scipy.optimize.fmin_cg#

scipy.optimize.fmin_cg(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)[source]#

Minimize a function using a nonlinear conjugate gradient algorithm.

Parameters

fcallable, f(x, *args): Objective function to be minimized. Here x must be a 1-D array of the variables that are to be changed in the search for a minimum, and args are the other (fixed) parameters of f.
x0ndarray: A user-supplied initial estimate of xopt, the optimal value of x. It must be a 1-D array of values.
fprimecallable, fprime(x, *args), optional: A function that returns the gradient of f at x. Here x and args are as described above for f. The returned value must be a 1-D array. Defaults to None, in which case the gradient is approximated numerically (see epsilon, below).
argstuple, optional: Parameter values passed to f and fprime. Must be supplied whenever additional fixed parameters are needed to completely specify the functions f and fprime.
gtolfloat, optional: Stop when the norm of the gradient is less than gtol.
normfloat, optional: Order to use for the norm of the gradient (-np.Inf is min, np.Inf is max).
epsilonfloat or ndarray, optional: Step size(s) to use when fprime is approximated numerically. Can be a scalar or a 1-D array. Defaults to sqrt(eps), with eps the floating point machine precision. Usually sqrt(eps) is about 1.5e-8.
maxiterint, optional: Maximum number of iterations to perform. Default is 200 * len(x0).
full_outputbool, optional: If True, return fopt, func_calls, grad_calls, and warnflag in addition to xopt. See the Returns section below for additional information on optional return values.
dispbool, optional: If True, return a convergence message, followed by xopt.
retallbool, optional: If True, add to the returned values the results of each iteration.
callbackcallable, optional: An optional user-supplied function, called after each iteration. Called as callback(xk), where xk is the current value of x0.

Returns

xoptndarray

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

foptfloat, optional

Minimum value found, f(xopt). Only returned if full_output is True.

func_callsint, optional

The number of function_calls made. Only returned if full_output is True.

grad_callsint, optional

The number of gradient calls made. Only returned if full_output is True.

warnflagint, optional

Integer value with warning status, only returned if full_output is True.

0 : Success.

1 : The maximum number of iterations was exceeded.

2Gradient and/or function calls were not changing. May indicate: that precision was lost, i.e., the routine did not converge.

3 : NaN result encountered.

allvecslist of ndarray, optional

List of arrays, containing the results at each iteration. Only returned if retall is True.

See also

minimize: common interface to all scipy.optimize algorithms for unconstrained and constrained minimization of multivariate functions. It provides an alternative way to call fmin_cg, by specifying method='CG'.

Notes

This conjugate gradient algorithm is based on that of Polak and Ribiere [1].

Conjugate gradient methods tend to work better when:

f has a unique global minimizing point, and no local minima or other stationary points,
f is, at least locally, reasonably well approximated by a quadratic function of the variables,
f is continuous and has a continuous gradient,
fprime is not too large, e.g., has a norm less than 1000,
The initial guess, x0, is reasonably close to f ‘s global minimizing point, xopt.

References

1: Wright & Nocedal, “Numerical Optimization”, 1999, pp. 120-122.

Examples

Example 1: seek the minimum value of the expression a*u**2 + b*u*v + c*v**2 + d*u + e*v + f for given values of the parameters and an initial guess (u, v) = (0, 0).

>>> args = (2, 3, 7, 8, 9, 10)  # parameter values
>>> def f(x, *args):
...     u, v = x
...     a, b, c, d, e, f = args
...     return a*u**2 + b*u*v + c*v**2 + d*u + e*v + f
>>> def gradf(x, *args):
...     u, v = x
...     a, b, c, d, e, f = args
...     gu = 2*a*u + b*v + d     # u-component of the gradient
...     gv = b*u + 2*c*v + e     # v-component of the gradient
...     return np.asarray((gu, gv))
>>> x0 = np.asarray((0, 0))  # Initial guess.
>>> from scipy import optimize
>>> res1 = optimize.fmin_cg(f, x0, fprime=gradf, args=args)
Optimization terminated successfully.
         Current function value: 1.617021
         Iterations: 4
         Function evaluations: 8
         Gradient evaluations: 8
>>> res1
array([-1.80851064, -0.25531915])

Example 2: solve the same problem using the minimize function. (This myopts dictionary shows all of the available options, although in practice only non-default values would be needed. The returned value will be a dictionary.)

>>> opts = {'maxiter' : None,    # default value.
...         'disp' : True,    # non-default value.
...         'gtol' : 1e-5,    # default value.
...         'norm' : np.inf,  # default value.
...         'eps' : 1.4901161193847656e-08}  # default value.
>>> res2 = optimize.minimize(f, x0, jac=gradf, args=args,
...                          method='CG', options=opts)
Optimization terminated successfully.
        Current function value: 1.617021
        Iterations: 4
        Function evaluations: 8
        Gradient evaluations: 8
>>> res2.x  # minimum found
array([-1.80851064, -0.25531915])