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def brute(func, ranges, args=(), Ns=20, full_output=0, finish=<function fmin at 0x0000020DFE4D0040>, disp=False, workers=1)
Minimize a function over a given range by brute force.
Uses the "brute force" method, i.e., computes the function's value
at each point of a multidimensional grid of points, to find the global
minimum of the function.
The function is evaluated everywhere in the range with the datatype of the
first call to the function, as enforced by the ``vectorize`` NumPy
function. The value and type of the function evaluation returned when
``full_output=True`` are affected in addition by the ``finish`` argument
(see Notes).
The brute force approach is inefficient because the number of grid points
increases exponentially - the number of grid points to evaluate is
``Ns ** len(x)``. Consequently, even with coarse grid spacing, even
moderately sized problems can take a long time to run, and/or run into
memory limitations.
Parameters
----------
func : callable
The objective function to be minimized. Must be in the
form ``f(x, *args)``, where ``x`` is the argument in
the form of a 1-D array and ``args`` is a tuple of any
additional fixed parameters needed to completely specify
the function.
ranges : tuple
Each component of the `ranges` tuple must be either a
"slice object" or a range tuple of the form ``(low, high)``.
The program uses these to create the grid of points on which
the objective function will be computed. See `Note 2` for
more detail.
args : tuple, optional
Any additional fixed parameters needed to completely specify
the function.
Ns : int, optional
Number of grid points along the axes, if not otherwise
specified. See `Note2`.
full_output : bool, optional
If True, return the evaluation grid and the objective function's
values on it.
finish : callable, optional
An optimization function that is called with the result of brute force
minimization as initial guess. `finish` should take `func` and
the initial guess as positional arguments, and take `args` as
keyword arguments. It may additionally take `full_output`
and/or `disp` as keyword arguments. Use None if no "polishing"
function is to be used. See Notes for more details.
disp : bool, optional
Set to True to print convergence messages from the `finish` callable.
workers : int or map-like callable, optional
If `workers` is an int the grid is subdivided into `workers`
sections and evaluated in parallel (uses
`multiprocessing.Pool <multiprocessing>`).
Supply `-1` to use all cores available to the Process.
Alternatively supply a map-like callable, such as
`multiprocessing.Pool.map` for evaluating the grid in parallel.
This evaluation is carried out as ``workers(func, iterable)``.
Requires that `func` be pickleable.
.. versionadded:: 1.3.0
Returns
-------
x0 : ndarray
A 1-D array containing the coordinates of a point at which the
objective function had its minimum value. (See `Note 1` for
which point is returned.)
fval : float
Function value at the point `x0`. (Returned when `full_output` is
True.)
grid : tuple
Representation of the evaluation grid. It has the same
length as `x0`. (Returned when `full_output` is True.)
Jout : ndarray
Function values at each point of the evaluation
grid, i.e., ``Jout = func(*grid)``. (Returned
when `full_output` is True.)
See Also
--------
basinhopping, differential_evolution
Notes
-----
*Note 1*: The program finds the gridpoint at which the lowest value
of the objective function occurs. If `finish` is None, that is the
point returned. When the global minimum occurs within (or not very far
outside) the grid's boundaries, and the grid is fine enough, that
point will be in the neighborhood of the global minimum.
However, users often employ some other optimization program to
"polish" the gridpoint values, i.e., to seek a more precise
(local) minimum near `brute's` best gridpoint.
The `brute` function's `finish` option provides a convenient way to do
that. Any polishing program used must take `brute's` output as its
initial guess as a positional argument, and take `brute's` input values
for `args` as keyword arguments, otherwise an error will be raised.
It may additionally take `full_output` and/or `disp` as keyword arguments.
`brute` assumes that the `finish` function returns either an
`OptimizeResult` object or a tuple in the form:
``(xmin, Jmin, ... , statuscode)``, where ``xmin`` is the minimizing
value of the argument, ``Jmin`` is the minimum value of the objective
function, "..." may be some other returned values (which are not used
by `brute`), and ``statuscode`` is the status code of the `finish` program.
Note that when `finish` is not None, the values returned are those
of the `finish` program, *not* the gridpoint ones. Consequently,
while `brute` confines its search to the input grid points,
the `finish` program's results usually will not coincide with any
gridpoint, and may fall outside the grid's boundary. Thus, if a
minimum only needs to be found over the provided grid points, make
sure to pass in ``finish=None``.
*Note 2*: The grid of points is a `numpy.mgrid` object.
For `brute` the `ranges` and `Ns` inputs have the following effect.
Each component of the `ranges` tuple can be either a slice object or a
two-tuple giving a range of values, such as (0, 5). If the component is a
slice object, `brute` uses it directly. If the component is a two-tuple
range, `brute` internally converts it to a slice object that interpolates
`Ns` points from its low-value to its high-value, inclusive.
Examples
--------
We illustrate the use of `brute` to seek the global minimum of a function
of two variables that is given as the sum of a positive-definite
quadratic and two deep "Gaussian-shaped" craters. Specifically, define
the objective function `f` as the sum of three other functions,
``f = f1 + f2 + f3``. We suppose each of these has a signature
``(z, *params)``, where ``z = (x, y)``, and ``params`` and the functions
are as defined below.
>>> import numpy as np
>>> params = (2, 3, 7, 8, 9, 10, 44, -1, 2, 26, 1, -2, 0.5)
>>> def f1(z, *params):
... x, y = z
... a, b, c, d, e, f, g, h, i, j, k, l, scale = params
... return (a * x**2 + b * x * y + c * y**2 + d*x + e*y + f)
>>> def f2(z, *params):
... x, y = z
... a, b, c, d, e, f, g, h, i, j, k, l, scale = params
... return (-g*np.exp(-((x-h)**2 + (y-i)**2) / scale))
>>> def f3(z, *params):
... x, y = z
... a, b, c, d, e, f, g, h, i, j, k, l, scale = params
... return (-j*np.exp(-((x-k)**2 + (y-l)**2) / scale))
>>> def f(z, *params):
... return f1(z, *params) + f2(z, *params) + f3(z, *params)
Thus, the objective function may have local minima near the minimum
of each of the three functions of which it is composed. To
use `fmin` to polish its gridpoint result, we may then continue as
follows:
>>> rranges = (slice(-4, 4, 0.25), slice(-4, 4, 0.25))
>>> from scipy import optimize
>>> resbrute = optimize.brute(f, rranges, args=params, full_output=True,
... finish=optimize.fmin)
>>> resbrute[0] # global minimum
array([-1.05665192, 1.80834843])
>>> resbrute[1] # function value at global minimum
-3.4085818767
Note that if `finish` had been set to None, we would have gotten the
gridpoint [-1.0 1.75] where the rounded function value is -2.892.
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