scipy.optimize.fmin_ncg

scipy.optimize.fmin_ncg(f, x0, fprime, fhess_p=None, fhess=None, args=(), avextol=1e-05, epsilon=1.4901161193847656e-08, maxiter=None, full_output=0, disp=1, retall=0, callback=None, c1=0.0001, c2=0.9)

Unconstrained minimization of a function using the Newton-CG method.

Parameters:

fcallable f(x, *args)

Objective function to be minimized.

x0ndarray

Initial guess.

fprimecallable f'(x, *args)

Gradient of f.

fhess_pcallable fhess_p(x, p, *args), optional

Function which computes the Hessian of f times an arbitrary vector, p.

fhesscallable fhess(x, *args), optional

Function to compute the Hessian matrix of f.

argstuple, optional

Extra arguments passed to f, fprime, fhess_p, and fhess (the same set of extra arguments is supplied to all of these functions).

epsilonfloat or ndarray, optional

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

callbackcallable, optional

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

avextolfloat, optional

Convergence is assumed when the average relative error in the minimizer falls below this amount.

maxiterint, optional

Maximum number of iterations to perform.

full_outputbool, optional

If True, return the optional outputs.

dispbool, optional

If True, print convergence message.

retallbool, optional

If True, return a list of results at each iteration.

c1float, default: 1e-4

Parameter for Armijo condition rule.

c2float, default: 0.9

Parameter for curvature condition rule

Returns:

xoptndarray

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

foptfloat

Value of the function at xopt, i.e., fopt = f(xopt).

fcallsint

Number of function calls made.

gcallsint

Number of gradient calls made.

hcallsint

Number of Hessian calls made.

warnflagint

Warnings generated by the algorithm. 1 : Maximum number of iterations exceeded. 2 : Line search failure (precision loss). 3 : NaN result encountered.

allvecslist

The result at each iteration, if retall is True (see below).

See also

minimize

Interface to minimization algorithms for multivariate functions. See the ‘Newton-CG’ method in particular.

Notes

Only one of fhess_p or fhess need to be given. If fhess is provided, then fhess_p will be ignored. If neither fhess nor fhess_p is provided, then the hessian product will be approximated using finite differences on fprime. fhess_p must compute the hessian times an arbitrary vector. If it is not given, finite-differences on fprime are used to compute it.

Newton-CG methods are also called truncated Newton methods. This function differs from scipy.optimize.fmin_tnc because

1. scipy.optimize.fmin_ncg is written purely in Python using NumPy

   and scipy while scipy.optimize.fmin_tnc calls a C function.

2. scipy.optimize.fmin_ncg is only for unconstrained minimization

   while scipy.optimize.fmin_tnc is for unconstrained minimization or box constrained minimization. (Box constraints give lower and upper bounds for each variable separately.)

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

References

Wright & Nocedal, ‘Numerical Optimization’, 1999, p. 140.