scipy.optimize.fmin_bfgs(f, x0, fprime=None, args=(), gtol=1e-05, norm=inf, epsilon=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)[source]#

Minimize a function using the BFGS algorithm.

fcallable f(x,*args)

Objective function to be minimized.


Initial guess, shape (n,)

fprimecallable f'(x,*args), optional

Gradient of f.

argstuple, optional

Extra arguments passed to f and fprime.

gtolfloat, optional

Terminate successfully if gradient norm is less than gtol

normfloat, optional

Order of norm (Inf is max, -Inf is min)

epsilonint or ndarray, optional

If fprime is approximated, use this value for the step size.

callbackcallable, optional

An optional user-supplied function to call after each iteration. Called as callback(xk), where xk is the current parameter vector.

maxiterint, optional

Maximum number of iterations to perform.

full_outputbool, optional

If True, return fopt, func_calls, grad_calls, and warnflag in addition to xopt.

dispbool, optional

Print convergence message if True.

retallbool, optional

Return a list of results at each iteration if True.

xrtolfloat, default: 0

Relative tolerance for x. Terminate successfully if step size is less than xk * xrtol where xk is the current parameter vector.

c1float, default: 1e-4

Parameter for Armijo condition rule.

c2float, default: 0.9

Parameter for curvature condition rule.

hess_inv0None or ndarray, optional``

Initial inverse hessian estimate, shape (n, n). If None (default) then the identity matrix is used.


Parameters which minimize f, i.e., f(xopt) == fopt.


Minimum value.


Value of gradient at minimum, f’(xopt), which should be near 0.


Value of 1/f’’(xopt), i.e., the inverse Hessian matrix.


Number of function_calls made.


Number of gradient calls made.


1 : Maximum number of iterations exceeded. 2 : Gradient and/or function calls not changing. 3 : NaN result encountered.


The value of xopt at each iteration. Only returned if retall is True.

See also


Interface to minimization algorithms for multivariate functions. See method='BFGS' in particular.


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.


Wright, and Nocedal ‘Numerical Optimization’, 1999, p. 198.


>>> 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])