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scipy.optimize.minimize_scalar#

scipy.optimize.minimize_scalar(fun, bracket=None, bounds=None, args=(), method=None, tol=None, options=None)[source]#

Local minimization of scalar function of one variable.

Parameters:
funcallable

Objective function. Scalar function, must return a scalar.

bracketsequence, optional

For methods ‘brent’ and ‘golden’, bracket defines the bracketing interval and is required. Either a triple (xa, xb, xc) satisfying xa < xb < xc and func(xb) < func(xa) and  func(xb) < func(xc), or a pair (xa, xb) to be used as initial points for a downhill bracket search (see scipy.optimize.bracket). The minimizer res.x will not necessarily satisfy xa <= res.x <= xb.

boundssequence, optional

For method ‘bounded’, bounds is mandatory and must have two finite items corresponding to the optimization bounds.

argstuple, optional

Extra arguments passed to the objective function.

methodstr or callable, optional

Type of solver. Should be one of:

Default is “Bounded” if bounds are provided and “Brent” otherwise. See the ‘Notes’ section for details of each solver.

tolfloat, optional

Tolerance for termination. For detailed control, use solver-specific options.

optionsdict, optional

A dictionary of solver options.

maxiterint

Maximum number of iterations to perform.

dispbool

Set to True to print convergence messages.

See show_options for solver-specific options.

Returns:
resOptimizeResult

The optimization result represented as a OptimizeResult object. Important attributes are: x the solution array, success a Boolean flag indicating if the optimizer exited successfully and message which describes the cause of the termination. See OptimizeResult for a description of other attributes.

See also

minimize

Interface to minimization algorithms for scalar multivariate functions

show_options

Additional options accepted by the solvers

Notes

This section describes the available solvers that can be selected by the ‘method’ parameter. The default method is the "Bounded" Brent method if bounds are passed and unbounded "Brent" otherwise.

Method Brent uses Brent’s algorithm [1] to find a local minimum. The algorithm uses inverse parabolic interpolation when possible to speed up convergence of the golden section method.

Method Golden uses the golden section search technique [1]. It uses analog of the bisection method to decrease the bracketed interval. It is usually preferable to use the Brent method.

Method Bounded can perform bounded minimization [2] [3]. It uses the Brent method to find a local minimum in the interval x1 < xopt < x2.

Note that the Brent and Golden methods do not guarantee success unless a valid bracket triple is provided. If a three-point bracket cannot be found, consider scipy.optimize.minimize. Also, all methods are intended only for local minimization. When the function of interest has more than one local minimum, consider Global optimization.

Custom minimizers

It may be useful to pass a custom minimization method, for example when using some library frontend to minimize_scalar. You can simply pass a callable as the method parameter.

The callable is called as method(fun, args, **kwargs, **options) where kwargs corresponds to any other parameters passed to minimize (such as bracket, tol, etc.), except the options dict, which has its contents also passed as method parameters pair by pair. The method shall return an OptimizeResult object.

The provided method callable must be able to accept (and possibly ignore) arbitrary parameters; the set of parameters accepted by minimize may expand in future versions and then these parameters will be passed to the method. You can find an example in the scipy.optimize tutorial.

Added in version 0.11.0.

References

[1] (1,2)

Press, W., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes in C. Cambridge University Press.

[2]

Forsythe, G.E., M. A. Malcolm, and C. B. Moler. “Computer Methods for Mathematical Computations.” Prentice-Hall Series in Automatic Computation 259 (1977).

[3]

Brent, Richard P. Algorithms for Minimization Without Derivatives. Courier Corporation, 2013.

Examples

Consider the problem of minimizing the following function.

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

Using the Brent method, we find the local minimum as:

>>> from scipy.optimize import minimize_scalar
>>> res = minimize_scalar(f)
>>> res.fun
-9.9149495908

The minimizer is:

>>> res.x
1.28077640403

Using the Bounded method, we find a local minimum with specified bounds as:

>>> res = minimize_scalar(f, bounds=(-3, -1), method='bounded')
>>> res.fun  # minimum
3.28365179850e-13
>>> res.x  # minimizer
-2.0000002026
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