scipy/optimize/lbfgsb.py

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

    fmin_l_bfgs_b

"""

## License for the Python wrapper
## ==============================

## Copyright (c) 2004 David M. Cooke <cookedm@physics.mcmaster.ca>

## Permission is hereby granted, free of charge, to any person obtaining a
## copy of this software and associated documentation files (the "Software"),
## to deal in the Software without restriction, including without limitation
## the rights to use, copy, modify, merge, publish, distribute, sublicense,
## and/or sell copies of the Software, and to permit persons to whom the
## Software is furnished to do so, subject to the following conditions:

## The above copyright notice and this permission notice shall be included in
## all copies or substantial portions of the Software.

## THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
## IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
## FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
## AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
## LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
## FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
## DEALINGS IN THE SOFTWARE.

## Modifications by Travis Oliphant and Enthought, Inc. for inclusion in SciPy

from __future__ import division, print_function, absolute_import

import numpy as np
from numpy import array, asarray, float64, int32, zeros
from . import _lbfgsb
from .optimize import (approx_fprime, MemoizeJac, OptimizeResult,
                       _check_unknown_options, wrap_function)

__all__ = ['fmin_l_bfgs_b']


def fmin_l_bfgs_b(func, x0, fprime=None, args=(),
                  approx_grad=0,
                  bounds=None, m=10, factr=1e7, pgtol=1e-5,
                  epsilon=1e-8,
                  iprint=-1, maxfun=15000, maxiter=15000, disp=None,
                  callback=None):
    """
    Minimize a function func using the L-BFGS-B algorithm.

    Parameters
    ----------
    func : callable f(x,*args)
        Function to minimise.
    x0 : ndarray
        Initial guess.
    fprime : callable fprime(x,*args)
        The gradient of `func`.  If None, then `func` returns the function
        value and the gradient (``f, g = func(x, *args)``), unless
        `approx_grad` is True in which case `func` returns only ``f``.
    args : sequence
        Arguments to pass to `func` and `fprime`.
    approx_grad : bool
        Whether to approximate the gradient numerically (in which case
        `func` returns only the function value).
    bounds : list
        ``(min, max)`` pairs for each element in ``x``, defining
        the bounds on that parameter. Use None for one of ``min`` or
        ``max`` when there is no bound in that direction.
    m : int
        The maximum number of variable metric corrections
        used to define the limited memory matrix. (The limited memory BFGS
        method does not store the full hessian but uses this many terms in an
        approximation to it.)
    factr : float
        The iteration stops when
        ``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``,
        where ``eps`` is the machine precision, which is automatically
        generated by the code. Typical values for `factr` are: 1e12 for
        low accuracy; 1e7 for moderate accuracy; 10.0 for extremely
        high accuracy.
    pgtol : float
        The iteration will stop when
        ``max{|proj g_i | i = 1, ..., n} <= pgtol``
        where ``pg_i`` is the i-th component of the projected gradient.
    epsilon : float
        Step size used when `approx_grad` is True, for numerically
        calculating the gradient
    iprint : int
        Controls the frequency of output. ``iprint < 0`` means no output;
        ``iprint == 0`` means write messages to stdout; ``iprint > 1`` in
        addition means write logging information to a file named
        ``iterate.dat`` in the current working directory.
    disp : int, optional
        If zero, then no output.  If a positive number, then this over-rides
        `iprint` (i.e., `iprint` gets the value of `disp`).
    maxfun : int
        Maximum number of function evaluations.
    maxiter : int
        Maximum number of iterations.
    callback : callable, optional
        Called after each iteration, as ``callback(xk)``, where ``xk`` is the
        current parameter vector.

    Returns
    -------
    x : array_like
        Estimated position of the minimum.
    f : float
        Value of `func` at the minimum.
    d : dict
        Information dictionary.

        * d['warnflag'] is

          - 0 if converged,
          - 1 if too many function evaluations or too many iterations,
          - 2 if stopped for another reason, given in d['task']

        * d['grad'] is the gradient at the minimum (should be 0 ish)
        * d['funcalls'] is the number of function calls made.
        * d['nit'] is the number of iterations.

    See also
    --------
    minimize: Interface to minimization algorithms for multivariate
        functions. See the 'L-BFGS-B' `method` in particular.

    Notes
    -----
    License of L-BFGS-B (FORTRAN code):

    The version included here (in fortran code) is 3.0
    (released April 25, 2011).  It was written by Ciyou Zhu, Richard Byrd,
    and Jorge Nocedal <nocedal@ece.nwu.edu>. It carries the following
    condition for use:

    This software is freely available, but we expect that all publications
    describing work using this software, or all commercial products using it,
    quote at least one of the references given below. This software is released
    under the BSD License.

    References
    ----------
    * R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound
      Constrained Optimization, (1995), SIAM Journal on Scientific and
      Statistical Computing, 16, 5, pp. 1190-1208.
    * C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B,
      FORTRAN routines for large scale bound constrained optimization (1997),
      ACM Transactions on Mathematical Software, 23, 4, pp. 550 - 560.
    * J.L. Morales and J. Nocedal. L-BFGS-B: Remark on Algorithm 778: L-BFGS-B,
      FORTRAN routines for large scale bound constrained optimization (2011),
      ACM Transactions on Mathematical Software, 38, 1.

    """
    # handle fprime/approx_grad
    if approx_grad:
        fun = func
        jac = None
    elif fprime is None:
        fun = MemoizeJac(func)
        jac = fun.derivative
    else:
        fun = func
        jac = fprime

    # build options
    if disp is None:
        disp = iprint
    opts = {'disp': disp,
            'iprint': iprint,
            'maxcor': m,
            'ftol': factr * np.finfo(float).eps,
            'gtol': pgtol,
            'eps': epsilon,
            'maxfun': maxfun,
            'maxiter': maxiter,
            'callback': callback}

    res = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=bounds,
                           **opts)
    d = {'grad': res['jac'],
         'task': res['message'],
         'funcalls': res['nfev'],
         'nit': res['nit'],
         'warnflag': res['status']}
    f = res['fun']
    x = res['x']

    return x, f, d


def _minimize_lbfgsb(fun, x0, args=(), jac=None, bounds=None,
                     disp=None, maxcor=10, ftol=2.2204460492503131e-09,
                     gtol=1e-5, eps=1e-8, maxfun=15000, maxiter=15000,
                     iprint=-1, callback=None, **unknown_options):
    """
    Minimize a scalar function of one or more variables using the L-BFGS-B
    algorithm.

    Options for the L-BFGS-B algorithm are:
        disp : bool
           Set to True to print convergence messages.
        maxcor : int
            The maximum number of variable metric corrections used to
            define the limited memory matrix. (The limited memory BFGS
            method does not store the full hessian but uses this many terms
            in an approximation to it.)
        factr : float
            The iteration stops when ``(f^k -
            f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``, where ``eps``
            is the machine precision, which is automatically generated by
            the code. Typical values for `factr` are: 1e12 for low
            accuracy; 1e7 for moderate accuracy; 10.0 for extremely high
            accuracy.
        gtol : float
            The iteration will stop when ``max{|proj g_i | i = 1, ..., n}
            <= gtol`` where ``pg_i`` is the i-th component of the
            projected gradient.
        eps : float
            Step size used for numerical approximation of the jacobian.
        disp : int
            Set to True to print convergence messages.
        maxfun : int
            Maximum number of function evaluations.
        maxiter : int
            Maximum number of iterations.

    This function is called by the `minimize` function with
    `method=L-BFGS-B`. It is not supposed to be called directly.
    """
    _check_unknown_options(unknown_options)
    m = maxcor
    epsilon = eps
    pgtol = gtol
    factr = ftol / np.finfo(float).eps

    x0 = asarray(x0).ravel()
    n, = x0.shape

    if bounds is None:
        bounds = [(None, None)] * n
    if len(bounds) != n:
        raise ValueError('length of x0 != length of bounds')

    if disp is not None:
        if disp == 0:
            iprint = -1
        else:
            iprint = disp

    n_function_evals, fun = wrap_function(fun, ())
    if jac is None:
        def func_and_grad(x):
            f = fun(x, *args)
            g = approx_fprime(x, fun, epsilon, *args)
            return f, g
    else:
        def func_and_grad(x):
            f = fun(x, *args)
            g = jac(x, *args)
            return f, g

    nbd = zeros(n, int32)
    low_bnd = zeros(n, float64)
    upper_bnd = zeros(n, float64)
    bounds_map = {(None, None): 0,
                  (1, None): 1,
                  (1, 1): 2,
                  (None, 1): 3}
    for i in range(0, n):
        l, u = bounds[i]
        if l is not None:
            low_bnd[i] = l
            l = 1
        if u is not None:
            upper_bnd[i] = u
            u = 1
        nbd[i] = bounds_map[l, u]

    x = array(x0, float64)
    f = array(0.0, float64)
    g = zeros((n,), float64)
    wa = zeros(2*m*n + 5*n + 11*m*m + 8*m, float64)
    iwa = zeros(3*n, int32)
    task = zeros(1, 'S60')
    csave = zeros(1, 'S60')
    lsave = zeros(4, int32)
    isave = zeros(44, int32)
    dsave = zeros(29, float64)

    task[:] = 'START'

    n_iterations = 0

    while 1:
#        x, f, g, wa, iwa, task, csave, lsave, isave, dsave = \
        _lbfgsb.setulb(m, x, low_bnd, upper_bnd, nbd, f, g, factr,
                       pgtol, wa, iwa, task, iprint, csave, lsave,
                       isave, dsave)
        task_str = task.tostring()
        if task_str.startswith(b'FG'):
            if n_function_evals[0] > maxfun:
                task[:] = ('STOP: TOTAL NO. of f AND g EVALUATIONS '
                           'EXCEEDS LIMIT')
            else:
                # minimization routine wants f and g at the current x
                # Overwrite f and g:
                f, g = func_and_grad(x)
        elif task_str.startswith(b'NEW_X'):
            # new iteration
            if n_iterations > maxiter:
                task[:] = 'STOP: TOTAL NO. of ITERATIONS EXCEEDS LIMIT'
            else:
                n_iterations += 1
                if callback is not None:
                    callback(x)
        else:
            break

    task_str = task.tostring().strip(b'\x00').strip()
    if task_str.startswith(b'CONV'):
        warnflag = 0
    elif n_function_evals[0] > maxfun:
        warnflag = 1
    elif n_iterations > maxiter:
        warnflag = 1
    else:
        warnflag = 2

    return OptimizeResult(fun=f, jac=g, nfev=n_function_evals[0],
                          nit=n_iterations, status=warnflag, message=task_str,
                          x=x, success=(warnflag == 0))


if __name__ == '__main__':
    def func(x):
        f = 0.25 * (x[0] - 1) ** 2
        for i in range(1, x.shape[0]):
            f += (x[i] - x[i-1] ** 2) ** 2
        f *= 4
        return f

    def grad(x):
        g = zeros(x.shape, float64)
        t1 = x[1] - x[0] ** 2
        g[0] = 2 * (x[0] - 1) - 16 * x[0] * t1
        for i in range(1, g.shape[0] - 1):
            t2 = t1
            t1 = x[i + 1] - x[i] ** 2
            g[i] = 8 * t2 - 16*x[i] * t1
        g[-1] = 8 * t1
        return g

    def func_and_grad(x):
        return func(x), grad(x)

    class Problem(object):
        def fun(self, x):
            return func_and_grad(x)

    factr = 1e7
    pgtol = 1e-5

    n = 25
    m = 10

    bounds = [(None, None)] * n
    for i in range(0, n, 2):
        bounds[i] = (1.0, 100)
    for i in range(1, n, 2):
        bounds[i] = (-100, 100)

    x0 = zeros((n,), float64)
    x0[:] = 3

    x, f, d = fmin_l_bfgs_b(func, x0, fprime=grad, m=m,
                            factr=factr, pgtol=pgtol)
    print(x)
    print(f)
    print(d)
    x, f, d = fmin_l_bfgs_b(func, x0, approx_grad=1,
                            m=m, factr=factr, pgtol=pgtol)
    print(x)
    print(f)
    print(d)
    x, f, d = fmin_l_bfgs_b(func_and_grad, x0, approx_grad=0,
                            m=m, factr=factr, pgtol=pgtol)
    print(x)
    print(f)
    print(d)
    p = Problem()
    x, f, d = fmin_l_bfgs_b(p.fun, x0, approx_grad=0,
                            m=m, factr=factr, pgtol=pgtol)
    print(x)
    print(f)
    print(d)