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Finds the global minimum of a function using the
DIRECT algorithm.
Parameters
----------
func : callable
The objective function to be minimized.
``func(x, *args) -> float``
where ``x`` is an 1-D array with shape (n,) and ``args`` is a tuple of
the fixed parameters needed to completely specify the function.
bounds : sequence or `Bounds`
Bounds for variables. There are two ways to specify the bounds:
1. Instance of `Bounds` class.
2. ``(min, max)`` pairs for each element in ``x``.
args : tuple, optional
Any additional fixed parameters needed to
completely specify the objective function.
eps : float, optional
Minimal required difference of the objective function values
between the current best hyperrectangle and the next potentially
optimal hyperrectangle to be divided. In consequence, `eps` serves as a
tradeoff between local and global search: the smaller, the more local
the search becomes. Default is 1e-4.
maxfun : int or None, optional
Approximate upper bound on objective function evaluations.
If `None`, will be automatically set to ``1000 * N`` where ``N``
represents the number of dimensions. Will be capped if necessary to
limit DIRECT's RAM usage to app. 1GiB. This will only occur for very
high dimensional problems and excessive `max_fun`. Default is `None`.
maxiter : int, optional
Maximum number of iterations. Default is 1000.
locally_biased : bool, optional
If `True` (default), use the locally biased variant of the
algorithm known as DIRECT_L. If `False`, use the original unbiased
DIRECT algorithm. For hard problems with many local minima,
`False` is recommended.
f_min : float, optional
Function value of the global optimum. Set this value only if the
global optimum is known. Default is ``-np.inf``, so that this
termination criterion is deactivated.
f_min_rtol : float, optional
Terminate the optimization once the relative error between the
current best minimum `f` and the supplied global minimum `f_min`
is smaller than `f_min_rtol`. This parameter is only used if
`f_min` is also set. Must lie between 0 and 1. Default is 1e-4.
vol_tol : float, optional
Terminate the optimization once the volume of the hyperrectangle
containing the lowest function value is smaller than `vol_tol`
of the complete search space. Must lie between 0 and 1.
Default is 1e-16.
len_tol : float, optional
If `locally_biased=True`, terminate the optimization once half of
the normalized maximal side length of the hyperrectangle containing
the lowest function value is smaller than `len_tol`.
If `locally_biased=False`, terminate the optimization once half of
the normalized diagonal of the hyperrectangle containing the lowest
function value is smaller than `len_tol`. Must lie between 0 and 1.
Default is 1e-6.
callback : callable, optional
A callback function with signature ``callback(xk)`` where ``xk``
represents the best function value found so far.
Returns
-------
res : OptimizeResult
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.
Notes
-----
DIviding RECTangles (DIRECT) is a deterministic global
optimization algorithm capable of minimizing a black box function with
its variables subject to lower and upper bound constraints by sampling
potential solutions in the search space [1]_. The algorithm starts by
normalising the search space to an n-dimensional unit hypercube.
It samples the function at the center of this hypercube and at 2n
(n is the number of variables) more points, 2 in each coordinate
direction. Using these function values, DIRECT then divides the
domain into hyperrectangles, each having exactly one of the sampling
points as its center. In each iteration, DIRECT chooses, using the `eps`
parameter which defaults to 1e-4, some of the existing hyperrectangles
to be further divided. This division process continues until either the
maximum number of iterations or maximum function evaluations allowed
are exceeded, or the hyperrectangle containing the minimal value found
so far becomes small enough. If `f_min` is specified, the optimization
will stop once this function value is reached within a relative tolerance.
The locally biased variant of DIRECT (originally called DIRECT_L) [2]_ is
used by default. It makes the search more locally biased and more
efficient for cases with only a few local minima.
A note about termination criteria: `vol_tol` refers to the volume of the
hyperrectangle containing the lowest function value found so far. This
volume decreases exponentially with increasing dimensionality of the
problem. Therefore `vol_tol` should be decreased to avoid premature
termination of the algorithm for higher dimensions. This does not hold
for `len_tol`: it refers either to half of the maximal side length
(for ``locally_biased=True``) or half of the diagonal of the
hyperrectangle (for ``locally_biased=False``).
This code is based on the DIRECT 2.0.4 Fortran code by Gablonsky et al. at
https://ctk.math.ncsu.edu/SOFTWARE/DIRECTv204.tar.gz .
This original version was initially converted via f2c and then cleaned up
and reorganized by Steven G. Johnson, August 2007, for the NLopt project.
The `direct` function wraps the C implementation.
.. versionadded:: 1.9.0
References
----------
.. [1] Jones, D.R., Perttunen, C.D. & Stuckman, B.E. Lipschitzian
optimization without the Lipschitz constant. J Optim Theory Appl
79, 157-181 (1993).
.. [2] Gablonsky, J., Kelley, C. A Locally-Biased form of the DIRECT
Algorithm. Journal of Global Optimization 21, 27-37 (2001).
Examples
--------
The following example is a 2-D problem with four local minima: minimizing
the Styblinski-Tang function
(https://en.wikipedia.org/wiki/Test_functions_for_optimization).
>>> from scipy.optimize import direct, Bounds
>>> def styblinski_tang(pos):
... x, y = pos
... return 0.5 * (x**4 - 16*x**2 + 5*x + y**4 - 16*y**2 + 5*y)
>>> bounds = Bounds([-4., -4.], [4., 4.])
>>> result = direct(styblinski_tang, bounds)
>>> result.x, result.fun, result.nfev
array([-2.90321597, -2.90321597]), -78.3323279095383, 2011
The correct global minimum was found but with a huge number of function
evaluations (2011). Loosening the termination tolerances `vol_tol` and
`len_tol` can be used to stop DIRECT earlier.
>>> result = direct(styblinski_tang, bounds, len_tol=1e-3)
>>> result.x, result.fun, result.nfev
array([-2.9044353, -2.9044353]), -78.33230330754142, 207
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