brute(func, ranges, args=(), Ns=20, full_output=0, finish=<function fmin>, disp=False)¶
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
vectorizeNumPy function. The value and type of the function evaluation returned when
full_output=Trueare affected in addition by the
finishargument (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.
func : callable
The objective function to be minimized. Must be in the form
f(x, *args), where
xis the argument in the form of a 1-D array and
argsis 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.
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.)
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
brutefunction’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.
bruteassumes that the finish function returns either an
OptimizeResultobject or a tuple in the form:
(xmin, Jmin, ... , statuscode), where
xminis the minimizing value of the argument,
Jminis the minimum value of the objective function, ”...” may be some other returned values (which are not used by
statuscodeis 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
bruteconfines 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
brutethe 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,
bruteuses it directly. If the component is a two-tuple range,
bruteinternally converts it to a slice object that interpolates Ns points from its low-value to its high-value, inclusive.
We illustrate the use of
bruteto 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
paramsand the functions are as defined below.
>>> 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
fminto 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 # global minimum array([-1.05665192, 1.80834843]) >>> resbrute # 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.