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def fmin_cg(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, c1=0.0001, c2=0.4)
Minimize a function using a nonlinear conjugate gradient algorithm.
Parameters
----------
f : callable, ``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`.
x0 : ndarray
A user-supplied initial estimate of `xopt`, the optimal value of `x`.
It must be a 1-D array of values.
fprime : callable, ``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).
args : tuple, 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`.
gtol : float, optional
Stop when the norm of the gradient is less than `gtol`.
norm : float, optional
Order to use for the norm of the gradient
(``-np.inf`` is min, ``np.inf`` is max).
epsilon : float 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.
maxiter : int, optional
Maximum number of iterations to perform. Default is ``200 * len(x0)``.
full_output : bool, 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.
disp : bool, optional
If True, return a convergence message, followed by `xopt`.
retall : bool, optional
If True, add to the returned values the results of each iteration.
callback : callable, optional
An optional user-supplied function, called after each iteration.
Called as ``callback(xk)``, where ``xk`` is the current value of `x0`.
c1 : float, default: 1e-4
Parameter for Armijo condition rule.
c2 : float, default: 0.4
Parameter for curvature condition rule.
Returns
-------
xopt : ndarray
Parameters which minimize f, i.e., ``f(xopt) == fopt``.
fopt : float, optional
Minimum value found, f(xopt). Only returned if `full_output` is True.
func_calls : int, optional
The number of function_calls made. Only returned if `full_output`
is True.
grad_calls : int, optional
The number of gradient calls made. Only returned if `full_output` is
True.
warnflag : int, optional
Integer value with warning status, only returned if `full_output` is
True.
0 : Success.
1 : The maximum number of iterations was exceeded.
2 : Gradient and/or function calls were not changing. May indicate
that precision was lost, i.e., the routine did not converge.
3 : NaN result encountered.
allvecs : list 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:
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`.
Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``.
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)``.
>>> import numpy as np
>>> 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])
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