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Module « scipy.optimize »

Fonction fmin_bfgs - module scipy.optimize

Signature de la fonction fmin_bfgs

def fmin_bfgs(f, x0, fprime=None, args=(), gtol=1e-05, norm=inf, epsilon=np.float64(1.4901161193847656e-08), maxiter=None, full_output=0, disp=1, retall=0, callback=None, xrtol=0, c1=0.0001, c2=0.9, hess_inv0=None) 

Description

help(scipy.optimize.fmin_bfgs)

Minimize a function using the BFGS algorithm.

Parameters
----------
f : callable ``f(x,*args)``
    Objective function to be minimized.
x0 : ndarray
    Initial guess, shape (n,)
fprime : callable ``f'(x,*args)``, optional
    Gradient of f.
args : tuple, optional
    Extra arguments passed to f and fprime.
gtol : float, optional
    Terminate successfully if gradient norm is less than `gtol`
norm : float, optional
    Order of norm (Inf is max, -Inf is min)
epsilon : int or ndarray, optional
    If `fprime` is approximated, use this value for the step size.
callback : callable, optional
    An optional user-supplied function to call after each
    iteration. Called as ``callback(xk)``, where ``xk`` is the
    current parameter vector.
maxiter : int, optional
    Maximum number of iterations to perform.
full_output : bool, optional
    If True, return ``fopt``, ``func_calls``, ``grad_calls``, and
    ``warnflag`` in addition to ``xopt``.
disp : bool, optional
    Print convergence message if True.
retall : bool, optional
    Return a list of results at each iteration if True.
xrtol : float, default: 0
    Relative tolerance for `x`. Terminate successfully if step
    size is less than ``xk * xrtol`` where ``xk`` is the current
    parameter vector.
c1 : float, default: 1e-4
    Parameter for Armijo condition rule.
c2 : float, default: 0.9
    Parameter for curvature condition rule.
hess_inv0 : None or ndarray, optional``
    Initial inverse hessian estimate, shape (n, n). If None (default) then
    the identity matrix is used.

Returns
-------
xopt : ndarray
    Parameters which minimize f, i.e., ``f(xopt) == fopt``.
fopt : float
    Minimum value.
gopt : ndarray
    Value of gradient at minimum, f'(xopt), which should be near 0.
Bopt : ndarray
    Value of 1/f''(xopt), i.e., the inverse Hessian matrix.
func_calls : int
    Number of function_calls made.
grad_calls : int
    Number of gradient calls made.
warnflag : integer
    1 : Maximum number of iterations exceeded.
    2 : Gradient and/or function calls not changing.
    3 : NaN result encountered.
allvecs : list
    The value of `xopt` at each iteration. Only returned if `retall` is
    True.

Notes
-----
Optimize the function, `f`, whose gradient is given by `fprime`
using the quasi-Newton method of Broyden, Fletcher, Goldfarb,
and Shanno (BFGS).

Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``.

See Also
--------
minimize: Interface to minimization algorithms for multivariate
    functions. See ``method='BFGS'`` in particular.

References
----------
Wright, and Nocedal 'Numerical Optimization', 1999, p. 198.

Examples
--------
>>> import numpy as np
>>> from scipy.optimize import fmin_bfgs
>>> def quadratic_cost(x, Q):
...     return x @ Q @ x
...
>>> x0 = np.array([-3, -4])
>>> cost_weight =  np.diag([1., 10.])
>>> # Note that a trailing comma is necessary for a tuple with single element
>>> fmin_bfgs(quadratic_cost, x0, args=(cost_weight,))
Optimization terminated successfully.
        Current function value: 0.000000
        Iterations: 7                   # may vary
        Function evaluations: 24        # may vary
        Gradient evaluations: 8         # may vary
array([ 2.85169950e-06, -4.61820139e-07])

>>> def quadratic_cost_grad(x, Q):
...     return 2 * Q @ x
...
>>> fmin_bfgs(quadratic_cost, x0, quadratic_cost_grad, args=(cost_weight,))
Optimization terminated successfully.
        Current function value: 0.000000
        Iterations: 7
        Function evaluations: 8
        Gradient evaluations: 8
array([ 2.85916637e-06, -4.54371951e-07])

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