line_search_strong_wolfe.py 13 KB
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# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.
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#
# Copyright(C) 2013-2017 Max-Planck-Society
#
# NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik
# and financially supported by the Studienstiftung des deutschen Volkes.
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import numpy as np

from .line_search import LineSearch
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from ...energies import LineEnergy
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class LineSearchStrongWolfe(LineSearch):
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    """Class for finding a step size that satisfies the strong Wolfe conditions.
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    Algorithm contains two stages. It begins with a trial step length and
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    keeps increasing it until it finds an acceptable step length or an
    interval. If it does not satisfy the Wolfe conditions, it performs the Zoom
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    algorithm (second stage). By interpolating it decreases the size of the
    interval until an acceptable step length is found.

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    Parameters
    ----------
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    c1 : float
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        Parameter for Armijo condition rule. (Default: 1e-4)
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    c2 : float
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        Parameter for curvature condition rule. (Default: 0.9)
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    max_step_size : float
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        Maximum step allowed in to be made in the descent direction.
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        (Default: 50)
    max_iterations : integer
        Maximum number of iterations performed by the line search algorithm.
        (Default: 10)
    max_zoom_iterations : integer
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        Maximum number of iterations performed by the zoom algorithm.
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        (Default: 10)
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    Attributes
    ----------
    c1 : float
        Parameter for Armijo condition rule.
    c2 : float
        Parameter for curvature condition rule.
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    max_step_size : float
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        Maximum step allowed in to be made in the descent direction.
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    max_iterations : integer
        Maximum number of iterations performed by the line search algorithm.
    max_zoom_iterations : integer
        Maximum number of iterations performed by the zoom algorithm.
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    """

    def __init__(self, c1=1e-4, c2=0.9,
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                 max_step_size=1000000000, max_iterations=100,
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                 max_zoom_iterations=30):
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        super(LineSearchStrongWolfe, self).__init__()

        self.c1 = np.float(c1)
        self.c2 = np.float(c2)
        self.max_step_size = max_step_size
        self.max_iterations = int(max_iterations)
        self.max_zoom_iterations = int(max_zoom_iterations)

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    def perform_line_search(self, energy, pk, f_k_minus_1=None):
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        """Performs the first stage of the algorithm.
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        It starts with a trial step size and it keeps increasing it until it
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        satisfies the strong Wolf conditions. It also performs the descent and
        returns the optimal step length and the new energy.
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        Parameters
        ----------
        energy : Energy object
            Energy object from which we will calculate the energy and the
            gradient at a specific point.
        pk : Field
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            Vector pointing into the search direction.
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        f_k_minus_1 : float
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            Value of the fuction (which is being minimized) at the k-1
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            iteration of the line search procedure. (Default: None)
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        Returns
        -------
        energy_star : Energy object
            The new Energy object on the new position.
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        """

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        le_0 = LineEnergy(0., energy, pk, 0.)
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        # initialize the zero phis
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        old_phi_0 = f_k_minus_1
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        phi_0 = le_0.value
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        phiprime_0 = le_0.directional_derivative
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        if phiprime_0 >= 0:
            self.logger.error("Input direction must be a descent direction")
            raise RuntimeError
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        # set alphas
        alpha0 = 0.
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        phi_alpha0 = phi_0
        phiprime_alpha0 = phiprime_0

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        if self.preferred_initial_step_size is not None:
            alpha1 = self.preferred_initial_step_size
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        elif old_phi_0 is not None and phiprime_0 != 0:
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            alpha1 = min(1.0, 1.01*2*(phi_0 - old_phi_0)/phiprime_0)
            if alpha1 < 0:
                alpha1 = 1.0
        else:
            alpha1 = 1.0

        # start the minimization loop
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        for i in xrange(self.max_iterations):
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            if alpha1 == 0:
                self.logger.warn("Increment size became 0.")
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                return le_0.energy

            le_alpha1 = le_0.at(alpha1)
            phi_alpha1 = le_alpha1.value
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            if (phi_alpha1 > phi_0 + self.c1*alpha1*phiprime_0) or \
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               ((phi_alpha1 >= phi_alpha0) and (i > 0)):
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                le_star = self._zoom(alpha0, alpha1, phi_0, phiprime_0,
                                     phi_alpha0, phiprime_alpha0, phi_alpha1,
                                     le_0)
                return le_star.energy
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            phiprime_alpha1 = le_alpha1.directional_derivative
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            if abs(phiprime_alpha1) <= -self.c2*phiprime_0:
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                return le_alpha1.energy
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            if phiprime_alpha1 >= 0:
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                le_star = self._zoom(alpha1, alpha0, phi_0, phiprime_0,
                                     phi_alpha1, phiprime_alpha1, phi_alpha0,
                                     le_0)
                return le_star.energy
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            # update alphas
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            alpha0, alpha1 = alpha1, min(2*alpha1, self.max_step_size)
            if alpha1 == self.max_step_size:
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                print "reached max step size, bailing out"
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                return le_alpha1.energy

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            phi_alpha0 = phi_alpha1
            phiprime_alpha0 = phiprime_alpha1

        else:
            # max_iterations was reached
            self.logger.error("The line search algorithm did not converge.")
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            return le_alpha1.energy
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    def _zoom(self, alpha_lo, alpha_hi, phi_0, phiprime_0,
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              phi_lo, phiprime_lo, phi_hi, le_0):
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        """Performs the second stage of the line search algorithm.
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        If the first stage was not successful then the Zoom algorithm tries to
        find a suitable step length by using bisection, quadratic, cubic
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        interpolation.
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        Parameters
        ----------
        alpha_lo : float
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            A boundary for the step length interval.
            Fulfills Wolfe condition 1.
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        alpha_hi : float
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            The other boundary for the step length interval.
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        phi_0 : float
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            Value of the energy at the starting point of the line search
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            algorithm.
        phiprime_0 : Field
            Gradient at the starting point of the line search algorithm.
        phi_lo : float
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            Value of the energy if we perform a step of length alpha_lo in
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            descent direction.
        phiprime_lo : Field
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            Gradient at the nwe position if we perform a step of length
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            alpha_lo in descent direction.
        phi_hi : float
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            Value of the energy if we perform a step of length alpha_hi in
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            descent direction.
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        Returns
        -------
        energy_star : Energy object
            The new Energy object on the new position.
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        """
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        # define the cubic and quadratic interpolant checks
        cubic_delta = 0.2  # cubic
        quad_delta = 0.1  # quadratic
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        alpha_recent = None
        phi_recent = None
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        assert phi_lo <= phi_0 + self.c1*alpha_lo*phiprime_0
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        assert phiprime_lo*(alpha_hi-alpha_lo) < 0.
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        for i in xrange(self.max_zoom_iterations):
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            # assert phi_lo <= phi_0 + self.c1*alpha_lo*phiprime_0
            # assert phiprime_lo*(alpha_hi-alpha_lo)<0.
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            delta_alpha = alpha_hi - alpha_lo
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            a, b = min(alpha_lo, alpha_hi), max(alpha_lo, alpha_hi)
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            # Try cubic interpolation
            if i > 0:
                cubic_check = cubic_delta * delta_alpha
                alpha_j = self._cubicmin(alpha_lo, phi_lo, phiprime_lo,
                                         alpha_hi, phi_hi,
                                         alpha_recent, phi_recent)
            # If cubic was not successful or not available, try quadratic
            if (i == 0) or (alpha_j is None) or (alpha_j > b - cubic_check) or\
               (alpha_j < a + cubic_check):
                quad_check = quad_delta * delta_alpha
                alpha_j = self._quadmin(alpha_lo, phi_lo, phiprime_lo,
                                        alpha_hi, phi_hi)
                # If quadratic was not successful, try bisection
                if (alpha_j is None) or (alpha_j > b - quad_check) or \
                   (alpha_j < a + quad_check):
                    alpha_j = alpha_lo + 0.5*delta_alpha
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            # Check if the current value of alpha_j is already sufficient
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            le_alphaj = le_0.at(alpha_j)
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            phi_alphaj = le_alphaj.value
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            # If the first Wolfe condition is not met replace alpha_hi
            # by alpha_j
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            if (phi_alphaj > phi_0 + self.c1*alpha_j*phiprime_0) or \
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               (phi_alphaj >= phi_lo):
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                alpha_recent, phi_recent = alpha_hi, phi_hi
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                alpha_hi, phi_hi = alpha_j, phi_alphaj
            else:
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                phiprime_alphaj = le_alphaj.directional_derivative
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                # If the second Wolfe condition is met, return the result
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                if abs(phiprime_alphaj) <= -self.c2*phiprime_0:
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                    return le_alphaj
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                # If not, check the sign of the slope
                if phiprime_alphaj*delta_alpha >= 0:
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                    alpha_recent, phi_recent = alpha_hi, phi_hi
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                    alpha_hi, phi_hi = alpha_lo, phi_lo
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                else:
                    alpha_recent, phi_recent = alpha_lo, phi_lo
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                # Replace alpha_lo by alpha_j
                (alpha_lo, phi_lo, phiprime_lo) = (alpha_j, phi_alphaj,
                                                   phiprime_alphaj)

        else:
            self.logger.error("The line search algorithm (zoom) did not "
                              "converge.")
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            return le_alphaj
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    def _cubicmin(self, a, fa, fpa, b, fb, c, fc):
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        """Estimating the minimum with cubic interpolation.
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        Finds the minimizer for a cubic polynomial that goes through the
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        points ( a,f(a) ), ( b,f(b) ), and ( c,f(c) ) with derivative at point
        a of fpa.
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        f(x) = A *(x-a)^3 + B*(x-a)^2 + C*(x-a) + D
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        If no minimizer can be found return None
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        Parameters
        ----------
        a : float
            Selected point.
        fa : float
            Value of polynomial at point a.
        fpa : Field
            Derivative at point a.
        b : float
            Selected point.
        fb : float
            Value of polynomial at point b.
        c : float
            Selected point.
        fc : float
            Value of polynomial at point c.
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        Returns
        -------
        xmin : float
            Position of the approximated minimum.
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        """

        with np.errstate(divide='raise', over='raise', invalid='raise'):
            try:
                C = fpa
                db = b - a
                dc = c - a
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                denom = db * db * dc * dc * (db - dc)
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                d1 = np.empty((2, 2))
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                d1[0, 0] = dc * dc
                d1[0, 1] = -(db*db)
                d1[1, 0] = -(dc*dc*dc)
                d1[1, 1] = db*db*db
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                [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(self, a, fa, fpa, b, fb):
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        """Estimating the minimum with quadratic interpolation.
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        Finds the minimizer for a quadratic polynomial that goes through
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        the points ( a,f(a) ), ( b,f(b) ) with derivative at point a of fpa.
        f(x) = B*(x-a)^2 + C*(x-a) + D
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        Parameters
        ----------
        a : float
            Selected point.
        fa : float
            Value of polynomial at point a.
        fpa : Field
            Derivative at point a.
        b : float
            Selected point.
        fb : float
            Value of polynomial at point b.
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        Returns
        -------
        xmin : float
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            Position of the approximated minimum.
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        """
        # 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