# 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.

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.

`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 .

Conjugate gradient methods tend to work better when:

1. f has a unique global minimizing point, and no local minima or other stationary points,

2. f is, at least locally, reasonably well approximated by a quadratic function of the variables,

3. f is continuous and has a continuous gradient,

4. fprime is not too large, e.g., has a norm less than 1000,

5. 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
...     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
>>> 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