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executable file 382 lines (331 sloc) 12.742 kB
"""
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
import numpy as np
from numpy import array, asarray, float64, int32, zeros
import _lbfgsb
from optimize import approx_fprime, MemoizeJac, Result, _check_unknown_options
from numpy.compat import asbytes
__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, disp=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.
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,
- 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.
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,
'maxiter': maxfun}
res = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=bounds,
**opts)
d = {'grad': res['jac'],
'task': res['message'],
'funcalls': res['nfev'],
'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, maxiter=15000, iprint=-1,
**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.
maxiter : int
Maximum number of function evaluations.
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
maxfun = maxiter
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
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_function_evals = 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(asbytes('FG')):
# minimization routine wants f and g at the current x
n_function_evals += 1
# Overwrite f and g:
f, g = func_and_grad(x)
elif task_str.startswith(asbytes('NEW_X')):
# new iteration
if n_function_evals > maxfun:
task[:] = 'STOP: TOTAL NO. of f AND g EVALUATIONS EXCEEDS LIMIT'
else:
break
task_str = task.tostring().strip(asbytes('\x00')).strip()
if task_str.startswith(asbytes('CONV')):
warnflag = 0
elif n_function_evals > maxfun:
warnflag = 1
else:
warnflag = 2
d = {'grad' : g,
'task' : task_str,
'funcalls' : n_function_evals,
'warnflag' : warnflag
}
return Result(fun=f, jac=g, nfev=n_function_evals, 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
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