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.
Notes
-----
License of L-BFGS-B (Fortran code):
The version included here (in fortran code) is 2.1 (released in 1997).
It was written by Ciyou Zhu, Richard Byrd, and Jorge Nocedal
<*****@*****.**>. 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.
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, Vol 23, Num. 4, pp. 550 - 560.
"""
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 approx_grad:
def func_and_grad(x):
f = func(x, *args)
g = approx_fprime(x, func, epsilon, *args)
return f, g
elif fprime is None:
def func_and_grad(x):
f, g = func(x, *args)
return f, g
else:
def func_and_grad(x):
f = func(x, *args)
g = fprime(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+4*n + 12*m**2 + 12*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 x, f, d
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 : tuple
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.
disp : int, optional
If zero, then no output. If positive number, then this over-rides
`iprint`.
maxfun : int
Maximum number of function evaluations.
Returns
-------
x : ndarray
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.
Notes
-----
License of L-BFGS-B (Fortran code):
The version included here (in fortran code) is 2.1 (released in 1997).
It was written by Ciyou Zhu, Richard Byrd, and Jorge Nocedal
<*****@*****.**>. 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.
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, Vol 23, Num. 4, pp. 550 - 560.
"""
n = len(x0)
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 approx_grad:
def func_and_grad(x):
f = func(x, *args)
g = approx_fprime(x, func, epsilon, *args)
return f, g
elif fprime is None:
def func_and_grad(x):
f, g = func(x, *args)
return f, g
else:
def func_and_grad(x):
f = func(x, *args)
g = fprime(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 + 4 * n + 12 * m ** 2 + 12 * 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 x, f, d
def _minimize_lbfgsb(fun, x0, args=(), jac=None, bounds=None, options={},
full_output=False):
"""
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.
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.
eps : float
Step size used for numerical approximation of the jacobian.
disp : int
Set to True to print convergence messages.
maxfev : 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.
"""
# retrieve useful options
disp = options.get('disp', None)
m = options.get('maxcor', 10)
factr = options.get('factr', 1e7)
pgtol = options.get('pgtol', 1e-5)
epsilon = options.get('eps', 1e-8)
maxfun = options.get('maxfev', 15000)
iprint = options.get('iprint', -1)
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
}
if full_output:
info = {'fun': f,
'jac': g,
'nfev': n_function_evals,
'status': warnflag,
'message': task_str,
'solution': x,
'success': warnflag==0}
return x, info
else:
return x
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
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
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(asbytes('FG')):
if n_function_evals > maxfun:
task[:] = 'STOP: TOTAL NO. of f AND g EVALUATIONS EXCEEDS LIMIT'
else:
# 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_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(asbytes('\x00')).strip()
if task_str.startswith(asbytes('CONV')):
warnflag = 0
elif n_function_evals > maxfun:
warnflag = 1
elif n_iterations > maxiter:
warnflag = 1
else:
warnflag = 2
return Result(fun=f,
jac=g,
nfev=n_function_evals,
nit=n_iterations,
status=warnflag,
message=task_str,
x=x,
success=(warnflag == 0))
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):
"""
Minimize a function func using the L-BFGS-B algorithm.
Arguments:
func -- function to minimize. Called as func(x, *args)
x0 -- initial guess to minimum
fprime -- gradient of func. If None, then func returns the function
value and the gradient ( f, g = func(x, *args) ), unless
approx_grad is True then func returns only f.
Called as fprime(x, *args)
args -- arguments to pass to function
approx_grad -- if true, approximate the gradient numerically and func returns
only function value.
bounds -- a list of (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 -- 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 -- The iteration stops when
(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch
where epsmch is the machine precision, which is automatically
generated by the code. Typical values for factr: 1e12 for
low accuracy; 1e7 for moderate accuracy; 10.0 for extremely
high accuracy.
pgtol -- The iteration will stop when
max{|proj g_i | i = 1, ..., n} <= pgtol
where pg_i is the ith component of the projected gradient.
epsilon -- step size used when approx_grad is true, for numerically
calculating the gradient
iprint -- controls the frequency of output. <0 means no output.
maxfun -- maximum number of function evaluations.
Returns:
x, f, d = fmin_lbfgs_b(func, x0, ...)
x -- position of the minimum
f -- value of func at the minimum
d -- dictionary of information from routine
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.
License of L-BFGS-B (Fortran code)
==================================
The version included here (in fortran code) is 2.1 (released in 1997). It was
written by Ciyou Zhu, Richard Byrd, and Jorge Nocedal <*****@*****.**>. 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.
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, Vol 23, Num. 4, pp. 550 - 560.
See also:
scikits.openopt, which offers a unified syntax to call this and other solvers
fmin, fmin_powell, fmin_cg,
fmin_bfgs, fmin_ncg -- multivariate local optimizers
leastsq -- nonlinear least squares minimizer
fmin_l_bfgs_b, fmin_tnc,
fmin_cobyla -- constrained multivariate optimizers
anneal, brute -- global optimizers
fminbound, brent, golden, bracket -- local scalar minimizers
fsolve -- n-dimenstional root-finding
brentq, brenth, ridder, bisect, newton -- one-dimensional root-finding
fixed_point -- scalar fixed-point finder
"""
n = len(x0)
if bounds is None:
bounds = [(None,None)] * n
if len(bounds) != n:
raise ValueError('length of x0 != length of bounds')
if approx_grad:
def func_and_grad(x):
f = func(x, *args)
g = approx_fprime(x, func, epsilon, *args)
return f, g
elif fprime is None:
def func_and_grad(x):
f, g = func(x, *args)
return f, g
else:
def func_and_grad(x):
f = func(x, *args)
g = fprime(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+4*n + 12*m**2 + 12*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('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('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('\x00').strip()
if task_str.startswith('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 x, f, d
def _minimize_lbfgsb(fun,
x0,
args=(),
jac=None,
bounds=None,
options={},
full_output=False):
"""
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.
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.
eps : float
Step size used for numerical approximation of the jacobian.
disp : int
Set to True to print convergence messages.
maxfev : 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.
"""
# retrieve useful options
disp = options.get('disp', None)
m = options.get('maxcor', 10)
factr = options.get('factr', 1e7)
pgtol = options.get('pgtol', 1e-5)
epsilon = options.get('eps', 1e-8)
maxfun = options.get('maxfev', 15000)
iprint = options.get('iprint', -1)
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 + 4 * n + 12 * m**2 + 12 * 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
}
if full_output:
info = {
'fun': f,
'jac': g,
'nfev': n_function_evals,
'status': warnflag,
'message': task_str,
'solution': x,
'success': warnflag == 0
}
return x, info
else:
return x
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
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
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(asbytes("FG")):
if n_function_evals > maxfun:
task[:] = "STOP: TOTAL NO. of f AND g EVALUATIONS EXCEEDS LIMIT"
else:
# 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_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(asbytes("\x00")).strip()
if task_str.startswith(asbytes("CONV")):
warnflag = 0
elif n_function_evals > maxfun:
warnflag = 1
elif n_iterations > maxiter:
warnflag = 1
else:
warnflag = 2
return Result(
fun=f,
jac=g,
nfev=n_function_evals,
nit=n_iterations,
status=warnflag,
message=task_str,
x=x,
success=(warnflag == 0),
)
