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"""
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,
_approx_fprime_helper)
from scipy.sparse.linalg import LinearOperator
__all__ = ['fmin_l_bfgs_b', 'LbfgsInvHessProduct']
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, maxls=20):
"""
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), optional
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, optional
Arguments to pass to `func` and `fprime`.
approx_grad : bool, optional
Whether to approximate the gradient numerically (in which case
`func` returns only the function value).
bounds : list, optional
``(min, max)`` pairs for each element in ``x``, defining
the bounds on that parameter. Use None or +-inf for one of ``min`` or
``max`` when there is no bound in that direction.
m : int, optional
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, optional
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, optional
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, optional
Step size used when `approx_grad` is True, for numerically
calculating the gradient
iprint : int, optional
Controls the frequency of output. ``iprint < 0`` means no output;
``iprint = 0`` print only one line at the last iteration;
``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations;
``iprint = 99`` print details of every iteration except n-vectors;
``iprint = 100`` print also the changes of active set and final x;
``iprint > 100`` print details of every iteration including x and g.
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, optional
Maximum number of function evaluations.
maxiter : int, optional
Maximum number of iterations.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
maxls : int, optional
Maximum number of line search steps (per iteration). Default is 20.
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,
'maxls': maxls}
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, maxls=20, **unknown_options):
"""
Minimize a scalar function of one or more variables using the L-BFGS-B
algorithm.
Options
-------
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.
ftol : float
The iteration stops when ``(f^k -
f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= ftol``.
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.
maxls : int, optional
Maximum number of line search steps (per iteration). Default is 20.
"""
_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')
# unbounded variables must use None, not +-inf, for optimizer to work properly
bounds = [(None if l == -np.inf else l, None if u == np.inf else u) for l, u in 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_helper(x, fun, epsilon, args=args, f0=f)
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]
if not maxls > 0:
raise ValueError('maxls must be positive.')
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, maxls)
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
# These two portions of the workspace are described in the mainlb
# subroutine in lbfgsb.f. See line 363.
s = wa[0: m*n].reshape(m, n)
y = wa[m*n: 2*m*n].reshape(m, n)
# See lbfgsb.f line 160 for this portion of the workspace.
# isave(31) = the total number of BFGS updates prior the current iteration;
n_bfgs_updates = isave[30]
n_corrs = min(n_bfgs_updates, maxcor)
hess_inv = LbfgsInvHessProduct(s[:n_corrs], y[:n_corrs])
return OptimizeResult(fun=f, jac=g, nfev=n_function_evals[0],
nit=n_iterations, status=warnflag, message=task_str,
x=x, success=(warnflag == 0), hess_inv=hess_inv)
class LbfgsInvHessProduct(LinearOperator):
"""Linear operator for the L-BFGS approximate inverse Hessian.
This operator computes the product of a vector with the approximate inverse
of the Hessian of the objective function, using the L-BFGS limited
memory approximation to the inverse Hessian, accumulated during the
optimization.
Objects of this class implement the ``scipy.sparse.linalg.LinearOperator``
interface.
Parameters
----------
sk : array_like, shape=(n_corr, n)
Array of `n_corr` most recent updates to the solution vector.
(See [1]).
yk : array_like, shape=(n_corr, n)
Array of `n_corr` most recent updates to the gradient. (See [1]).
References
----------
.. [1] Nocedal, Jorge. "Updating quasi-Newton matrices with limited
storage." Mathematics of computation 35.151 (1980): 773-782.
"""
def __init__(self, sk, yk):
"""Construct the operator."""
if sk.shape != yk.shape or sk.ndim != 2:
raise ValueError('sk and yk must have matching shape, (n_corrs, n)')
n_corrs, n = sk.shape
super(LbfgsInvHessProduct, self).__init__(
dtype=np.float64, shape=(n, n))
self.sk = sk
self.yk = yk
self.n_corrs = n_corrs
self.rho = 1 / np.einsum('ij,ij->i', sk, yk)
def _matvec(self, x):
"""Efficient matrix-vector multiply with the BFGS matrices.
This calculation is described in Section (4) of [1].
Parameters
----------
x : ndarray
An array with shape (n,) or (n,1).
Returns
-------
y : ndarray
The matrix-vector product
"""
s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
q = np.array(x, dtype=self.dtype, copy=True)
if q.ndim == 2 and q.shape[1] == 1:
q = q.reshape(-1)
alpha = np.zeros(n_corrs)
for i in range(n_corrs-1, -1, -1):
alpha[i] = rho[i] * np.dot(s[i], q)
q = q - alpha[i]*y[i]
r = q
for i in range(n_corrs):
beta = rho[i] * np.dot(y[i], r)
r = r + s[i] * (alpha[i] - beta)
return r
def todense(self):
"""Return a dense array representation of this operator.
Returns
-------
arr : ndarray, shape=(n, n)
An array with the same shape and containing
the same data represented by this `LinearOperator`.
"""
s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
I = np.eye(*self.shape, dtype=self.dtype)
Hk = I
for i in range(n_corrs):
A1 = I - s[i][:, np.newaxis] * y[i][np.newaxis, :] * rho[i]
A2 = I - y[i][:, np.newaxis] * s[i][np.newaxis, :] * rho[i]
Hk = np.dot(A1, np.dot(Hk, A2)) + (rho[i] * s[i][:, np.newaxis] *
s[i][np.newaxis, :])
return Hk
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)
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