scipy.optimize.fmin

scipy.optimize.fmin(func, x0, args=(), xtol=0.0001, ftol=0.0001, maxiter=None, maxfun=None, full_output=0, disp=1, retall=0, callback=None, initial_simplex=None)

Minimize a function using the downhill simplex algorithm.

This algorithm only uses function values, not derivatives or second derivatives.

Parameters:

func : callable func(x,*args)

The objective function to be minimized.

x0 : ndarray

Initial guess.

args : tuple, optional

Extra arguments passed to func, i.e. f(x,*args).

xtol : float, optional

Absolute error in xopt between iterations that is acceptable for convergence.

ftol : number, optional

Absolute error in func(xopt) between iterations that is acceptable for convergence.

maxiter : int, optional

Maximum number of iterations to perform.

maxfun : number, optional

Maximum number of function evaluations to make.

full_output : bool, optional

Set to True if fopt and warnflag outputs are desired.

disp : bool, optional

Set to True to print convergence messages.

retall : bool, optional

Set to True to return list of solutions at each iteration.

callback : callable, optional

Called after each iteration, as callback(xk), where xk is the current parameter vector.

initial_simplex : array_like of shape (N + 1, N), optional

Initial simplex. If given, overrides x0. initial_simplex[j,:] should contain the coordinates of the j-th vertex of the N+1 vertices in the simplex, where N is the dimension.

Returns:

xopt : ndarray

Parameter that minimizes function.

fopt : float

Value of function at minimum: fopt = func(xopt).

iter : int

Number of iterations performed.

funcalls : int

Number of function calls made.

warnflag : int

1 : Maximum number of function evaluations made. 2 : Maximum number of iterations reached.

allvecs : list

Solution at each iteration.

See also

minimize

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

Notes

Uses a Nelder-Mead simplex algorithm to find the minimum of function of one or more variables.

This algorithm has a long history of successful use in applications. But it will usually be slower than an algorithm that uses first or second derivative information. In practice it can have poor performance in high-dimensional problems and is not robust to minimizing complicated functions. Additionally, there currently is no complete theory describing when the algorithm will successfully converge to the minimum, or how fast it will if it does. Both the ftol and xtol criteria must be met for convergence.

References

[R198]

Nelder, J.A. and Mead, R. (1965), “A simplex method for function minimization”, The Computer Journal, 7, pp. 308-313

[R199]

Wright, M.H. (1996), “Direct Search Methods: Once Scorned, Now Respectable”, in Numerical Analysis 1995, Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis, D.F. Griffiths and G.A. Watson (Eds.), Addison Wesley Longman, Harlow, UK, pp. 191-208.

Examples

>>> def f(x):
...     return x**2

>>> from scipy import optimize

>>> minimum = optimize.fmin(f, 1)
Optimization terminated successfully.
         Current function value: 0.000000
         Iterations: 17
         Function evaluations: 34
>>> minimum[0]
-8.8817841970012523e-16