find_minimum#
- scipy.optimize.elementwise.find_minimum(f, init, /, *, args=(), tolerances=None, maxiter=100, callback=None)[source]#
Find the minimum of an unimodal, real-valued function of a real variable.
For each element of the output of f,
find_minimum
seeks the scalar minimizer that minimizes the element. This function currently uses Chandrupatla’s bracketing minimization algorithm [1] and therefore requires argument init to provide a three-point minimization bracket:x1 < x2 < x3
such thatfunc(x1) >= func(x2) <= func(x3)
, where one of the inequalities is strict.Provided a valid bracket,
find_minimum
is guaranteed to converge to a local minimum that satisfies the provided tolerances if the function is continuous within the bracket.This function works elementwise when init and args contain (broadcastable) arrays.
- Parameters:
- fcallable
The function whose minimizer is desired. The signature must be:
f(x: array, *args) -> array
where each element of
x
is a finite real andargs
is a tuple, which may contain an arbitrary number of arrays that are broadcastable withx
.f must be an elementwise function: each element
f(x)[i]
must equalf(x[i])
for all indicesi
. It must not mutate the arrayx
or the arrays inargs
.find_minimum
seeks an arrayx
such thatf(x)
is an array of local minima.- init3-tuple of float array_like
The abscissae of a standard scalar minimization bracket. A bracket is valid if arrays
x1, x2, x3 = init
satisfyx1 < x2 < x3
andfunc(x1) >= func(x2) <= func(x3)
, where one of the inequalities is strict. Arrays must be broadcastable with one another and the arrays of args.- argstuple of array_like, optional
Additional positional array arguments to be passed to f. Arrays must be broadcastable with one another and the arrays of init. If the callable for which the root is desired requires arguments that are not broadcastable with x, wrap that callable with f such that f accepts only x and broadcastable
*args
.- tolerancesdictionary of floats, optional
Absolute and relative tolerances on the root and function value. Valid keys of the dictionary are:
xatol
- absolute tolerance on the rootxrtol
- relative tolerance on the rootfatol
- absolute tolerance on the function valuefrtol
- relative tolerance on the function value
See Notes for default values and explicit termination conditions.
- maxiterint, default: 100
The maximum number of iterations of the algorithm to perform.
- callbackcallable, optional
An optional user-supplied function to be called before the first iteration and after each iteration. Called as
callback(res)
, whereres
is a_RichResult
similar to that returned byfind_minimum
(but containing the current iterate’s values of all variables). If callback raises aStopIteration
, the algorithm will terminate immediately andfind_root
will return a result. callback must not mutate res or its attributes.
- Returns:
- res_RichResult
An object similar to an instance of
scipy.optimize.OptimizeResult
with the following attributes. The descriptions are written as though the values will be scalars; however, if f returns an array, the outputs will be arrays of the same shape.- successbool array
True
where the algorithm terminated successfully (status0
);False
otherwise.- statusint array
An integer representing the exit status of the algorithm.
0
: The algorithm converged to the specified tolerances.-1
: The algorithm encountered an invalid bracket.-2
: The maximum number of iterations was reached.-3
: A non-finite value was encountered.-4
: Iteration was terminated by callback.1
: The algorithm is proceeding normally (in callback only).
- xfloat array
The minimizer of the function, if the algorithm terminated successfully.
- f_xfloat array
The value of f evaluated at x.
- nfevint array
The number of abscissae at which f was evaluated to find the root. This is distinct from the number of times f is called because the the function may evaluated at multiple points in a single call.
- nitint array
The number of iterations of the algorithm that were performed.
- brackettuple of float arrays
The final three-point bracket.
- f_brackettuple of float arrays
The value of f evaluated at the bracket points.
See also
Notes
Implemented based on Chandrupatla’s original paper [1].
If
xl < xm < xr
are the points of the bracket andfl >= fm <= fr
(where one of the inequalities is strict) are the values of f evaluated at those points, then the algorithm is considered to have converged when:xr - xl <= abs(xm)*xrtol + xatol
or(fl - 2*fm + fr)/2 <= abs(fm)*frtol + fatol
.
Note that first of these differs from the termination conditions described in [1].
The default value of xrtol is the square root of the precision of the appropriate dtype, and
xatol = fatol = frtol
is the smallest normal number of the appropriate dtype.References
[1] (1,2,3)Chandrupatla, Tirupathi R. (1998). “An efficient quadratic fit-sectioning algorithm for minimization without derivatives”. Computer Methods in Applied Mechanics and Engineering, 152 (1-2), 211-217. https://doi.org/10.1016/S0045-7825(97)00190-4
Examples
Suppose we wish to minimize the following function.
>>> def f(x, c=1): ... return (x - c)**2 + 2
First, we must find a valid bracket. The function is unimodal, so bracket_minium will easily find a bracket.
>>> from scipy.optimize import elementwise >>> res_bracket = elementwise.bracket_minimum(f, 0) >>> res_bracket.success True >>> res_bracket.bracket (0.0, 0.5, 1.5)
Indeed, the bracket points are ordered and the function value at the middle bracket point is less than at the surrounding points.
>>> xl, xm, xr = res_bracket.bracket >>> fl, fm, fr = res_bracket.f_bracket >>> (xl < xm < xr) and (fl > fm <= fr) True
Once we have a valid bracket,
find_minimum
can be used to provide an estimate of the minimizer.>>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket) >>> res_minimum.x 1.0000000149011612
The function value changes by only a few ULPs within the bracket, so the minimizer cannot be determined much more precisely by evaluating the function alone (i.e. we would need its derivative to do better).
>>> import numpy as np >>> fl, fm, fr = res_minimum.f_bracket >>> (fl - fm) / np.spacing(fm), (fr - fm) / np.spacing(fm) (0.0, 2.0)
Therefore, a precise minimum of the function is given by:
>>> res_minimum.f_x 2.0
bracket_minimum
andfind_minimum
accept arrays for most arguments. For instance, to find the minimizers and minima for a few values of the parameterc
at once:>>> c = np.asarray([1, 1.5, 2]) >>> res_bracket = elementwise.bracket_minimum(f, 0, args=(c,)) >>> res_bracket.bracket (array([0. , 0.5, 0.5]), array([0.5, 1.5, 1.5]), array([1.5, 2.5, 2.5])) >>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket, args=(c,)) >>> res_minimum.x array([1.00000001, 1.5 , 2. ]) >>> res_minimum.f_x array([2., 2., 2.])