0.)

    alpha0 = 0
    if old_phi0 is not None and derphi0 != 0:
        alpha1 = min(1.0, 1.01*2*(phi0 - old_phi0)/derphi0)
    else:
        alpha1 = 1.0

    if alpha1 < 0:
        alpha1 = 1.0

    if amax is not None:
        alpha1 = min(alpha1, amax)

    phi_a1 = phi(alpha1)
    #derphi_a1 = derphi(alpha1) evaluated below

    phi_a0 = phi0
    derphi_a0 = derphi0

    if extra_condition is None:
        extra_condition = lambda alpha, phi: True

    for i in range(maxiter):
        if alpha1 == 0 or (amax is not None and alpha0 == amax):
            # alpha1 == 0: This shouldn't happen. Perhaps the increment has
            # slipped below machine precision?
            alpha_star = None
            phi_star = phi0
            phi0 = old_phi0
            derphi_star = None

            if alpha1 == 0:
                msg = 'Rounding errors prevent the line search from converging'
            else:
                msg = "The line search algorithm could not find a solution " + \
                      "less than or equal to amax: %s" % amax

            warn(msg, LineSearchWarning)
            break

        not_first_iteration = i > 0
        if (phi_a1 > phi0 + c1 * alpha1 * derphi0) or \
           ((phi_a1 >= phi_a0) and not_first_iteration):
            alpha_star, phi_star, derphi_star = \
                        _zoom(alpha0, alpha1, phi_a0,
                              phi_a1, derphi_a0, phi, derphi,
                              phi0, derphi0, c1, c2, extra_condition)
            break

        derphi_a1 = derphi(alpha1)
        if (abs(derphi_a1) <= -c2*derphi0):
            if extra_condition(alpha1, phi_a1):
                alpha_star = alpha1
                phi_star = phi_a1
                derphi_star = derphi_a1
                break

        if (derphi_a1 >= 0):
            alpha_star, phi_star, derphi_star = \
                        _zoom(alpha1, alpha0, phi_a1,
                              phi_a0, derphi_a1, phi, derphi,
                              phi0, derphi0, c1, c2, extra_condition)
            break

        alpha2 = 2 * alpha1  # increase by factor of two on each iteration
        if amax is not None:
            alpha2 = min(alpha2, amax)
        alpha0 = alpha1
        alpha1 = alpha2
        phi_a0 = phi_a1
        phi_a1 = phi(alpha1)
        derphi_a0 = derphi_a1

    else:
        # stopping test maxiter reached
        alpha_star = alpha1
        phi_star = phi_a1
        derphi_star = None
        warn('The line search algorithm did not converge', LineSearchWarning)

    return alpha_star, phi_star, phi0, derphi_star


def _cubicmin(a, fa, fpa, b, fb, c, fc):
    """
    Finds the minimizer for a cubic polynomial that goes through the
    points (a,fa), (b,fb), and (c,fc) with derivative at a of fpa.

    If no minimizer can be found, return None.

    """
    # f(x) = A *(x-a)^3 + B*(x-a)^2 + C*(x-a) + D

    with np.errstate(divide='raise', over='raise', invalid='raise'):
        try:
            C = fpa
            db = b - a
            dc = c - a
            denom = (db * dc) ** 2 * (db - dc)
            d1 = np.empty((2, 2))
            d1[0, 0] = dc ** 2
            d1[0, 1] = -db ** 2
            d1[1, 0] = -dc ** 3
            d1[1, 1] = db ** 3
            [A, B] = np.dot(d1, np.asarray([fb - fa - C * db,
                                            fc - fa - C * dc]).flatten())
            A /= denom
            B /= denom
            radical = B * B - 3 * A * C
            xmin = a + (-B + np.sqrt(radical)) / (3 * A)
        except ArithmeticError:
            return None
    if not np.isfinite(xmin):
        return None
    return xmin


def _quadmin(a, fa, fpa, b, fb):
    """
    Finds the minimizer for a quadratic polynomial that goes through
    the points (a,fa), (b,fb) with derivative at a of fpa.

    """
    # f(x) = B*(x-a)^2 + C*(x-a) + D
    with np.errstate(divide='raise', over='raise', invalid='raise'):
        try:
            D = fa
            C = fpa
            db = b - a * 1.0
            B = (fb - D - C * db) / (db * db)
            xmin = a - C / (2.0 * B)
        except ArithmeticError:
            return None
    if not np.isfinite(xmin):
        return None
    return xmin


def _zoom(a_lo, a_hi, phi_lo, phi_hi, derphi_lo,
          phi, derphi, phi0, derphi0, c1, c2, extra_condition):
    """Zoom stage of approximate linesearch satisfying strong Wolfe conditions.
    
    Part of the optimization algorithm in `scalar_search_wolfe2`.
    
    Notes
    -----
    Implements Algorithm 3.6 (zoom) in Wright and Nocedal,
    'Numerical Optimization', 1999, pp. 61.

    """

    maxiter = 10
    i = 0
    delta1 = 0.2  # cubic interpolant check
    delta2 = 0.1  # quadratic interpolant check
    phi_rec = phi0
    a_rec = 0
    while True:
        # interpolate to find a trial step length between a_lo and
        # a_hi Need to choose interpolation here. Use cubic
        # interpolation and then if the result is within delta *
        # dalpha or outside of the interval bounded by a_lo or a_hi
        # then use quadratic interpolation, if the result is still too
        # close, then use bisection

        dalpha = a_hi - a_lo
        if dalpha < 0:
            a, b = a_hi, a_lo
        else:
            a, b = a_lo, a_hi

        # minimizer of cubic interpolant
        # (uses phi_lo, derphi_lo, phi_hi, and the most recent value of phi)
        #
        # if the result is too close to the end points (or out of the
        # interval), then use quadratic interpolation with phi_lo,
        # derphi_lo and phi_hi if the result is still too close to the
        # end points (or out of the interval) then use bisection

        if (i > 0):
            cchk = delta1 * dalpha
            a_j = _cubicmin(a_lo, phi_lo, derphi_lo, a_hi, phi_hi,
                            a_rec, phi_rec)
        if (i == 0) or (a_j is None) or (a_j > b - cchk) or (a_j < a + cchk):
            qchk = delta2 * dalpha
            a_j = _quadmin(a_lo, phi_lo, derphi_lo, a_hi, phi_hi)
            if (a_j is None) or (a_j > b-qchk) or (a_j < a+qchk):
                a_j = a_lo + 0.5*dalpha

        # Check new value of a_j

        phi_aj = phi(a_j)
        if (phi_aj > phi0 + c1*a_j*derphi0) or (phi_aj >= phi_lo):
            phi_rec = phi_hi
            a_rec = a_hi
            a_hi = a_j
            phi_hi = phi_aj
        else:
            derphi_aj = derphi(a_j)
            if abs(derphi_aj) <= -c2*derphi0 and extra_condition(a_j, phi_aj):
                a_star = a_j
                val_star = phi_aj
                valprime_star = derphi_aj
                break
            if derphi_aj*(a_hi - a_lo) >= 0:
                phi_rec = phi_hi
                a_rec = a_hi
                a_hi = a_lo
                phi_hi = phi_lo
            else:
                phi_rec = phi_lo
                a_rec = a_lo
            a_lo = a_j
            phi_lo = phi_aj
            derphi_lo = derphi_aj
        i += 1
        if (i > maxiter):
            # Failed to find a conforming step size
            a_star = None
            val_star = None
            valprime_star = None
            break
    return a_star, val_star, valprime_star


#------------------------------------------------------------------------------
# Armijo line and scalar searches
#------------------------------------------------------------------------------

def line_search_armijo(f, xk, pk, gfk, old_fval, args=(), c1=1e-4, alpha0=1):
    """Minimize over alpha, the function ``f(xk+alpha pk)``.

    Parameters
    ----------
    f : callable
        Function to be minimized.
    xk : array_like
        Current point.
    pk : array_like
        Search direction.
    gfk : array_like
        Gradient of `f` at point `xk`.
    old_fval : float
        Value of `f` at point `xk`.
    args : tuple, optional
        Optional arguments.
    c1 : float, optional
        Value to control stopping criterion.
    alpha0 : scalar, optional
        Value of `alpha` at start of the optimization.

    Returns
    -------
    alpha
    f_count
    f_val_at_alpha

    Notes
    -----
    Uses the interpolation algorithm (Armijo backtracking) as suggested by
    Wright and Nocedal in 'Numerical Optimization', 1999, pp. 56-57

    """
    xk = np.atleast_1d(xk)
    fc = [0]

    def phi(alpha1):
        fc[0] += 1
        return f(xk + alpha1*pk, *args)

    if old_fval is None:
        phi0 = phi(0.)
    else:
        phi0 = old_fval  # compute f(xk) -- done in past loop

    derphi0 = np.dot(gfk, pk)
    alpha, phi1 = scalar_search_armijo(phi, phi0, derphi0, c1=c1,
                                       alpha0=alpha0)
    return alpha, fc[0], phi1


def line_search_BFGS(f, xk, pk, gfk, old_fval, args=(), c1=1e-4, alpha0=1):
    """
    Compatibility wrapper for `line_search_armijo`
    """
    r = line_search_armijo(f, xk, pk, gfk, old_fval, args=args, c1=c1,
                           alpha0=alpha0)
    return r[0], r[1], 0, r[2]


def scalar_search_armijo(phi, phi0, derphi0, c1=1e-4, alpha0=1, amin=0):
    """Minimize over alpha, the function ``phi(alpha)``.

    Uses the interpolation algorithm (Armijo backtracking) as suggested by
    Wright and Nocedal in 'Numerical Optimization', 1999, pp. 56-57

    alpha > 0 is assumed to be a descent direction.

    Returns
    -------
    alpha
    phi1

    """
    phi_a0 = phi(alpha0)
    if phi_a0 <= phi0 + c1*alpha0*derphi0:
        return alpha0, phi_a0

    # Otherwise, compute the minimizer of a quadratic interpolant:

    alpha1 = -(derphi0) * alpha0**2 / 2.0 / (phi_a0 - phi0 - derphi0 * alpha0)
    phi_a1 = phi(alpha1)

    if (phi_a1 <= phi0 + c1*alpha1*derphi0):
        return alpha1, phi_a1

    # Otherwise, loop with cubic interpolation until we find an alpha which
    # satisfies the first Wolfe condition (since we are backtracking, we will
    # assume that the value of alpha is not too small and satisfies the second
    # condition.

    while alpha1 > amin:       # we are assuming alpha>0 is a descent direction
        factor = alpha0**2 * alpha1**2 * (alpha1-alpha0)
        a = alpha0**2 * (phi_a1 - phi0 - derphi0*alpha1) - \
            alpha1**2 * (phi_a0 - phi0 - derphi0*alpha0)
        a = a / factor
        b = -alpha0**3 * (phi_a1 - phi0 - derphi0*alpha1) + \
            alpha1**3 * (phi_a0 - phi0 - derphi0*alpha0)
        b = b / factor

        alpha2 = (-b + np.sqrt(abs(b**2 - 3 * a * derphi0))) / (3.0*a)
        phi_a2 = phi(alpha2)

        if (phi_a2 <= phi0 + c1*alpha2*derphi0):
            return alpha2, phi_a2

        if (alpha1 - alpha2) > alpha1 / 2.0 or (1 - alpha2/alpha1) < 0.96:
            alpha2 = alpha1 / 2.0

        alpha0 = alpha1
        alpha1 = alpha2
        phi_a0 = phi_a1
        phi_a1 = phi_a2

    # Failed to find a suitable step length
    return None, phi_a1


#------------------------------------------------------------------------------
# Non-monotone line search for DF-SANE
#------------------------------------------------------------------------------

def _nonmonotone_line_search_cruz(f, x_k, d, prev_fs, eta,
                                  gamma=1e-4, tau_min=0.1, tau_max=0.5):
    """
    Nonmonotone backtracking line search as described in [1]_

    Parameters
    ----------
    f : callable
        Function returning a tuple ``(f, F)`` where ``f`` is the value
        of a merit function and ``F`` the residual.
    x_k : ndarray
        Initial position.
    d : ndarray
        Search direction.
    prev_fs : float
        List of previous merit function values. Should have ``len(prev_fs) <= M``
        where ``M`` is the nonmonotonicity window parameter.
    eta : float
        Allowed merit function increase, see [1]_
    gamma, tau_min, tau_max : float, optional
        Search parameters, see [1]_

    Returns
    -------
    alpha : float
        Step length
    xp : ndarray
        Next position
    fp : float
        Merit function value at next position
    Fp : ndarray
        Residual at next position

    References
    ----------
    [1] "Spectral residual method without gradient information for solving
        large-scale nonlinear systems of equations." W. La Cruz,
        J.M. Martinez, M. Raydan. Math. Comp. **75**, 1429 (2006).

    """
    f_k = prev_fs[-1]
    f_bar = max(prev_fs)

    alpha_p = 1
    alpha_m = 1
    alpha = 1

    while True:
        xp = x_k + alpha_p * d
        fp, Fp = f(xp)

        if fp <= f_bar + eta - gamma * alpha_p**2 * f_k:
            alpha = alpha_p
            break

        alpha_tp = alpha_p**2 * f_k / (fp + (2*alpha_p - 1)*f_k)

        xp = x_k - alpha_m * d
        fp, Fp = f(xp)

        if fp <= f_bar + eta - gamma * alpha_m**2 * f_k:
            alpha = -alpha_m
            break

        alpha_tm = alpha_m**2 * f_k / (fp + (2*alpha_m - 1)*f_k)

        alpha_p = np.clip(alpha_tp, tau_min * alpha_p, tau_max * alpha_p)
        alpha_m = np.clip(alpha_tm, tau_min * alpha_m, tau_max * alpha_m)

    return alpha, xp, fp, Fp


def _nonmonotone_line_search_cheng(f, x_k, d, f_k, C, Q, eta,
                                   gamma=1e-4, tau_min=0.1, tau_max=0.5,
                                   nu=0.85):
    """
    Nonmonotone line search from [1]

    Parameters
    ----------
    f : callable
        Function returning a tuple ``(f, F)`` where ``f`` is the value
        of a merit function and ``F`` the residual.
    x_k : ndarray
        Initial position.
    d : ndarray
        Search direction.
    f_k : float
        Initial merit function value.
    C, Q : float
        Control parameters. On the first iteration, give values
        Q=1.0, C=f_k
    eta : float
        Allowed merit function increase, see [1]_
    nu, gamma, tau_min, tau_max : float, optional
        Search parameters, see [1]_

    Returns
    -------
    alpha : float
        Step length
    xp : ndarray
        Next position
    fp : float
        Merit function value at next position
    Fp : ndarray
        Residual at next position
    C : float
        New value for the control parameter C
    Q : float
        New value for the control parameter Q

    References
    ----------
    .. [1] W. Cheng & D.-H. Li, ''A derivative-free nonmonotone line
           search and its application to the spectral residual
           method'', IMA J. Numer. Anal. 29, 814 (2009).

    """
    alpha_p = 1
    alpha_m = 1
    alpha = 1

    while True:
        xp = x_k + alpha_p * d
        fp, Fp = f(xp)

        if fp <= C + eta - gamma * alpha_p**2 * f_k:
            alpha = alpha_p
            break

        alpha_tp = alpha_p**2 * f_k / (fp + (2*alpha_p - 1)*f_k)

        xp = x_k - alpha_m * d
        fp, Fp = f(xp)

        if fp <= C + eta - gamma * alpha_m**2 * f_k:
            alpha = -alpha_m
            break

        alpha_tm = alpha_m**2 * f_k / (fp + (2*alpha_m - 1)*f_k)

        alpha_p = np.clip(alpha_tp, tau_min * alpha_p, tau_max * alpha_p)
        alpha_m = np.clip(alpha_tm, tau_min * alpha_m, tau_max * alpha_m)

    # Update C and Q
    Q_next = nu * Q + 1
    C = (nu * Q * (C + eta) + fp) / Q_next
    Q = Q_next

    return alpha, xp, fp, Fp, C, Q