280 lines
12 KiB
Python
280 lines
12 KiB
Python
|
from __future__ import annotations
|
||
|
|
from typing import (
|
||
|
|
Any, Callable, Iterable, Optional, Tuple, TYPE_CHECKING, Union
|
||
|
|
)
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
from scipy.optimize import OptimizeResult
|
||
|
|
from ._constraints import old_bound_to_new, Bounds
|
||
|
|
from ._direct import direct as _direct # type: ignore
|
||
|
|
|
||
|
|
if TYPE_CHECKING:
|
||
|
|
import numpy.typing as npt
|
||
|
|
from _typeshed import NoneType
|
||
|
|
|
||
|
|
__all__ = ['direct']
|
||
|
|
|
||
|
|
ERROR_MESSAGES = (
|
||
|
|
"Number of function evaluations done is larger than maxfun={}",
|
||
|
|
"Number of iterations is larger than maxiter={}",
|
||
|
|
"u[i] < l[i] for some i",
|
||
|
|
"maxfun is too large",
|
||
|
|
"Initialization failed",
|
||
|
|
"There was an error in the creation of the sample points",
|
||
|
|
"An error occured while the function was sampled",
|
||
|
|
"Maximum number of levels has been reached.",
|
||
|
|
"Forced stop",
|
||
|
|
"Invalid arguments",
|
||
|
|
"Out of memory",
|
||
|
|
)
|
||
|
|
|
||
|
|
SUCCESS_MESSAGES = (
|
||
|
|
("The best function value found is within a relative error={} "
|
||
|
|
"of the (known) global optimum f_min"),
|
||
|
|
("The volume of the hyperrectangle containing the lowest function value "
|
||
|
|
"found is below vol_tol={}"),
|
||
|
|
("The side length measure of the hyperrectangle containing the lowest "
|
||
|
|
"function value found is below len_tol={}"),
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
def direct(
|
||
|
|
func: Callable[[npt.ArrayLike, Tuple[Any]], float],
|
||
|
|
bounds: Union[Iterable, Bounds],
|
||
|
|
*,
|
||
|
|
args: tuple = (),
|
||
|
|
eps: float = 1e-4,
|
||
|
|
maxfun: Union[int, None] = None,
|
||
|
|
maxiter: int = 1000,
|
||
|
|
locally_biased: bool = True,
|
||
|
|
f_min: float = -np.inf,
|
||
|
|
f_min_rtol: float = 1e-4,
|
||
|
|
vol_tol: float = 1e-16,
|
||
|
|
len_tol: float = 1e-6,
|
||
|
|
callback: Optional[Callable[[npt.ArrayLike], NoneType]] = None
|
||
|
|
) -> OptimizeResult:
|
||
|
|
"""
|
||
|
|
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
|
||
|
|
|
||
|
|
"""
|
||
|
|
# convert bounds to new Bounds class if necessary
|
||
|
|
if not isinstance(bounds, Bounds):
|
||
|
|
if isinstance(bounds, list) or isinstance(bounds, tuple):
|
||
|
|
lb, ub = old_bound_to_new(bounds)
|
||
|
|
bounds = Bounds(lb, ub)
|
||
|
|
else:
|
||
|
|
message = ("bounds must be a sequence or "
|
||
|
|
"instance of Bounds class")
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
lb = np.ascontiguousarray(bounds.lb, dtype=np.float64)
|
||
|
|
ub = np.ascontiguousarray(bounds.ub, dtype=np.float64)
|
||
|
|
|
||
|
|
# validate bounds
|
||
|
|
# check that lower bounds are smaller than upper bounds
|
||
|
|
if not np.all(lb < ub):
|
||
|
|
raise ValueError('Bounds are not consistent min < max')
|
||
|
|
# check for infs
|
||
|
|
if (np.any(np.isinf(lb)) or np.any(np.isinf(ub))):
|
||
|
|
raise ValueError("Bounds must not be inf.")
|
||
|
|
|
||
|
|
# validate tolerances
|
||
|
|
if (vol_tol < 0 or vol_tol > 1):
|
||
|
|
raise ValueError("vol_tol must be between 0 and 1.")
|
||
|
|
if (len_tol < 0 or len_tol > 1):
|
||
|
|
raise ValueError("len_tol must be between 0 and 1.")
|
||
|
|
if (f_min_rtol < 0 or f_min_rtol > 1):
|
||
|
|
raise ValueError("f_min_rtol must be between 0 and 1.")
|
||
|
|
|
||
|
|
# validate maxfun and maxiter
|
||
|
|
if maxfun is None:
|
||
|
|
maxfun = 1000 * lb.shape[0]
|
||
|
|
if not isinstance(maxfun, int):
|
||
|
|
raise ValueError("maxfun must be of type int.")
|
||
|
|
if maxfun < 0:
|
||
|
|
raise ValueError("maxfun must be > 0.")
|
||
|
|
if not isinstance(maxiter, int):
|
||
|
|
raise ValueError("maxiter must be of type int.")
|
||
|
|
if maxiter < 0:
|
||
|
|
raise ValueError("maxiter must be > 0.")
|
||
|
|
|
||
|
|
# validate boolean parameters
|
||
|
|
if not isinstance(locally_biased, bool):
|
||
|
|
raise ValueError("locally_biased must be True or False.")
|
||
|
|
|
||
|
|
def _func_wrap(x, args=None):
|
||
|
|
x = np.asarray(x)
|
||
|
|
if args is None:
|
||
|
|
f = func(x)
|
||
|
|
else:
|
||
|
|
f = func(x, *args)
|
||
|
|
# always return a float
|
||
|
|
return np.asarray(f).item()
|
||
|
|
|
||
|
|
# TODO: fix disp argument
|
||
|
|
x, fun, ret_code, nfev, nit = _direct(
|
||
|
|
_func_wrap,
|
||
|
|
np.asarray(lb), np.asarray(ub),
|
||
|
|
args,
|
||
|
|
False, eps, maxfun, maxiter,
|
||
|
|
locally_biased,
|
||
|
|
f_min, f_min_rtol,
|
||
|
|
vol_tol, len_tol, callback
|
||
|
|
)
|
||
|
|
|
||
|
|
format_val = (maxfun, maxiter, f_min_rtol, vol_tol, len_tol)
|
||
|
|
if ret_code > 2:
|
||
|
|
message = SUCCESS_MESSAGES[ret_code - 3].format(
|
||
|
|
format_val[ret_code - 1])
|
||
|
|
elif 0 < ret_code <= 2:
|
||
|
|
message = ERROR_MESSAGES[ret_code - 1].format(format_val[ret_code - 1])
|
||
|
|
elif 0 > ret_code > -100:
|
||
|
|
message = ERROR_MESSAGES[abs(ret_code) + 1]
|
||
|
|
else:
|
||
|
|
message = ERROR_MESSAGES[ret_code + 99]
|
||
|
|
|
||
|
|
return OptimizeResult(x=np.asarray(x), fun=fun, status=ret_code,
|
||
|
|
success=ret_code > 2, message=message,
|
||
|
|
nfev=nfev, nit=nit)
|