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 : 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.
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.
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 [R92].
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
[R92] (1, 2) 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) >>> print 'res1 = ', res1 Optimization terminated successfully. Current function value: 1.617021 Iterations: 2 Function evaluations: 5 Gradient evaluations: 5 res1 = [-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: 2 Function evaluations: 5 Gradient evaluations: 5 >>> res2.x # minimum found array([-1.80851064 -0.25531915])