Chimie Théorique » QMX

#__docformat__ = "restructuredtext en"

# ******NOTICE***************

# optimize.py module by Travis E. Oliphant

#

# You may copy and use this module as you see fit with no

# guarantee implied provided you keep this notice in all copies.

# *****END NOTICE************

import numpy as np

from numpy import atleast_1d, eye, mgrid, argmin, zeros, shape, empty, \

     squeeze, vectorize, asarray, absolute, sqrt, Inf, asfarray, isinf

from ase.utils.linesearch import LineSearch

from ase.optimize.optimize import Optimizer

from numpy import arange

# These have been copied from Numeric's MLab.py

# I don't think they made the transition to scipy_core

# Modified from scipy_optimize

abs = absolute

import __builtin__

pymin = __builtin__.min

pymax = __builtin__.max

__version__="0.1"

class BFGSLineSearch(Optimizer):

    def __init__(self, atoms, restart=None, logfile='-', maxstep=.2,

                 trajectory=None, c1=.23, c2=0.46, alpha=10., stpmax=50.):

        """Minimize a function using the BFGS algorithm.

        Notes:

            Optimize the function, f, whose gradient is given by fprime

            using the quasi-Newton method of Broyden, Fletcher, Goldfarb,

            and Shanno (BFGS) See Wright, and Nocedal 'Numerical

            Optimization', 1999, pg. 198.

        *See Also*:

          scikits.openopt : SciKit which offers a unified syntax to call

                            this and other solvers.

        """

        self.maxstep = maxstep

        self.stpmax = stpmax

        self.alpha = alpha

        self.H = None

        self.c1 = c1

        self.c2 = c2

        self.force_calls = 0

        self.function_calls = 0

        self.r0 = None

        self.g0 = None

        self.e0 = None

        self.load_restart = False

        self.task = 'START'

        self.rep_count = 0

        self.p = None

        self.alpha_k = None

        self.no_update = False

        self.replay = False

        Optimizer.__init__(self, atoms, restart, logfile, trajectory)

    def read(self):

        self.r0, self.g0, self.e0, self.task, self.H = self.load()

        self.load_restart = True

    def reset(self):

        print 'reset'

        self.H = None

        self.r0 = None

        self.g0 = None

        self.e0 = None

        self.rep_count = 0

    def step(self, f):

        atoms = self.atoms

        r = atoms.get_positions()

        r = r.reshape(-1)

        g = -f.reshape(-1) / self.alpha

        #g = -f.reshape(-1)

        p0 = self.p

        self.update(r, g, self.r0, self.g0, p0)

        e = atoms.get_potential_energy() / self.alpha

        #e = self.func(r)

        self.p = -np.dot(self.H,g)

        p_size = np.sqrt((self.p **2).sum())

        if self.nsteps != 0:

            p0_size = np.sqrt((p0 **2).sum())

            delta_p = self.p/p_size + p0/p0_size

        if p_size <= np.sqrt(len(atoms) * 1e-10):

            self.p /= (p_size / np.sqrt(len(atoms)*1e-10))

        ls = LineSearch()

        self.alpha_k, e, self.e0, self.no_update = \

           ls._line_search(self.func, self.fprime, r, self.p, g, e, self.e0,

                           maxstep=self.maxstep, c1=self.c1,

                           c2=self.c2, stpmax=self.stpmax)

        #if alpha_k is None:  # line search failed try different one.

        #    alpha_k, fc, gc, e, e0, gfkp1 = \

        #             line_search(self.func, self.fprime,r,p,g,

        #                         e,self.e0)

        #if abs(e - self.e0) < 0.000001:

        #    self.rep_count += 1

        #else:

        #    self.rep_count = 0

        #if (alpha_k is None) or (self.rep_count >= 3):

        #    # If the line search fails, reset the Hessian matrix and

        #    # start a new line search.

        #    self.reset()

        #    return

        dr = self.alpha_k * self.p

        atoms.set_positions((r+dr).reshape(len(atoms),-1))

        self.r0 = r

        self.g0 = g

        self.dump((self.r0, self.g0, self.e0, self.task, self.H))

    def update(self, r, g, r0, g0, p0):

        self.I = eye(len(self.atoms) * 3, dtype=int)

        if self.H is None:

            self.H = eye(3 * len(self.atoms))

            #self.H = eye(3 * len(self.atoms)) / self.alpha

            return

        else:

            dr = r - r0

            dg = g - g0

            if not ((self.alpha_k > 0 and abs(np.dot(g,p0))-abs(np.dot(g0,p0)) < 0) \

                or self.replay):

                return

            if self.no_update == True:

                print 'skip update'

                return

            try: # this was handled in numeric, let it remaines for more safety

                rhok = 1.0 / (np.dot(dg,dr))

            except ZeroDivisionError:

                rhok = 1000.0

                print "Divide-by-zero encountered: rhok assumed large"

            if isinf(rhok): # this is patch for np

                rhok = 1000.0

                print "Divide-by-zero encountered: rhok assumed large"

            A1 = self.I - dr[:, np.newaxis] * dg[np.newaxis, :] * rhok

            A2 = self.I - dg[:, np.newaxis] * dr[np.newaxis, :] * rhok

            H0 = self.H

            self.H = np.dot(A1, np.dot(self.H, A2)) + rhok * dr[:, np.newaxis] \

                     * dr[np.newaxis, :]

            #self.B = np.linalg.inv(self.H)

            #omega, V = np.linalg.eigh(self.B)

            #eighfile = open('eigh.log','w')

    def func(self, x):

        """Objective function for use of the optimizers"""

        self.atoms.set_positions(x.reshape(-1, 3))

        self.function_calls += 1

        # Scale the problem as SciPy uses I as initial Hessian.

        return self.atoms.get_potential_energy() / self.alpha

        #return self.atoms.get_potential_energy()

    def fprime(self, x):

        """Gradient of the objective function for use of the optimizers"""

        self.atoms.set_positions(x.reshape(-1, 3))

        self.force_calls += 1

        # Remember that forces are minus the gradient!

        # Scale the problem as SciPy uses I as initial Hessian.

        return - self.atoms.get_forces().reshape(-1) / self.alpha

        #return - self.atoms.get_forces().reshape(-1)

    def replay_trajectory(self, traj):

        """Initialize hessian from old trajectory."""

        self.replay = True

        if isinstance(traj, str):

            from ase.io.trajectory import PickleTrajectory

            traj = PickleTrajectory(traj, 'r')

        atoms = traj[0]

        r0 = None

        g0 = None

        for i in range(0, len(traj) - 1):

            r = traj[i].get_positions().ravel()

            g = - traj[i].get_forces().ravel() / self.alpha

            self.update(r, g, r0, g0, self.p)

            self.p = -np.dot(self.H,g)

            r0 = r.copy()

            g0 = g.copy()

        self.r0 = r0

        self.g0 = g0

        #self.r0 = traj[-2].get_positions().ravel()

        #self.g0 = - traj[-2].get_forces().ravel()

def wrap_function(function, args):

    ncalls = [0]

    def function_wrapper(x):

        ncalls[0] += 1

        return function(x, *args)

    return ncalls, function_wrapper

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

    C = fpa

    D = fa

    db = b-a

    dc = c-a

    if (db == 0) or (dc == 0) or (b==c): return None

    denom = (db*dc)**2 * (db-dc)

    d1 = 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,asarray([fb-fa-C*db,fc-fa-C*dc]).flatten())

    A /= denom

    B /= denom

    radical = B*B-3*A*C

    if radical < 0:  return None

    if (A == 0): return None

    xmin = a + (-B + sqrt(radical))/(3*A)

    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

    D = fa

    C = fpa

    db = b-a*1.0

    if (db==0): return None

    B = (fb-D-C*db)/(db*db)

    if (B <= 0): return None

    xmin = a  - C / (2.0*B)

    return xmin

def zoom(a_lo, a_hi, phi_lo, phi_hi, derphi_lo,

         phi, derphi, phi0, derphi0, c1, c2):

    maxiter = 10

    i = 0

    delta1 = 0.2  # cubic interpolant check

    delta2 = 0.1  # quadratic interpolant check

    phi_rec = phi0

    a_rec = 0

    while 1:

        # 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 stil 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

#                print "Using bisection."

#            else: print "Using quadratic."

#        else: print "Using cubic."

        # 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:

                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):

            a_star = a_j

            val_star = phi_aj

            valprime_star = None

            break

    return a_star, val_star, valprime_star

def line_search(f, myfprime, xk, pk, gfk, old_fval, old_old_fval,

                args=(), c1=1e-4, c2=0.9, amax=50):

    """Find alpha that satisfies strong Wolfe conditions.

    Parameters:

        f : callable f(x,*args)

            Objective function.

        myfprime : callable f'(x,*args)

            Objective function gradient (can be None).

        xk : ndarray

            Starting point.

        pk : ndarray

            Search direction.

        gfk : ndarray

            Gradient value for x=xk (xk being the current parameter

            estimate).

        args : tuple

            Additional arguments passed to objective function.

        c1 : float

            Parameter for Armijo condition rule.

        c2 : float

            Parameter for curvature condition rule.

    Returns:

        alpha0 : float

            Alpha for which ``x_new = x0 + alpha * pk``.

        fc : int

            Number of function evaluations made.

        gc : int

            Number of gradient evaluations made.

    Notes:

        Uses the line search algorithm to enforce strong Wolfe

        conditions.  See Wright and Nocedal, 'Numerical Optimization',

        1999, pg. 59-60.

        For the zoom phase it uses an algorithm by [...].

    """

    global _ls_fc, _ls_gc, _ls_ingfk

    _ls_fc = 0

    _ls_gc = 0

    _ls_ingfk = None

    def phi(alpha):

        global _ls_fc

        _ls_fc += 1

        return f(xk+alpha*pk,*args)

    if isinstance(myfprime,type(())):

        def phiprime(alpha):

            global _ls_fc, _ls_ingfk

            _ls_fc += len(xk)+1

            eps = myfprime[1]

            fprime = myfprime[0]

            newargs = (f,eps) + args

            _ls_ingfk = fprime(xk+alpha*pk,*newargs)  # store for later use

            return np.dot(_ls_ingfk,pk)

    else:

        fprime = myfprime

        def phiprime(alpha):

            global _ls_gc, _ls_ingfk

            _ls_gc += 1

            _ls_ingfk = fprime(xk+alpha*pk,*args)  # store for later use

            return np.dot(_ls_ingfk,pk)

    alpha0 = 0

    phi0 = old_fval

    derphi0 = np.dot(gfk,pk)

    alpha1 = pymin(1.,1.01*2*(phi0-old_old_fval)/derphi0)

    if alpha1 == 0:

        # This shouldn't happen. Perhaps the increment has slipped below

        # machine precision?  For now, set the return variables skip the

        # useless while loop, and raise warnflag=2 due to possible imprecision.

        alpha_star = None

        fval_star = old_fval

        old_fval = old_old_fval

        fprime_star = None

    phi_a1 = phi(alpha1)

    #derphi_a1 = phiprime(alpha1)  evaluated below

    phi_a0 = phi0

    derphi_a0 = derphi0

    i = 1

    maxiter = 10

    while 1:         # bracketing phase

        if alpha1 == 0:

            break

        if (phi_a1 > phi0 + c1*alpha1*derphi0) or \

           ((phi_a1 >= phi_a0) and (i > 1)):

            alpha_star, fval_star, fprime_star = \

                        zoom(alpha0, alpha1, phi_a0,

                             phi_a1, derphi_a0, phi, phiprime,

                             phi0, derphi0, c1, c2)

            break

        derphi_a1 = phiprime(alpha1)

        if (abs(derphi_a1) <= -c2*derphi0):

            alpha_star = alpha1

            fval_star = phi_a1

            fprime_star = derphi_a1

            break

        if (derphi_a1 >= 0):

            alpha_star, fval_star, fprime_star = \

                        zoom(alpha1, alpha0, phi_a1,

                             phi_a0, derphi_a1, phi, phiprime,

                             phi0, derphi0, c1, c2)

            break

        alpha2 = 2 * alpha1   # increase by factor of two on each iteration

        i = i + 1

        alpha0 = alpha1

        alpha1 = alpha2

        phi_a0 = phi_a1

        phi_a1 = phi(alpha1)

        derphi_a0 = derphi_a1

        # stopping test if lower function not found

        if (i > maxiter):

            alpha_star = alpha1

            fval_star = phi_a1

            fprime_star = None

            break

    if fprime_star is not None:

        # fprime_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.

        fprime_star = _ls_ingfk

    return alpha_star, _ls_fc, _ls_gc, fval_star, old_fval, fprime_star