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3136 lines
103 KiB
Python
3136 lines
103 KiB
Python
6 years ago
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#__docformat__ = "restructuredtext en"
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# ******NOTICE***************
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# optimize.py module by Travis E. Oliphant
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#
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# You may copy and use this module as you see fit with no
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# guarantee implied provided you keep this notice in all copies.
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# *****END NOTICE************
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# A collection of optimization algorithms. Version 0.5
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# CHANGES
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# Added fminbound (July 2001)
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# Added brute (Aug. 2002)
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# Finished line search satisfying strong Wolfe conditions (Mar. 2004)
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# Updated strong Wolfe conditions line search to use
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# cubic-interpolation (Mar. 2004)
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from __future__ import division, print_function, absolute_import
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# Minimization routines
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__all__ = ['fmin', 'fmin_powell', 'fmin_bfgs', 'fmin_ncg', 'fmin_cg',
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'fminbound', 'brent', 'golden', 'bracket', 'rosen', 'rosen_der',
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'rosen_hess', 'rosen_hess_prod', 'brute', 'approx_fprime',
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'line_search', 'check_grad', 'OptimizeResult', 'show_options',
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'OptimizeWarning']
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__docformat__ = "restructuredtext en"
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import warnings
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import sys
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import numpy
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from scipy._lib.six import callable, xrange
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from numpy import (atleast_1d, eye, mgrid, argmin, zeros, shape, squeeze,
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vectorize, asarray, sqrt, Inf, asfarray, isinf)
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import numpy as np
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from .linesearch import (line_search_wolfe1, line_search_wolfe2,
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line_search_wolfe2 as line_search,
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LineSearchWarning)
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from scipy._lib._util import getargspec_no_self as _getargspec
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# standard status messages of optimizers
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_status_message = {'success': 'Optimization terminated successfully.',
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'maxfev': 'Maximum number of function evaluations has '
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'been exceeded.',
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'maxiter': 'Maximum number of iterations has been '
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'exceeded.',
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'pr_loss': 'Desired error not necessarily achieved due '
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'to precision loss.'}
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class MemoizeJac(object):
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""" Decorator that caches the value gradient of function each time it
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is called. """
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def __init__(self, fun):
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self.fun = fun
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self.jac = None
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self.x = None
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def __call__(self, x, *args):
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self.x = numpy.asarray(x).copy()
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fg = self.fun(x, *args)
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self.jac = fg[1]
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return fg[0]
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def derivative(self, x, *args):
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if self.jac is not None and numpy.alltrue(x == self.x):
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return self.jac
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else:
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self(x, *args)
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return self.jac
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class OptimizeResult(dict):
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""" Represents the optimization result.
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Attributes
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----------
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x : ndarray
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The solution of the optimization.
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success : bool
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Whether or not the optimizer exited successfully.
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status : int
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Termination status of the optimizer. Its value depends on the
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underlying solver. Refer to `message` for details.
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message : str
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Description of the cause of the termination.
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fun, jac, hess: ndarray
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Values of objective function, its Jacobian and its Hessian (if
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available). The Hessians may be approximations, see the documentation
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of the function in question.
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hess_inv : object
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Inverse of the objective function's Hessian; may be an approximation.
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Not available for all solvers. The type of this attribute may be
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either np.ndarray or scipy.sparse.linalg.LinearOperator.
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nfev, njev, nhev : int
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Number of evaluations of the objective functions and of its
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Jacobian and Hessian.
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nit : int
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Number of iterations performed by the optimizer.
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maxcv : float
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The maximum constraint violation.
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Notes
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-----
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There may be additional attributes not listed above depending of the
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specific solver. Since this class is essentially a subclass of dict
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with attribute accessors, one can see which attributes are available
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using the `keys()` method.
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"""
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def __getattr__(self, name):
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try:
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return self[name]
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except KeyError:
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raise AttributeError(name)
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__setattr__ = dict.__setitem__
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__delattr__ = dict.__delitem__
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def __repr__(self):
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if self.keys():
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m = max(map(len, list(self.keys()))) + 1
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return '\n'.join([k.rjust(m) + ': ' + repr(v)
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for k, v in sorted(self.items())])
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else:
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return self.__class__.__name__ + "()"
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def __dir__(self):
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return list(self.keys())
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class OptimizeWarning(UserWarning):
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pass
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def _check_unknown_options(unknown_options):
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if unknown_options:
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msg = ", ".join(map(str, unknown_options.keys()))
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# Stack level 4: this is called from _minimize_*, which is
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# called from another function in Scipy. Level 4 is the first
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# level in user code.
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warnings.warn("Unknown solver options: %s" % msg, OptimizeWarning, 4)
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def is_array_scalar(x):
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"""Test whether `x` is either a scalar or an array scalar.
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"""
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return np.size(x) == 1
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_epsilon = sqrt(numpy.finfo(float).eps)
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def vecnorm(x, ord=2):
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if ord == Inf:
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return numpy.amax(numpy.abs(x))
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elif ord == -Inf:
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return numpy.amin(numpy.abs(x))
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else:
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return numpy.sum(numpy.abs(x)**ord, axis=0)**(1.0 / ord)
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def rosen(x):
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"""
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The Rosenbrock function.
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The function computed is::
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sum(100.0*(x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0)
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Parameters
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----------
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x : array_like
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1-D array of points at which the Rosenbrock function is to be computed.
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Returns
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-------
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f : float
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The value of the Rosenbrock function.
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See Also
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--------
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rosen_der, rosen_hess, rosen_hess_prod
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Examples
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--------
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>>> from scipy.optimize import rosen
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>>> X = 0.1 * np.arange(10)
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>>> rosen(X)
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76.56
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"""
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x = asarray(x)
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r = numpy.sum(100.0 * (x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0,
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axis=0)
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return r
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def rosen_der(x):
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"""
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The derivative (i.e. gradient) of the Rosenbrock function.
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Parameters
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----------
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x : array_like
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1-D array of points at which the derivative is to be computed.
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Returns
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-------
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rosen_der : (N,) ndarray
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The gradient of the Rosenbrock function at `x`.
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See Also
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--------
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rosen, rosen_hess, rosen_hess_prod
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"""
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x = asarray(x)
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xm = x[1:-1]
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xm_m1 = x[:-2]
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xm_p1 = x[2:]
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der = numpy.zeros_like(x)
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der[1:-1] = (200 * (xm - xm_m1**2) -
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400 * (xm_p1 - xm**2) * xm - 2 * (1 - xm))
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der[0] = -400 * x[0] * (x[1] - x[0]**2) - 2 * (1 - x[0])
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der[-1] = 200 * (x[-1] - x[-2]**2)
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return der
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def rosen_hess(x):
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"""
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The Hessian matrix of the Rosenbrock function.
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Parameters
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----------
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x : array_like
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1-D array of points at which the Hessian matrix is to be computed.
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Returns
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-------
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rosen_hess : ndarray
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The Hessian matrix of the Rosenbrock function at `x`.
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See Also
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--------
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rosen, rosen_der, rosen_hess_prod
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"""
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x = atleast_1d(x)
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H = numpy.diag(-400 * x[:-1], 1) - numpy.diag(400 * x[:-1], -1)
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diagonal = numpy.zeros(len(x), dtype=x.dtype)
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diagonal[0] = 1200 * x[0]**2 - 400 * x[1] + 2
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diagonal[-1] = 200
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diagonal[1:-1] = 202 + 1200 * x[1:-1]**2 - 400 * x[2:]
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H = H + numpy.diag(diagonal)
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return H
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def rosen_hess_prod(x, p):
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"""
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Product of the Hessian matrix of the Rosenbrock function with a vector.
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Parameters
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----------
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x : array_like
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1-D array of points at which the Hessian matrix is to be computed.
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p : array_like
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1-D array, the vector to be multiplied by the Hessian matrix.
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Returns
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-------
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rosen_hess_prod : ndarray
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The Hessian matrix of the Rosenbrock function at `x` multiplied
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by the vector `p`.
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See Also
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--------
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rosen, rosen_der, rosen_hess
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"""
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x = atleast_1d(x)
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Hp = numpy.zeros(len(x), dtype=x.dtype)
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Hp[0] = (1200 * x[0]**2 - 400 * x[1] + 2) * p[0] - 400 * x[0] * p[1]
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Hp[1:-1] = (-400 * x[:-2] * p[:-2] +
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(202 + 1200 * x[1:-1]**2 - 400 * x[2:]) * p[1:-1] -
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400 * x[1:-1] * p[2:])
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Hp[-1] = -400 * x[-2] * p[-2] + 200*p[-1]
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return Hp
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def wrap_function(function, args):
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ncalls = [0]
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if function is None:
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return ncalls, None
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def function_wrapper(*wrapper_args):
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ncalls[0] += 1
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return function(*(wrapper_args + args))
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return ncalls, function_wrapper
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def fmin(func, x0, args=(), xtol=1e-4, ftol=1e-4, maxiter=None, maxfun=None,
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full_output=0, disp=1, retall=0, callback=None, initial_simplex=None):
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"""
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Minimize a function using the downhill simplex algorithm.
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This algorithm only uses function values, not derivatives or second
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derivatives.
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Parameters
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----------
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func : callable func(x,*args)
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The objective function to be minimized.
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x0 : ndarray
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Initial guess.
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args : tuple, optional
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Extra arguments passed to func, i.e. ``f(x,*args)``.
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xtol : float, optional
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Absolute error in xopt between iterations that is acceptable for
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convergence.
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ftol : number, optional
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Absolute error in func(xopt) between iterations that is acceptable for
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convergence.
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maxiter : int, optional
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Maximum number of iterations to perform.
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maxfun : number, optional
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Maximum number of function evaluations to make.
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full_output : bool, optional
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Set to True if fopt and warnflag outputs are desired.
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disp : bool, optional
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Set to True to print convergence messages.
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retall : bool, optional
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Set to True to return list of solutions at each iteration.
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callback : callable, optional
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Called after each iteration, as callback(xk), where xk is the
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current parameter vector.
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initial_simplex : array_like of shape (N + 1, N), optional
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Initial simplex. If given, overrides `x0`.
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``initial_simplex[j,:]`` should contain the coordinates of
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the j-th vertex of the ``N+1`` vertices in the simplex, where
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``N`` is the dimension.
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Returns
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||
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-------
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||
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xopt : ndarray
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Parameter that minimizes function.
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fopt : float
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Value of function at minimum: ``fopt = func(xopt)``.
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iter : int
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Number of iterations performed.
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funcalls : int
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Number of function calls made.
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||
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warnflag : int
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1 : Maximum number of function evaluations made.
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2 : Maximum number of iterations reached.
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allvecs : list
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Solution at each iteration.
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||
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|
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See also
|
||
|
|
--------
|
||
|
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minimize: Interface to minimization algorithms for multivariate
|
||
|
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functions. See the 'Nelder-Mead' `method` in particular.
|
||
|
|
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||
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Notes
|
||
|
|
-----
|
||
|
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Uses a Nelder-Mead simplex algorithm to find the minimum of function of
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one or more variables.
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||
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|
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This algorithm has a long history of successful use in applications.
|
||
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But it will usually be slower than an algorithm that uses first or
|
||
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second derivative information. In practice it can have poor
|
||
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performance in high-dimensional problems and is not robust to
|
||
|
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minimizing complicated functions. Additionally, there currently is no
|
||
|
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complete theory describing when the algorithm will successfully
|
||
|
|
converge to the minimum, or how fast it will if it does. Both the ftol and
|
||
|
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xtol criteria must be met for convergence.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> def f(x):
|
||
|
|
... return x**2
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||
|
|
|
||
|
|
>>> from scipy import optimize
|
||
|
|
|
||
|
|
>>> minimum = optimize.fmin(f, 1)
|
||
|
|
Optimization terminated successfully.
|
||
|
|
Current function value: 0.000000
|
||
|
|
Iterations: 17
|
||
|
|
Function evaluations: 34
|
||
|
|
>>> minimum[0]
|
||
|
|
-8.8817841970012523e-16
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Nelder, J.A. and Mead, R. (1965), "A simplex method for function
|
||
|
|
minimization", The Computer Journal, 7, pp. 308-313
|
||
|
|
|
||
|
|
.. [2] Wright, M.H. (1996), "Direct Search Methods: Once Scorned, Now
|
||
|
|
Respectable", in Numerical Analysis 1995, Proceedings of the
|
||
|
|
1995 Dundee Biennial Conference in Numerical Analysis, D.F.
|
||
|
|
Griffiths and G.A. Watson (Eds.), Addison Wesley Longman,
|
||
|
|
Harlow, UK, pp. 191-208.
|
||
|
|
|
||
|
|
"""
|
||
|
|
opts = {'xatol': xtol,
|
||
|
|
'fatol': ftol,
|
||
|
|
'maxiter': maxiter,
|
||
|
|
'maxfev': maxfun,
|
||
|
|
'disp': disp,
|
||
|
|
'return_all': retall,
|
||
|
|
'initial_simplex': initial_simplex}
|
||
|
|
|
||
|
|
res = _minimize_neldermead(func, x0, args, callback=callback, **opts)
|
||
|
|
if full_output:
|
||
|
|
retlist = res['x'], res['fun'], res['nit'], res['nfev'], res['status']
|
||
|
|
if retall:
|
||
|
|
retlist += (res['allvecs'], )
|
||
|
|
return retlist
|
||
|
|
else:
|
||
|
|
if retall:
|
||
|
|
return res['x'], res['allvecs']
|
||
|
|
else:
|
||
|
|
return res['x']
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_neldermead(func, x0, args=(), callback=None,
|
||
|
|
maxiter=None, maxfev=None, disp=False,
|
||
|
|
return_all=False, initial_simplex=None,
|
||
|
|
xatol=1e-4, fatol=1e-4, adaptive=False,
|
||
|
|
**unknown_options):
|
||
|
|
"""
|
||
|
|
Minimization of scalar function of one or more variables using the
|
||
|
|
Nelder-Mead algorithm.
|
||
|
|
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
disp : bool
|
||
|
|
Set to True to print convergence messages.
|
||
|
|
maxiter, maxfev : int
|
||
|
|
Maximum allowed number of iterations and function evaluations.
|
||
|
|
Will default to ``N*200``, where ``N`` is the number of
|
||
|
|
variables, if neither `maxiter` or `maxfev` is set. If both
|
||
|
|
`maxiter` and `maxfev` are set, minimization will stop at the
|
||
|
|
first reached.
|
||
|
|
initial_simplex : array_like of shape (N + 1, N)
|
||
|
|
Initial simplex. If given, overrides `x0`.
|
||
|
|
``initial_simplex[j,:]`` should contain the coordinates of
|
||
|
|
the j-th vertex of the ``N+1`` vertices in the simplex, where
|
||
|
|
``N`` is the dimension.
|
||
|
|
xatol : float, optional
|
||
|
|
Absolute error in xopt between iterations that is acceptable for
|
||
|
|
convergence.
|
||
|
|
fatol : number, optional
|
||
|
|
Absolute error in func(xopt) between iterations that is acceptable for
|
||
|
|
convergence.
|
||
|
|
adaptive : bool, optional
|
||
|
|
Adapt algorithm parameters to dimensionality of problem. Useful for
|
||
|
|
high-dimensional minimization [1]_.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Gao, F. and Han, L.
|
||
|
|
Implementing the Nelder-Mead simplex algorithm with adaptive
|
||
|
|
parameters. 2012. Computational Optimization and Applications.
|
||
|
|
51:1, pp. 259-277
|
||
|
|
|
||
|
|
"""
|
||
|
|
if 'ftol' in unknown_options:
|
||
|
|
warnings.warn("ftol is deprecated for Nelder-Mead,"
|
||
|
|
" use fatol instead. If you specified both, only"
|
||
|
|
" fatol is used.",
|
||
|
|
DeprecationWarning)
|
||
|
|
if (np.isclose(fatol, 1e-4) and
|
||
|
|
not np.isclose(unknown_options['ftol'], 1e-4)):
|
||
|
|
# only ftol was probably specified, use it.
|
||
|
|
fatol = unknown_options['ftol']
|
||
|
|
unknown_options.pop('ftol')
|
||
|
|
if 'xtol' in unknown_options:
|
||
|
|
warnings.warn("xtol is deprecated for Nelder-Mead,"
|
||
|
|
" use xatol instead. If you specified both, only"
|
||
|
|
" xatol is used.",
|
||
|
|
DeprecationWarning)
|
||
|
|
if (np.isclose(xatol, 1e-4) and
|
||
|
|
not np.isclose(unknown_options['xtol'], 1e-4)):
|
||
|
|
# only xtol was probably specified, use it.
|
||
|
|
xatol = unknown_options['xtol']
|
||
|
|
unknown_options.pop('xtol')
|
||
|
|
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
maxfun = maxfev
|
||
|
|
retall = return_all
|
||
|
|
|
||
|
|
fcalls, func = wrap_function(func, args)
|
||
|
|
|
||
|
|
if adaptive:
|
||
|
|
dim = float(len(x0))
|
||
|
|
rho = 1
|
||
|
|
chi = 1 + 2/dim
|
||
|
|
psi = 0.75 - 1/(2*dim)
|
||
|
|
sigma = 1 - 1/dim
|
||
|
|
else:
|
||
|
|
rho = 1
|
||
|
|
chi = 2
|
||
|
|
psi = 0.5
|
||
|
|
sigma = 0.5
|
||
|
|
|
||
|
|
nonzdelt = 0.05
|
||
|
|
zdelt = 0.00025
|
||
|
|
|
||
|
|
x0 = asfarray(x0).flatten()
|
||
|
|
|
||
|
|
if initial_simplex is None:
|
||
|
|
N = len(x0)
|
||
|
|
|
||
|
|
sim = numpy.zeros((N + 1, N), dtype=x0.dtype)
|
||
|
|
sim[0] = x0
|
||
|
|
for k in range(N):
|
||
|
|
y = numpy.array(x0, copy=True)
|
||
|
|
if y[k] != 0:
|
||
|
|
y[k] = (1 + nonzdelt)*y[k]
|
||
|
|
else:
|
||
|
|
y[k] = zdelt
|
||
|
|
sim[k + 1] = y
|
||
|
|
else:
|
||
|
|
sim = np.asfarray(initial_simplex).copy()
|
||
|
|
if sim.ndim != 2 or sim.shape[0] != sim.shape[1] + 1:
|
||
|
|
raise ValueError("`initial_simplex` should be an array of shape (N+1,N)")
|
||
|
|
if len(x0) != sim.shape[1]:
|
||
|
|
raise ValueError("Size of `initial_simplex` is not consistent with `x0`")
|
||
|
|
N = sim.shape[1]
|
||
|
|
|
||
|
|
if retall:
|
||
|
|
allvecs = [sim[0]]
|
||
|
|
|
||
|
|
# If neither are set, then set both to default
|
||
|
|
if maxiter is None and maxfun is None:
|
||
|
|
maxiter = N * 200
|
||
|
|
maxfun = N * 200
|
||
|
|
elif maxiter is None:
|
||
|
|
# Convert remaining Nones, to np.inf, unless the other is np.inf, in
|
||
|
|
# which case use the default to avoid unbounded iteration
|
||
|
|
if maxfun == np.inf:
|
||
|
|
maxiter = N * 200
|
||
|
|
else:
|
||
|
|
maxiter = np.inf
|
||
|
|
elif maxfun is None:
|
||
|
|
if maxiter == np.inf:
|
||
|
|
maxfun = N * 200
|
||
|
|
else:
|
||
|
|
maxfun = np.inf
|
||
|
|
|
||
|
|
one2np1 = list(range(1, N + 1))
|
||
|
|
fsim = numpy.zeros((N + 1,), float)
|
||
|
|
|
||
|
|
for k in range(N + 1):
|
||
|
|
fsim[k] = func(sim[k])
|
||
|
|
|
||
|
|
ind = numpy.argsort(fsim)
|
||
|
|
fsim = numpy.take(fsim, ind, 0)
|
||
|
|
# sort so sim[0,:] has the lowest function value
|
||
|
|
sim = numpy.take(sim, ind, 0)
|
||
|
|
|
||
|
|
iterations = 1
|
||
|
|
|
||
|
|
while (fcalls[0] < maxfun and iterations < maxiter):
|
||
|
|
if (numpy.max(numpy.ravel(numpy.abs(sim[1:] - sim[0]))) <= xatol and
|
||
|
|
numpy.max(numpy.abs(fsim[0] - fsim[1:])) <= fatol):
|
||
|
|
break
|
||
|
|
|
||
|
|
xbar = numpy.add.reduce(sim[:-1], 0) / N
|
||
|
|
xr = (1 + rho) * xbar - rho * sim[-1]
|
||
|
|
fxr = func(xr)
|
||
|
|
doshrink = 0
|
||
|
|
|
||
|
|
if fxr < fsim[0]:
|
||
|
|
xe = (1 + rho * chi) * xbar - rho * chi * sim[-1]
|
||
|
|
fxe = func(xe)
|
||
|
|
|
||
|
|
if fxe < fxr:
|
||
|
|
sim[-1] = xe
|
||
|
|
fsim[-1] = fxe
|
||
|
|
else:
|
||
|
|
sim[-1] = xr
|
||
|
|
fsim[-1] = fxr
|
||
|
|
else: # fsim[0] <= fxr
|
||
|
|
if fxr < fsim[-2]:
|
||
|
|
sim[-1] = xr
|
||
|
|
fsim[-1] = fxr
|
||
|
|
else: # fxr >= fsim[-2]
|
||
|
|
# Perform contraction
|
||
|
|
if fxr < fsim[-1]:
|
||
|
|
xc = (1 + psi * rho) * xbar - psi * rho * sim[-1]
|
||
|
|
fxc = func(xc)
|
||
|
|
|
||
|
|
if fxc <= fxr:
|
||
|
|
sim[-1] = xc
|
||
|
|
fsim[-1] = fxc
|
||
|
|
else:
|
||
|
|
doshrink = 1
|
||
|
|
else:
|
||
|
|
# Perform an inside contraction
|
||
|
|
xcc = (1 - psi) * xbar + psi * sim[-1]
|
||
|
|
fxcc = func(xcc)
|
||
|
|
|
||
|
|
if fxcc < fsim[-1]:
|
||
|
|
sim[-1] = xcc
|
||
|
|
fsim[-1] = fxcc
|
||
|
|
else:
|
||
|
|
doshrink = 1
|
||
|
|
|
||
|
|
if doshrink:
|
||
|
|
for j in one2np1:
|
||
|
|
sim[j] = sim[0] + sigma * (sim[j] - sim[0])
|
||
|
|
fsim[j] = func(sim[j])
|
||
|
|
|
||
|
|
ind = numpy.argsort(fsim)
|
||
|
|
sim = numpy.take(sim, ind, 0)
|
||
|
|
fsim = numpy.take(fsim, ind, 0)
|
||
|
|
if callback is not None:
|
||
|
|
callback(sim[0])
|
||
|
|
iterations += 1
|
||
|
|
if retall:
|
||
|
|
allvecs.append(sim[0])
|
||
|
|
|
||
|
|
x = sim[0]
|
||
|
|
fval = numpy.min(fsim)
|
||
|
|
warnflag = 0
|
||
|
|
|
||
|
|
if fcalls[0] >= maxfun:
|
||
|
|
warnflag = 1
|
||
|
|
msg = _status_message['maxfev']
|
||
|
|
if disp:
|
||
|
|
print('Warning: ' + msg)
|
||
|
|
elif iterations >= maxiter:
|
||
|
|
warnflag = 2
|
||
|
|
msg = _status_message['maxiter']
|
||
|
|
if disp:
|
||
|
|
print('Warning: ' + msg)
|
||
|
|
else:
|
||
|
|
msg = _status_message['success']
|
||
|
|
if disp:
|
||
|
|
print(msg)
|
||
|
|
print(" Current function value: %f" % fval)
|
||
|
|
print(" Iterations: %d" % iterations)
|
||
|
|
print(" Function evaluations: %d" % fcalls[0])
|
||
|
|
|
||
|
|
result = OptimizeResult(fun=fval, nit=iterations, nfev=fcalls[0],
|
||
|
|
status=warnflag, success=(warnflag == 0),
|
||
|
|
message=msg, x=x, final_simplex=(sim, fsim))
|
||
|
|
if retall:
|
||
|
|
result['allvecs'] = allvecs
|
||
|
|
return result
|
||
|
|
|
||
|
|
|
||
|
|
def _approx_fprime_helper(xk, f, epsilon, args=(), f0=None):
|
||
|
|
"""
|
||
|
|
See ``approx_fprime``. An optional initial function value arg is added.
|
||
|
|
|
||
|
|
"""
|
||
|
|
if f0 is None:
|
||
|
|
f0 = f(*((xk,) + args))
|
||
|
|
grad = numpy.zeros((len(xk),), float)
|
||
|
|
ei = numpy.zeros((len(xk),), float)
|
||
|
|
for k in range(len(xk)):
|
||
|
|
ei[k] = 1.0
|
||
|
|
d = epsilon * ei
|
||
|
|
grad[k] = (f(*((xk + d,) + args)) - f0) / d[k]
|
||
|
|
ei[k] = 0.0
|
||
|
|
return grad
|
||
|
|
|
||
|
|
|
||
|
|
def approx_fprime(xk, f, epsilon, *args):
|
||
|
|
"""Finite-difference approximation of the gradient of a scalar function.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
xk : array_like
|
||
|
|
The coordinate vector at which to determine the gradient of `f`.
|
||
|
|
f : callable
|
||
|
|
The function of which to determine the gradient (partial derivatives).
|
||
|
|
Should take `xk` as first argument, other arguments to `f` can be
|
||
|
|
supplied in ``*args``. Should return a scalar, the value of the
|
||
|
|
function at `xk`.
|
||
|
|
epsilon : array_like
|
||
|
|
Increment to `xk` to use for determining the function gradient.
|
||
|
|
If a scalar, uses the same finite difference delta for all partial
|
||
|
|
derivatives. If an array, should contain one value per element of
|
||
|
|
`xk`.
|
||
|
|
\\*args : args, optional
|
||
|
|
Any other arguments that are to be passed to `f`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
grad : ndarray
|
||
|
|
The partial derivatives of `f` to `xk`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
check_grad : Check correctness of gradient function against approx_fprime.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The function gradient is determined by the forward finite difference
|
||
|
|
formula::
|
||
|
|
|
||
|
|
f(xk[i] + epsilon[i]) - f(xk[i])
|
||
|
|
f'[i] = ---------------------------------
|
||
|
|
epsilon[i]
|
||
|
|
|
||
|
|
The main use of `approx_fprime` is in scalar function optimizers like
|
||
|
|
`fmin_bfgs`, to determine numerically the Jacobian of a function.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy import optimize
|
||
|
|
>>> def func(x, c0, c1):
|
||
|
|
... "Coordinate vector `x` should be an array of size two."
|
||
|
|
... return c0 * x[0]**2 + c1*x[1]**2
|
||
|
|
|
||
|
|
>>> x = np.ones(2)
|
||
|
|
>>> c0, c1 = (1, 200)
|
||
|
|
>>> eps = np.sqrt(np.finfo(float).eps)
|
||
|
|
>>> optimize.approx_fprime(x, func, [eps, np.sqrt(200) * eps], c0, c1)
|
||
|
|
array([ 2. , 400.00004198])
|
||
|
|
|
||
|
|
"""
|
||
|
|
return _approx_fprime_helper(xk, f, epsilon, args=args)
|
||
|
|
|
||
|
|
|
||
|
|
def check_grad(func, grad, x0, *args, **kwargs):
|
||
|
|
"""Check the correctness of a gradient function by comparing it against a
|
||
|
|
(forward) finite-difference approximation of the gradient.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable ``func(x0, *args)``
|
||
|
|
Function whose derivative is to be checked.
|
||
|
|
grad : callable ``grad(x0, *args)``
|
||
|
|
Gradient of `func`.
|
||
|
|
x0 : ndarray
|
||
|
|
Points to check `grad` against forward difference approximation of grad
|
||
|
|
using `func`.
|
||
|
|
args : \\*args, optional
|
||
|
|
Extra arguments passed to `func` and `grad`.
|
||
|
|
epsilon : float, optional
|
||
|
|
Step size used for the finite difference approximation. It defaults to
|
||
|
|
``sqrt(numpy.finfo(float).eps)``, which is approximately 1.49e-08.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
err : float
|
||
|
|
The square root of the sum of squares (i.e. the 2-norm) of the
|
||
|
|
difference between ``grad(x0, *args)`` and the finite difference
|
||
|
|
approximation of `grad` using func at the points `x0`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
approx_fprime
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> def func(x):
|
||
|
|
... return x[0]**2 - 0.5 * x[1]**3
|
||
|
|
>>> def grad(x):
|
||
|
|
... return [2 * x[0], -1.5 * x[1]**2]
|
||
|
|
>>> from scipy.optimize import check_grad
|
||
|
|
>>> check_grad(func, grad, [1.5, -1.5])
|
||
|
|
2.9802322387695312e-08
|
||
|
|
|
||
|
|
"""
|
||
|
|
step = kwargs.pop('epsilon', _epsilon)
|
||
|
|
if kwargs:
|
||
|
|
raise ValueError("Unknown keyword arguments: %r" %
|
||
|
|
(list(kwargs.keys()),))
|
||
|
|
return sqrt(sum((grad(x0, *args) -
|
||
|
|
approx_fprime(x0, func, step, *args))**2))
|
||
|
|
|
||
|
|
|
||
|
|
def approx_fhess_p(x0, p, fprime, epsilon, *args):
|
||
|
|
f2 = fprime(*((x0 + epsilon*p,) + args))
|
||
|
|
f1 = fprime(*((x0,) + args))
|
||
|
|
return (f2 - f1) / epsilon
|
||
|
|
|
||
|
|
|
||
|
|
class _LineSearchError(RuntimeError):
|
||
|
|
pass
|
||
|
|
|
||
|
|
|
||
|
|
def _line_search_wolfe12(f, fprime, xk, pk, gfk, old_fval, old_old_fval,
|
||
|
|
**kwargs):
|
||
|
|
"""
|
||
|
|
Same as line_search_wolfe1, but fall back to line_search_wolfe2 if
|
||
|
|
suitable step length is not found, and raise an exception if a
|
||
|
|
suitable step length is not found.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
_LineSearchError
|
||
|
|
If no suitable step size is found
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
extra_condition = kwargs.pop('extra_condition', None)
|
||
|
|
|
||
|
|
ret = line_search_wolfe1(f, fprime, xk, pk, gfk,
|
||
|
|
old_fval, old_old_fval,
|
||
|
|
**kwargs)
|
||
|
|
|
||
|
|
if ret[0] is not None and extra_condition is not None:
|
||
|
|
xp1 = xk + ret[0] * pk
|
||
|
|
if not extra_condition(ret[0], xp1, ret[3], ret[5]):
|
||
|
|
# Reject step if extra_condition fails
|
||
|
|
ret = (None,)
|
||
|
|
|
||
|
|
if ret[0] is None:
|
||
|
|
# line search failed: try different one.
|
||
|
|
with warnings.catch_warnings():
|
||
|
|
warnings.simplefilter('ignore', LineSearchWarning)
|
||
|
|
kwargs2 = {}
|
||
|
|
for key in ('c1', 'c2', 'amax'):
|
||
|
|
if key in kwargs:
|
||
|
|
kwargs2[key] = kwargs[key]
|
||
|
|
ret = line_search_wolfe2(f, fprime, xk, pk, gfk,
|
||
|
|
old_fval, old_old_fval,
|
||
|
|
extra_condition=extra_condition,
|
||
|
|
**kwargs2)
|
||
|
|
|
||
|
|
if ret[0] is None:
|
||
|
|
raise _LineSearchError()
|
||
|
|
|
||
|
|
return ret
|
||
|
|
|
||
|
|
|
||
|
|
def fmin_bfgs(f, x0, fprime=None, args=(), gtol=1e-5, norm=Inf,
|
||
|
|
epsilon=_epsilon, maxiter=None, full_output=0, disp=1,
|
||
|
|
retall=0, callback=None):
|
||
|
|
"""
|
||
|
|
Minimize a function using the BFGS algorithm.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
f : callable f(x,*args)
|
||
|
|
Objective function to be minimized.
|
||
|
|
x0 : ndarray
|
||
|
|
Initial guess.
|
||
|
|
fprime : callable f'(x,*args), optional
|
||
|
|
Gradient of f.
|
||
|
|
args : tuple, optional
|
||
|
|
Extra arguments passed to f and fprime.
|
||
|
|
gtol : float, optional
|
||
|
|
Gradient norm must be less than gtol before successful termination.
|
||
|
|
norm : float, optional
|
||
|
|
Order of norm (Inf is max, -Inf is min)
|
||
|
|
epsilon : int or ndarray, optional
|
||
|
|
If fprime is approximated, use this value for the step size.
|
||
|
|
callback : callable, optional
|
||
|
|
An optional user-supplied function to call after each
|
||
|
|
iteration. Called as callback(xk), where xk is the
|
||
|
|
current parameter vector.
|
||
|
|
maxiter : int, optional
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
full_output : bool, optional
|
||
|
|
If True,return fopt, func_calls, grad_calls, and warnflag
|
||
|
|
in addition to xopt.
|
||
|
|
disp : bool, optional
|
||
|
|
Print convergence message if True.
|
||
|
|
retall : bool, optional
|
||
|
|
Return a list of results at each iteration if True.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
xopt : ndarray
|
||
|
|
Parameters which minimize f, i.e. f(xopt) == fopt.
|
||
|
|
fopt : float
|
||
|
|
Minimum value.
|
||
|
|
gopt : ndarray
|
||
|
|
Value of gradient at minimum, f'(xopt), which should be near 0.
|
||
|
|
Bopt : ndarray
|
||
|
|
Value of 1/f''(xopt), i.e. the inverse hessian matrix.
|
||
|
|
func_calls : int
|
||
|
|
Number of function_calls made.
|
||
|
|
grad_calls : int
|
||
|
|
Number of gradient calls made.
|
||
|
|
warnflag : integer
|
||
|
|
1 : Maximum number of iterations exceeded.
|
||
|
|
2 : Gradient and/or function calls not changing.
|
||
|
|
allvecs : list
|
||
|
|
The value of xopt at each iteration. Only returned if retall is True.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
minimize: Interface to minimization algorithms for multivariate
|
||
|
|
functions. See the 'BFGS' `method` in particular.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Optimize the function, f, whose gradient is given by fprime
|
||
|
|
using the quasi-Newton method of Broyden, Fletcher, Goldfarb,
|
||
|
|
and Shanno (BFGS)
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
Wright, and Nocedal 'Numerical Optimization', 1999, pg. 198.
|
||
|
|
|
||
|
|
"""
|
||
|
|
opts = {'gtol': gtol,
|
||
|
|
'norm': norm,
|
||
|
|
'eps': epsilon,
|
||
|
|
'disp': disp,
|
||
|
|
'maxiter': maxiter,
|
||
|
|
'return_all': retall}
|
||
|
|
|
||
|
|
res = _minimize_bfgs(f, x0, args, fprime, callback=callback, **opts)
|
||
|
|
|
||
|
|
if full_output:
|
||
|
|
retlist = (res['x'], res['fun'], res['jac'], res['hess_inv'],
|
||
|
|
res['nfev'], res['njev'], res['status'])
|
||
|
|
if retall:
|
||
|
|
retlist += (res['allvecs'], )
|
||
|
|
return retlist
|
||
|
|
else:
|
||
|
|
if retall:
|
||
|
|
return res['x'], res['allvecs']
|
||
|
|
else:
|
||
|
|
return res['x']
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_bfgs(fun, x0, args=(), jac=None, callback=None,
|
||
|
|
gtol=1e-5, norm=Inf, eps=_epsilon, maxiter=None,
|
||
|
|
disp=False, return_all=False,
|
||
|
|
**unknown_options):
|
||
|
|
"""
|
||
|
|
Minimization of scalar function of one or more variables using the
|
||
|
|
BFGS algorithm.
|
||
|
|
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
disp : bool
|
||
|
|
Set to True to print convergence messages.
|
||
|
|
maxiter : int
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
gtol : float
|
||
|
|
Gradient norm must be less than `gtol` before successful
|
||
|
|
termination.
|
||
|
|
norm : float
|
||
|
|
Order of norm (Inf is max, -Inf is min).
|
||
|
|
eps : float or ndarray
|
||
|
|
If `jac` is approximated, use this value for the step size.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
f = fun
|
||
|
|
fprime = jac
|
||
|
|
epsilon = eps
|
||
|
|
retall = return_all
|
||
|
|
|
||
|
|
x0 = asarray(x0).flatten()
|
||
|
|
if x0.ndim == 0:
|
||
|
|
x0.shape = (1,)
|
||
|
|
if maxiter is None:
|
||
|
|
maxiter = len(x0) * 200
|
||
|
|
func_calls, f = wrap_function(f, args)
|
||
|
|
if fprime is None:
|
||
|
|
grad_calls, myfprime = wrap_function(approx_fprime, (f, epsilon))
|
||
|
|
else:
|
||
|
|
grad_calls, myfprime = wrap_function(fprime, args)
|
||
|
|
gfk = myfprime(x0)
|
||
|
|
k = 0
|
||
|
|
N = len(x0)
|
||
|
|
I = numpy.eye(N, dtype=int)
|
||
|
|
Hk = I
|
||
|
|
|
||
|
|
# Sets the initial step guess to dx ~ 1
|
||
|
|
old_fval = f(x0)
|
||
|
|
old_old_fval = old_fval + np.linalg.norm(gfk) / 2
|
||
|
|
|
||
|
|
xk = x0
|
||
|
|
if retall:
|
||
|
|
allvecs = [x0]
|
||
|
|
warnflag = 0
|
||
|
|
gnorm = vecnorm(gfk, ord=norm)
|
||
|
|
while (gnorm > gtol) and (k < maxiter):
|
||
|
|
pk = -numpy.dot(Hk, gfk)
|
||
|
|
try:
|
||
|
|
alpha_k, fc, gc, old_fval, old_old_fval, gfkp1 = \
|
||
|
|
_line_search_wolfe12(f, myfprime, xk, pk, gfk,
|
||
|
|
old_fval, old_old_fval, amin=1e-100, amax=1e100)
|
||
|
|
except _LineSearchError:
|
||
|
|
# Line search failed to find a better solution.
|
||
|
|
warnflag = 2
|
||
|
|
break
|
||
|
|
|
||
|
|
xkp1 = xk + alpha_k * pk
|
||
|
|
if retall:
|
||
|
|
allvecs.append(xkp1)
|
||
|
|
sk = xkp1 - xk
|
||
|
|
xk = xkp1
|
||
|
|
if gfkp1 is None:
|
||
|
|
gfkp1 = myfprime(xkp1)
|
||
|
|
|
||
|
|
yk = gfkp1 - gfk
|
||
|
|
gfk = gfkp1
|
||
|
|
if callback is not None:
|
||
|
|
callback(xk)
|
||
|
|
k += 1
|
||
|
|
gnorm = vecnorm(gfk, ord=norm)
|
||
|
|
if (gnorm <= gtol):
|
||
|
|
break
|
||
|
|
|
||
|
|
if not numpy.isfinite(old_fval):
|
||
|
|
# We correctly found +-Inf as optimal value, or something went
|
||
|
|
# wrong.
|
||
|
|
warnflag = 2
|
||
|
|
break
|
||
|
|
|
||
|
|
try: # this was handled in numeric, let it remaines for more safety
|
||
|
|
rhok = 1.0 / (numpy.dot(yk, sk))
|
||
|
|
except ZeroDivisionError:
|
||
|
|
rhok = 1000.0
|
||
|
|
if disp:
|
||
|
|
print("Divide-by-zero encountered: rhok assumed large")
|
||
|
|
if isinf(rhok): # this is patch for numpy
|
||
|
|
rhok = 1000.0
|
||
|
|
if disp:
|
||
|
|
print("Divide-by-zero encountered: rhok assumed large")
|
||
|
|
A1 = I - sk[:, numpy.newaxis] * yk[numpy.newaxis, :] * rhok
|
||
|
|
A2 = I - yk[:, numpy.newaxis] * sk[numpy.newaxis, :] * rhok
|
||
|
|
Hk = numpy.dot(A1, numpy.dot(Hk, A2)) + (rhok * sk[:, numpy.newaxis] *
|
||
|
|
sk[numpy.newaxis, :])
|
||
|
|
|
||
|
|
fval = old_fval
|
||
|
|
if np.isnan(fval):
|
||
|
|
# This can happen if the first call to f returned NaN;
|
||
|
|
# the loop is then never entered.
|
||
|
|
warnflag = 2
|
||
|
|
|
||
|
|
if warnflag == 2:
|
||
|
|
msg = _status_message['pr_loss']
|
||
|
|
elif k >= maxiter:
|
||
|
|
warnflag = 1
|
||
|
|
msg = _status_message['maxiter']
|
||
|
|
else:
|
||
|
|
msg = _status_message['success']
|
||
|
|
|
||
|
|
if disp:
|
||
|
|
print("%s%s" % ("Warning: " if warnflag != 0 else "", msg))
|
||
|
|
print(" Current function value: %f" % fval)
|
||
|
|
print(" Iterations: %d" % k)
|
||
|
|
print(" Function evaluations: %d" % func_calls[0])
|
||
|
|
print(" Gradient evaluations: %d" % grad_calls[0])
|
||
|
|
|
||
|
|
result = OptimizeResult(fun=fval, jac=gfk, hess_inv=Hk, nfev=func_calls[0],
|
||
|
|
njev=grad_calls[0], status=warnflag,
|
||
|
|
success=(warnflag == 0), message=msg, x=xk,
|
||
|
|
nit=k)
|
||
|
|
if retall:
|
||
|
|
result['allvecs'] = allvecs
|
||
|
|
return result
|
||
|
|
|
||
|
|
|
||
|
|
def fmin_cg(f, x0, fprime=None, args=(), gtol=1e-5, norm=Inf, epsilon=_epsilon,
|
||
|
|
maxiter=None, full_output=0, disp=1, retall=0, callback=None):
|
||
|
|
"""
|
||
|
|
Minimize a function using a nonlinear conjugate gradient algorithm.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
f : callable, ``f(x, *args)``
|
||
|
|
Objective function to be minimized. Here `x` must be a 1-D array of
|
||
|
|
the variables that are to be changed in the search for a minimum, and
|
||
|
|
`args` are the other (fixed) parameters of `f`.
|
||
|
|
x0 : ndarray
|
||
|
|
A user-supplied initial estimate of `xopt`, the optimal value of `x`.
|
||
|
|
It must be a 1-D array of values.
|
||
|
|
fprime : callable, ``fprime(x, *args)``, optional
|
||
|
|
A function that returns the gradient of `f` at `x`. Here `x` and `args`
|
||
|
|
are as described above for `f`. The returned value must be a 1-D array.
|
||
|
|
Defaults to None, in which case the gradient is approximated
|
||
|
|
numerically (see `epsilon`, below).
|
||
|
|
args : tuple, optional
|
||
|
|
Parameter values passed to `f` and `fprime`. Must be supplied whenever
|
||
|
|
additional fixed parameters are needed to completely specify the
|
||
|
|
functions `f` and `fprime`.
|
||
|
|
gtol : float, optional
|
||
|
|
Stop when the norm of the gradient is less than `gtol`.
|
||
|
|
norm : float, optional
|
||
|
|
Order to use for the norm of the gradient
|
||
|
|
(``-np.Inf`` is min, ``np.Inf`` is max).
|
||
|
|
epsilon : float or ndarray, optional
|
||
|
|
Step size(s) to use when `fprime` is approximated numerically. Can be a
|
||
|
|
scalar or a 1-D array. Defaults to ``sqrt(eps)``, with eps the
|
||
|
|
floating point machine precision. Usually ``sqrt(eps)`` is about
|
||
|
|
1.5e-8.
|
||
|
|
maxiter : int, optional
|
||
|
|
Maximum number of iterations to perform. Default is ``200 * len(x0)``.
|
||
|
|
full_output : bool, optional
|
||
|
|
If True, return `fopt`, `func_calls`, `grad_calls`, and `warnflag` in
|
||
|
|
addition to `xopt`. See the Returns section below for additional
|
||
|
|
information on optional return values.
|
||
|
|
disp : bool, optional
|
||
|
|
If True, return a convergence message, followed by `xopt`.
|
||
|
|
retall : bool, optional
|
||
|
|
If True, add to the returned values the results of each iteration.
|
||
|
|
callback : callable, optional
|
||
|
|
An optional user-supplied function, called after each iteration.
|
||
|
|
Called as ``callback(xk)``, where ``xk`` is the current value of `x0`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
xopt : ndarray
|
||
|
|
Parameters which minimize f, i.e. ``f(xopt) == fopt``.
|
||
|
|
fopt : float, optional
|
||
|
|
Minimum value found, f(xopt). Only returned if `full_output` is True.
|
||
|
|
func_calls : int, optional
|
||
|
|
The number of function_calls made. Only returned if `full_output`
|
||
|
|
is True.
|
||
|
|
grad_calls : int, optional
|
||
|
|
The number of gradient calls made. Only returned if `full_output` is
|
||
|
|
True.
|
||
|
|
warnflag : int, optional
|
||
|
|
Integer value with warning status, only returned if `full_output` is
|
||
|
|
True.
|
||
|
|
|
||
|
|
0 : Success.
|
||
|
|
|
||
|
|
1 : The maximum number of iterations was exceeded.
|
||
|
|
|
||
|
|
2 : Gradient and/or function calls were not changing. May indicate
|
||
|
|
that precision was lost, i.e., the routine did not converge.
|
||
|
|
|
||
|
|
allvecs : list of ndarray, optional
|
||
|
|
List of arrays, containing the results at each iteration.
|
||
|
|
Only returned if `retall` is True.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
minimize : common interface to all `scipy.optimize` algorithms for
|
||
|
|
unconstrained and constrained minimization of multivariate
|
||
|
|
functions. It provides an alternative way to call
|
||
|
|
``fmin_cg``, by specifying ``method='CG'``.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This conjugate gradient algorithm is based on that of Polak and Ribiere
|
||
|
|
[1]_.
|
||
|
|
|
||
|
|
Conjugate gradient methods tend to work better when:
|
||
|
|
|
||
|
|
1. `f` has a unique global minimizing point, and no local minima or
|
||
|
|
other stationary points,
|
||
|
|
2. `f` is, at least locally, reasonably well approximated by a
|
||
|
|
quadratic function of the variables,
|
||
|
|
3. `f` is continuous and has a continuous gradient,
|
||
|
|
4. `fprime` is not too large, e.g., has a norm less than 1000,
|
||
|
|
5. The initial guess, `x0`, is reasonably close to `f` 's global
|
||
|
|
minimizing point, `xopt`.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Wright & Nocedal, "Numerical Optimization", 1999, pp. 120-122.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
Example 1: seek the minimum value of the expression
|
||
|
|
``a*u**2 + b*u*v + c*v**2 + d*u + e*v + f`` for given values
|
||
|
|
of the parameters and an initial guess ``(u, v) = (0, 0)``.
|
||
|
|
|
||
|
|
>>> args = (2, 3, 7, 8, 9, 10) # parameter values
|
||
|
|
>>> def f(x, *args):
|
||
|
|
... u, v = x
|
||
|
|
... a, b, c, d, e, f = args
|
||
|
|
... return a*u**2 + b*u*v + c*v**2 + d*u + e*v + f
|
||
|
|
>>> def gradf(x, *args):
|
||
|
|
... u, v = x
|
||
|
|
... a, b, c, d, e, f = args
|
||
|
|
... gu = 2*a*u + b*v + d # u-component of the gradient
|
||
|
|
... gv = b*u + 2*c*v + e # v-component of the gradient
|
||
|
|
... return np.asarray((gu, gv))
|
||
|
|
>>> x0 = np.asarray((0, 0)) # Initial guess.
|
||
|
|
>>> from scipy import optimize
|
||
|
|
>>> res1 = optimize.fmin_cg(f, x0, fprime=gradf, args=args)
|
||
|
|
Optimization terminated successfully.
|
||
|
|
Current function value: 1.617021
|
||
|
|
Iterations: 4
|
||
|
|
Function evaluations: 8
|
||
|
|
Gradient evaluations: 8
|
||
|
|
>>> res1
|
||
|
|
array([-1.80851064, -0.25531915])
|
||
|
|
|
||
|
|
Example 2: solve the same problem using the `minimize` function.
|
||
|
|
(This `myopts` dictionary shows all of the available options,
|
||
|
|
although in practice only non-default values would be needed.
|
||
|
|
The returned value will be a dictionary.)
|
||
|
|
|
||
|
|
>>> opts = {'maxiter' : None, # default value.
|
||
|
|
... 'disp' : True, # non-default value.
|
||
|
|
... 'gtol' : 1e-5, # default value.
|
||
|
|
... 'norm' : np.inf, # default value.
|
||
|
|
... 'eps' : 1.4901161193847656e-08} # default value.
|
||
|
|
>>> res2 = optimize.minimize(f, x0, jac=gradf, args=args,
|
||
|
|
... method='CG', options=opts)
|
||
|
|
Optimization terminated successfully.
|
||
|
|
Current function value: 1.617021
|
||
|
|
Iterations: 4
|
||
|
|
Function evaluations: 8
|
||
|
|
Gradient evaluations: 8
|
||
|
|
>>> res2.x # minimum found
|
||
|
|
array([-1.80851064, -0.25531915])
|
||
|
|
|
||
|
|
"""
|
||
|
|
opts = {'gtol': gtol,
|
||
|
|
'norm': norm,
|
||
|
|
'eps': epsilon,
|
||
|
|
'disp': disp,
|
||
|
|
'maxiter': maxiter,
|
||
|
|
'return_all': retall}
|
||
|
|
|
||
|
|
res = _minimize_cg(f, x0, args, fprime, callback=callback, **opts)
|
||
|
|
|
||
|
|
if full_output:
|
||
|
|
retlist = res['x'], res['fun'], res['nfev'], res['njev'], res['status']
|
||
|
|
if retall:
|
||
|
|
retlist += (res['allvecs'], )
|
||
|
|
return retlist
|
||
|
|
else:
|
||
|
|
if retall:
|
||
|
|
return res['x'], res['allvecs']
|
||
|
|
else:
|
||
|
|
return res['x']
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_cg(fun, x0, args=(), jac=None, callback=None,
|
||
|
|
gtol=1e-5, norm=Inf, eps=_epsilon, maxiter=None,
|
||
|
|
disp=False, return_all=False,
|
||
|
|
**unknown_options):
|
||
|
|
"""
|
||
|
|
Minimization of scalar function of one or more variables using the
|
||
|
|
conjugate gradient algorithm.
|
||
|
|
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
disp : bool
|
||
|
|
Set to True to print convergence messages.
|
||
|
|
maxiter : int
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
gtol : float
|
||
|
|
Gradient norm must be less than `gtol` before successful
|
||
|
|
termination.
|
||
|
|
norm : float
|
||
|
|
Order of norm (Inf is max, -Inf is min).
|
||
|
|
eps : float or ndarray
|
||
|
|
If `jac` is approximated, use this value for the step size.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
f = fun
|
||
|
|
fprime = jac
|
||
|
|
epsilon = eps
|
||
|
|
retall = return_all
|
||
|
|
|
||
|
|
x0 = asarray(x0).flatten()
|
||
|
|
if maxiter is None:
|
||
|
|
maxiter = len(x0) * 200
|
||
|
|
func_calls, f = wrap_function(f, args)
|
||
|
|
if fprime is None:
|
||
|
|
grad_calls, myfprime = wrap_function(approx_fprime, (f, epsilon))
|
||
|
|
else:
|
||
|
|
grad_calls, myfprime = wrap_function(fprime, args)
|
||
|
|
gfk = myfprime(x0)
|
||
|
|
k = 0
|
||
|
|
xk = x0
|
||
|
|
|
||
|
|
# Sets the initial step guess to dx ~ 1
|
||
|
|
old_fval = f(xk)
|
||
|
|
old_old_fval = old_fval + np.linalg.norm(gfk) / 2
|
||
|
|
|
||
|
|
if retall:
|
||
|
|
allvecs = [xk]
|
||
|
|
warnflag = 0
|
||
|
|
pk = -gfk
|
||
|
|
gnorm = vecnorm(gfk, ord=norm)
|
||
|
|
|
||
|
|
sigma_3 = 0.01
|
||
|
|
|
||
|
|
while (gnorm > gtol) and (k < maxiter):
|
||
|
|
deltak = numpy.dot(gfk, gfk)
|
||
|
|
|
||
|
|
cached_step = [None]
|
||
|
|
|
||
|
|
def polak_ribiere_powell_step(alpha, gfkp1=None):
|
||
|
|
xkp1 = xk + alpha * pk
|
||
|
|
if gfkp1 is None:
|
||
|
|
gfkp1 = myfprime(xkp1)
|
||
|
|
yk = gfkp1 - gfk
|
||
|
|
beta_k = max(0, numpy.dot(yk, gfkp1) / deltak)
|
||
|
|
pkp1 = -gfkp1 + beta_k * pk
|
||
|
|
gnorm = vecnorm(gfkp1, ord=norm)
|
||
|
|
return (alpha, xkp1, pkp1, gfkp1, gnorm)
|
||
|
|
|
||
|
|
def descent_condition(alpha, xkp1, fp1, gfkp1):
|
||
|
|
# Polak-Ribiere+ needs an explicit check of a sufficient
|
||
|
|
# descent condition, which is not guaranteed by strong Wolfe.
|
||
|
|
#
|
||
|
|
# See Gilbert & Nocedal, "Global convergence properties of
|
||
|
|
# conjugate gradient methods for optimization",
|
||
|
|
# SIAM J. Optimization 2, 21 (1992).
|
||
|
|
cached_step[:] = polak_ribiere_powell_step(alpha, gfkp1)
|
||
|
|
alpha, xk, pk, gfk, gnorm = cached_step
|
||
|
|
|
||
|
|
# Accept step if it leads to convergence.
|
||
|
|
if gnorm <= gtol:
|
||
|
|
return True
|
||
|
|
|
||
|
|
# Accept step if sufficient descent condition applies.
|
||
|
|
return numpy.dot(pk, gfk) <= -sigma_3 * numpy.dot(gfk, gfk)
|
||
|
|
|
||
|
|
try:
|
||
|
|
alpha_k, fc, gc, old_fval, old_old_fval, gfkp1 = \
|
||
|
|
_line_search_wolfe12(f, myfprime, xk, pk, gfk, old_fval,
|
||
|
|
old_old_fval, c2=0.4, amin=1e-100, amax=1e100,
|
||
|
|
extra_condition=descent_condition)
|
||
|
|
except _LineSearchError:
|
||
|
|
# Line search failed to find a better solution.
|
||
|
|
warnflag = 2
|
||
|
|
break
|
||
|
|
|
||
|
|
# Reuse already computed results if possible
|
||
|
|
if alpha_k == cached_step[0]:
|
||
|
|
alpha_k, xk, pk, gfk, gnorm = cached_step
|
||
|
|
else:
|
||
|
|
alpha_k, xk, pk, gfk, gnorm = polak_ribiere_powell_step(alpha_k, gfkp1)
|
||
|
|
|
||
|
|
if retall:
|
||
|
|
allvecs.append(xk)
|
||
|
|
if callback is not None:
|
||
|
|
callback(xk)
|
||
|
|
k += 1
|
||
|
|
|
||
|
|
fval = old_fval
|
||
|
|
if warnflag == 2:
|
||
|
|
msg = _status_message['pr_loss']
|
||
|
|
elif k >= maxiter:
|
||
|
|
warnflag = 1
|
||
|
|
msg = _status_message['maxiter']
|
||
|
|
else:
|
||
|
|
msg = _status_message['success']
|
||
|
|
|
||
|
|
if disp:
|
||
|
|
print("%s%s" % ("Warning: " if warnflag != 0 else "", msg))
|
||
|
|
print(" Current function value: %f" % fval)
|
||
|
|
print(" Iterations: %d" % k)
|
||
|
|
print(" Function evaluations: %d" % func_calls[0])
|
||
|
|
print(" Gradient evaluations: %d" % grad_calls[0])
|
||
|
|
|
||
|
|
result = OptimizeResult(fun=fval, jac=gfk, nfev=func_calls[0],
|
||
|
|
njev=grad_calls[0], status=warnflag,
|
||
|
|
success=(warnflag == 0), message=msg, x=xk,
|
||
|
|
nit=k)
|
||
|
|
if retall:
|
||
|
|
result['allvecs'] = allvecs
|
||
|
|
return result
|
||
|
|
|
||
|
|
|
||
|
|
def fmin_ncg(f, x0, fprime, fhess_p=None, fhess=None, args=(), avextol=1e-5,
|
||
|
|
epsilon=_epsilon, maxiter=None, full_output=0, disp=1, retall=0,
|
||
|
|
callback=None):
|
||
|
|
"""
|
||
|
|
Unconstrained minimization of a function using the Newton-CG method.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
f : callable ``f(x, *args)``
|
||
|
|
Objective function to be minimized.
|
||
|
|
x0 : ndarray
|
||
|
|
Initial guess.
|
||
|
|
fprime : callable ``f'(x, *args)``
|
||
|
|
Gradient of f.
|
||
|
|
fhess_p : callable ``fhess_p(x, p, *args)``, optional
|
||
|
|
Function which computes the Hessian of f times an
|
||
|
|
arbitrary vector, p.
|
||
|
|
fhess : callable ``fhess(x, *args)``, optional
|
||
|
|
Function to compute the Hessian matrix of f.
|
||
|
|
args : tuple, optional
|
||
|
|
Extra arguments passed to f, fprime, fhess_p, and fhess
|
||
|
|
(the same set of extra arguments is supplied to all of
|
||
|
|
these functions).
|
||
|
|
epsilon : float or ndarray, optional
|
||
|
|
If fhess is approximated, use this value for the step size.
|
||
|
|
callback : callable, optional
|
||
|
|
An optional user-supplied function which is called after
|
||
|
|
each iteration. Called as callback(xk), where xk is the
|
||
|
|
current parameter vector.
|
||
|
|
avextol : float, optional
|
||
|
|
Convergence is assumed when the average relative error in
|
||
|
|
the minimizer falls below this amount.
|
||
|
|
maxiter : int, optional
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
full_output : bool, optional
|
||
|
|
If True, return the optional outputs.
|
||
|
|
disp : bool, optional
|
||
|
|
If True, print convergence message.
|
||
|
|
retall : bool, optional
|
||
|
|
If True, return a list of results at each iteration.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
xopt : ndarray
|
||
|
|
Parameters which minimize f, i.e. ``f(xopt) == fopt``.
|
||
|
|
fopt : float
|
||
|
|
Value of the function at xopt, i.e. ``fopt = f(xopt)``.
|
||
|
|
fcalls : int
|
||
|
|
Number of function calls made.
|
||
|
|
gcalls : int
|
||
|
|
Number of gradient calls made.
|
||
|
|
hcalls : int
|
||
|
|
Number of hessian calls made.
|
||
|
|
warnflag : int
|
||
|
|
Warnings generated by the algorithm.
|
||
|
|
1 : Maximum number of iterations exceeded.
|
||
|
|
allvecs : list
|
||
|
|
The result at each iteration, if retall is True (see below).
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
minimize: Interface to minimization algorithms for multivariate
|
||
|
|
functions. See the 'Newton-CG' `method` in particular.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Only one of `fhess_p` or `fhess` need to be given. If `fhess`
|
||
|
|
is provided, then `fhess_p` will be ignored. If neither `fhess`
|
||
|
|
nor `fhess_p` is provided, then the hessian product will be
|
||
|
|
approximated using finite differences on `fprime`. `fhess_p`
|
||
|
|
must compute the hessian times an arbitrary vector. If it is not
|
||
|
|
given, finite-differences on `fprime` are used to compute
|
||
|
|
it.
|
||
|
|
|
||
|
|
Newton-CG methods are also called truncated Newton methods. This
|
||
|
|
function differs from scipy.optimize.fmin_tnc because
|
||
|
|
|
||
|
|
1. scipy.optimize.fmin_ncg is written purely in python using numpy
|
||
|
|
and scipy while scipy.optimize.fmin_tnc calls a C function.
|
||
|
|
2. scipy.optimize.fmin_ncg is only for unconstrained minimization
|
||
|
|
while scipy.optimize.fmin_tnc is for unconstrained minimization
|
||
|
|
or box constrained minimization. (Box constraints give
|
||
|
|
lower and upper bounds for each variable separately.)
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
Wright & Nocedal, 'Numerical Optimization', 1999, pg. 140.
|
||
|
|
|
||
|
|
"""
|
||
|
|
opts = {'xtol': avextol,
|
||
|
|
'eps': epsilon,
|
||
|
|
'maxiter': maxiter,
|
||
|
|
'disp': disp,
|
||
|
|
'return_all': retall}
|
||
|
|
|
||
|
|
res = _minimize_newtoncg(f, x0, args, fprime, fhess, fhess_p,
|
||
|
|
callback=callback, **opts)
|
||
|
|
|
||
|
|
if full_output:
|
||
|
|
retlist = (res['x'], res['fun'], res['nfev'], res['njev'],
|
||
|
|
res['nhev'], res['status'])
|
||
|
|
if retall:
|
||
|
|
retlist += (res['allvecs'], )
|
||
|
|
return retlist
|
||
|
|
else:
|
||
|
|
if retall:
|
||
|
|
return res['x'], res['allvecs']
|
||
|
|
else:
|
||
|
|
return res['x']
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_newtoncg(fun, x0, args=(), jac=None, hess=None, hessp=None,
|
||
|
|
callback=None, xtol=1e-5, eps=_epsilon, maxiter=None,
|
||
|
|
disp=False, return_all=False,
|
||
|
|
**unknown_options):
|
||
|
|
"""
|
||
|
|
Minimization of scalar function of one or more variables using the
|
||
|
|
Newton-CG algorithm.
|
||
|
|
|
||
|
|
Note that the `jac` parameter (Jacobian) is required.
|
||
|
|
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
disp : bool
|
||
|
|
Set to True to print convergence messages.
|
||
|
|
xtol : float
|
||
|
|
Average relative error in solution `xopt` acceptable for
|
||
|
|
convergence.
|
||
|
|
maxiter : int
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
eps : float or ndarray
|
||
|
|
If `jac` is approximated, use this value for the step size.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
if jac is None:
|
||
|
|
raise ValueError('Jacobian is required for Newton-CG method')
|
||
|
|
f = fun
|
||
|
|
fprime = jac
|
||
|
|
fhess_p = hessp
|
||
|
|
fhess = hess
|
||
|
|
avextol = xtol
|
||
|
|
epsilon = eps
|
||
|
|
retall = return_all
|
||
|
|
|
||
|
|
def terminate(warnflag, msg):
|
||
|
|
if disp:
|
||
|
|
print(msg)
|
||
|
|
print(" Current function value: %f" % old_fval)
|
||
|
|
print(" Iterations: %d" % k)
|
||
|
|
print(" Function evaluations: %d" % fcalls[0])
|
||
|
|
print(" Gradient evaluations: %d" % gcalls[0])
|
||
|
|
print(" Hessian evaluations: %d" % hcalls)
|
||
|
|
fval = old_fval
|
||
|
|
result = OptimizeResult(fun=fval, jac=gfk, nfev=fcalls[0],
|
||
|
|
njev=gcalls[0], nhev=hcalls, status=warnflag,
|
||
|
|
success=(warnflag == 0), message=msg, x=xk,
|
||
|
|
nit=k)
|
||
|
|
if retall:
|
||
|
|
result['allvecs'] = allvecs
|
||
|
|
return result
|
||
|
|
|
||
|
|
x0 = asarray(x0).flatten()
|
||
|
|
fcalls, f = wrap_function(f, args)
|
||
|
|
gcalls, fprime = wrap_function(fprime, args)
|
||
|
|
hcalls = 0
|
||
|
|
if maxiter is None:
|
||
|
|
maxiter = len(x0)*200
|
||
|
|
cg_maxiter = 20*len(x0)
|
||
|
|
|
||
|
|
xtol = len(x0) * avextol
|
||
|
|
update = [2 * xtol]
|
||
|
|
xk = x0
|
||
|
|
if retall:
|
||
|
|
allvecs = [xk]
|
||
|
|
k = 0
|
||
|
|
gfk = None
|
||
|
|
old_fval = f(x0)
|
||
|
|
old_old_fval = None
|
||
|
|
float64eps = numpy.finfo(numpy.float64).eps
|
||
|
|
while numpy.add.reduce(numpy.abs(update)) > xtol:
|
||
|
|
if k >= maxiter:
|
||
|
|
msg = "Warning: " + _status_message['maxiter']
|
||
|
|
return terminate(1, msg)
|
||
|
|
# Compute a search direction pk by applying the CG method to
|
||
|
|
# del2 f(xk) p = - grad f(xk) starting from 0.
|
||
|
|
b = -fprime(xk)
|
||
|
|
maggrad = numpy.add.reduce(numpy.abs(b))
|
||
|
|
eta = numpy.min([0.5, numpy.sqrt(maggrad)])
|
||
|
|
termcond = eta * maggrad
|
||
|
|
xsupi = zeros(len(x0), dtype=x0.dtype)
|
||
|
|
ri = -b
|
||
|
|
psupi = -ri
|
||
|
|
i = 0
|
||
|
|
dri0 = numpy.dot(ri, ri)
|
||
|
|
|
||
|
|
if fhess is not None: # you want to compute hessian once.
|
||
|
|
A = fhess(*(xk,) + args)
|
||
|
|
hcalls = hcalls + 1
|
||
|
|
|
||
|
|
for k2 in xrange(cg_maxiter):
|
||
|
|
if numpy.add.reduce(numpy.abs(ri)) <= termcond:
|
||
|
|
break
|
||
|
|
if fhess is None:
|
||
|
|
if fhess_p is None:
|
||
|
|
Ap = approx_fhess_p(xk, psupi, fprime, epsilon)
|
||
|
|
else:
|
||
|
|
Ap = fhess_p(xk, psupi, *args)
|
||
|
|
hcalls = hcalls + 1
|
||
|
|
else:
|
||
|
|
Ap = numpy.dot(A, psupi)
|
||
|
|
# check curvature
|
||
|
|
Ap = asarray(Ap).squeeze() # get rid of matrices...
|
||
|
|
curv = numpy.dot(psupi, Ap)
|
||
|
|
if 0 <= curv <= 3 * float64eps:
|
||
|
|
break
|
||
|
|
elif curv < 0:
|
||
|
|
if (i > 0):
|
||
|
|
break
|
||
|
|
else:
|
||
|
|
# fall back to steepest descent direction
|
||
|
|
xsupi = dri0 / (-curv) * b
|
||
|
|
break
|
||
|
|
alphai = dri0 / curv
|
||
|
|
xsupi = xsupi + alphai * psupi
|
||
|
|
ri = ri + alphai * Ap
|
||
|
|
dri1 = numpy.dot(ri, ri)
|
||
|
|
betai = dri1 / dri0
|
||
|
|
psupi = -ri + betai * psupi
|
||
|
|
i = i + 1
|
||
|
|
dri0 = dri1 # update numpy.dot(ri,ri) for next time.
|
||
|
|
else:
|
||
|
|
# curvature keeps increasing, bail out
|
||
|
|
msg = ("Warning: CG iterations didn't converge. The Hessian is not "
|
||
|
|
"positive definite.")
|
||
|
|
return terminate(3, msg)
|
||
|
|
|
||
|
|
pk = xsupi # search direction is solution to system.
|
||
|
|
gfk = -b # gradient at xk
|
||
|
|
|
||
|
|
try:
|
||
|
|
alphak, fc, gc, old_fval, old_old_fval, gfkp1 = \
|
||
|
|
_line_search_wolfe12(f, fprime, xk, pk, gfk,
|
||
|
|
old_fval, old_old_fval)
|
||
|
|
except _LineSearchError:
|
||
|
|
# Line search failed to find a better solution.
|
||
|
|
msg = "Warning: " + _status_message['pr_loss']
|
||
|
|
return terminate(2, msg)
|
||
|
|
|
||
|
|
update = alphak * pk
|
||
|
|
xk = xk + update # upcast if necessary
|
||
|
|
if callback is not None:
|
||
|
|
callback(xk)
|
||
|
|
if retall:
|
||
|
|
allvecs.append(xk)
|
||
|
|
k += 1
|
||
|
|
else:
|
||
|
|
msg = _status_message['success']
|
||
|
|
return terminate(0, msg)
|
||
|
|
|
||
|
|
|
||
|
|
def fminbound(func, x1, x2, args=(), xtol=1e-5, maxfun=500,
|
||
|
|
full_output=0, disp=1):
|
||
|
|
"""Bounded minimization for scalar functions.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable f(x,*args)
|
||
|
|
Objective function to be minimized (must accept and return scalars).
|
||
|
|
x1, x2 : float or array scalar
|
||
|
|
The optimization bounds.
|
||
|
|
args : tuple, optional
|
||
|
|
Extra arguments passed to function.
|
||
|
|
xtol : float, optional
|
||
|
|
The convergence tolerance.
|
||
|
|
maxfun : int, optional
|
||
|
|
Maximum number of function evaluations allowed.
|
||
|
|
full_output : bool, optional
|
||
|
|
If True, return optional outputs.
|
||
|
|
disp : int, optional
|
||
|
|
If non-zero, print messages.
|
||
|
|
0 : no message printing.
|
||
|
|
1 : non-convergence notification messages only.
|
||
|
|
2 : print a message on convergence too.
|
||
|
|
3 : print iteration results.
|
||
|
|
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
xopt : ndarray
|
||
|
|
Parameters (over given interval) which minimize the
|
||
|
|
objective function.
|
||
|
|
fval : number
|
||
|
|
The function value at the minimum point.
|
||
|
|
ierr : int
|
||
|
|
An error flag (0 if converged, 1 if maximum number of
|
||
|
|
function calls reached).
|
||
|
|
numfunc : int
|
||
|
|
The number of function calls made.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
minimize_scalar: Interface to minimization algorithms for scalar
|
||
|
|
univariate functions. See the 'Bounded' `method` in particular.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Finds a local minimizer of the scalar function `func` in the
|
||
|
|
interval x1 < xopt < x2 using Brent's method. (See `brent`
|
||
|
|
for auto-bracketing).
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
`fminbound` finds the minimum of the function in the given range.
|
||
|
|
The following examples illustrate the same
|
||
|
|
|
||
|
|
>>> def f(x):
|
||
|
|
... return x**2
|
||
|
|
|
||
|
|
>>> from scipy import optimize
|
||
|
|
|
||
|
|
>>> minimum = optimize.fminbound(f, -1, 2)
|
||
|
|
>>> minimum
|
||
|
|
0.0
|
||
|
|
>>> minimum = optimize.fminbound(f, 1, 2)
|
||
|
|
>>> minimum
|
||
|
|
1.0000059608609866
|
||
|
|
"""
|
||
|
|
options = {'xatol': xtol,
|
||
|
|
'maxiter': maxfun,
|
||
|
|
'disp': disp}
|
||
|
|
|
||
|
|
res = _minimize_scalar_bounded(func, (x1, x2), args, **options)
|
||
|
|
if full_output:
|
||
|
|
return res['x'], res['fun'], res['status'], res['nfev']
|
||
|
|
else:
|
||
|
|
return res['x']
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_scalar_bounded(func, bounds, args=(),
|
||
|
|
xatol=1e-5, maxiter=500, disp=0,
|
||
|
|
**unknown_options):
|
||
|
|
"""
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
maxiter : int
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
disp: int, optional
|
||
|
|
If non-zero, print messages.
|
||
|
|
0 : no message printing.
|
||
|
|
1 : non-convergence notification messages only.
|
||
|
|
2 : print a message on convergence too.
|
||
|
|
3 : print iteration results.
|
||
|
|
xatol : float
|
||
|
|
Absolute error in solution `xopt` acceptable for convergence.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
maxfun = maxiter
|
||
|
|
# Test bounds are of correct form
|
||
|
|
if len(bounds) != 2:
|
||
|
|
raise ValueError('bounds must have two elements.')
|
||
|
|
x1, x2 = bounds
|
||
|
|
|
||
|
|
if not (is_array_scalar(x1) and is_array_scalar(x2)):
|
||
|
|
raise ValueError("Optimisation bounds must be scalars"
|
||
|
|
" or array scalars.")
|
||
|
|
if x1 > x2:
|
||
|
|
raise ValueError("The lower bound exceeds the upper bound.")
|
||
|
|
|
||
|
|
flag = 0
|
||
|
|
header = ' Func-count x f(x) Procedure'
|
||
|
|
step = ' initial'
|
||
|
|
|
||
|
|
sqrt_eps = sqrt(2.2e-16)
|
||
|
|
golden_mean = 0.5 * (3.0 - sqrt(5.0))
|
||
|
|
a, b = x1, x2
|
||
|
|
fulc = a + golden_mean * (b - a)
|
||
|
|
nfc, xf = fulc, fulc
|
||
|
|
rat = e = 0.0
|
||
|
|
x = xf
|
||
|
|
fx = func(x, *args)
|
||
|
|
num = 1
|
||
|
|
fmin_data = (1, xf, fx)
|
||
|
|
|
||
|
|
ffulc = fnfc = fx
|
||
|
|
xm = 0.5 * (a + b)
|
||
|
|
tol1 = sqrt_eps * numpy.abs(xf) + xatol / 3.0
|
||
|
|
tol2 = 2.0 * tol1
|
||
|
|
|
||
|
|
if disp > 2:
|
||
|
|
print(" ")
|
||
|
|
print(header)
|
||
|
|
print("%5.0f %12.6g %12.6g %s" % (fmin_data + (step,)))
|
||
|
|
|
||
|
|
while (numpy.abs(xf - xm) > (tol2 - 0.5 * (b - a))):
|
||
|
|
golden = 1
|
||
|
|
# Check for parabolic fit
|
||
|
|
if numpy.abs(e) > tol1:
|
||
|
|
golden = 0
|
||
|
|
r = (xf - nfc) * (fx - ffulc)
|
||
|
|
q = (xf - fulc) * (fx - fnfc)
|
||
|
|
p = (xf - fulc) * q - (xf - nfc) * r
|
||
|
|
q = 2.0 * (q - r)
|
||
|
|
if q > 0.0:
|
||
|
|
p = -p
|
||
|
|
q = numpy.abs(q)
|
||
|
|
r = e
|
||
|
|
e = rat
|
||
|
|
|
||
|
|
# Check for acceptability of parabola
|
||
|
|
if ((numpy.abs(p) < numpy.abs(0.5*q*r)) and (p > q*(a - xf)) and
|
||
|
|
(p < q * (b - xf))):
|
||
|
|
rat = (p + 0.0) / q
|
||
|
|
x = xf + rat
|
||
|
|
step = ' parabolic'
|
||
|
|
|
||
|
|
if ((x - a) < tol2) or ((b - x) < tol2):
|
||
|
|
si = numpy.sign(xm - xf) + ((xm - xf) == 0)
|
||
|
|
rat = tol1 * si
|
||
|
|
else: # do a golden section step
|
||
|
|
golden = 1
|
||
|
|
|
||
|
|
if golden: # Do a golden-section step
|
||
|
|
if xf >= xm:
|
||
|
|
e = a - xf
|
||
|
|
else:
|
||
|
|
e = b - xf
|
||
|
|
rat = golden_mean*e
|
||
|
|
step = ' golden'
|
||
|
|
|
||
|
|
si = numpy.sign(rat) + (rat == 0)
|
||
|
|
x = xf + si * numpy.max([numpy.abs(rat), tol1])
|
||
|
|
fu = func(x, *args)
|
||
|
|
num += 1
|
||
|
|
fmin_data = (num, x, fu)
|
||
|
|
if disp > 2:
|
||
|
|
print("%5.0f %12.6g %12.6g %s" % (fmin_data + (step,)))
|
||
|
|
|
||
|
|
if fu <= fx:
|
||
|
|
if x >= xf:
|
||
|
|
a = xf
|
||
|
|
else:
|
||
|
|
b = xf
|
||
|
|
fulc, ffulc = nfc, fnfc
|
||
|
|
nfc, fnfc = xf, fx
|
||
|
|
xf, fx = x, fu
|
||
|
|
else:
|
||
|
|
if x < xf:
|
||
|
|
a = x
|
||
|
|
else:
|
||
|
|
b = x
|
||
|
|
if (fu <= fnfc) or (nfc == xf):
|
||
|
|
fulc, ffulc = nfc, fnfc
|
||
|
|
nfc, fnfc = x, fu
|
||
|
|
elif (fu <= ffulc) or (fulc == xf) or (fulc == nfc):
|
||
|
|
fulc, ffulc = x, fu
|
||
|
|
|
||
|
|
xm = 0.5 * (a + b)
|
||
|
|
tol1 = sqrt_eps * numpy.abs(xf) + xatol / 3.0
|
||
|
|
tol2 = 2.0 * tol1
|
||
|
|
|
||
|
|
if num >= maxfun:
|
||
|
|
flag = 1
|
||
|
|
break
|
||
|
|
|
||
|
|
fval = fx
|
||
|
|
if disp > 0:
|
||
|
|
_endprint(x, flag, fval, maxfun, xatol, disp)
|
||
|
|
|
||
|
|
result = OptimizeResult(fun=fval, status=flag, success=(flag == 0),
|
||
|
|
message={0: 'Solution found.',
|
||
|
|
1: 'Maximum number of function calls '
|
||
|
|
'reached.'}.get(flag, ''),
|
||
|
|
x=xf, nfev=num)
|
||
|
|
|
||
|
|
return result
|
||
|
|
|
||
|
|
|
||
|
|
class Brent:
|
||
|
|
#need to rethink design of __init__
|
||
|
|
def __init__(self, func, args=(), tol=1.48e-8, maxiter=500,
|
||
|
|
full_output=0):
|
||
|
|
self.func = func
|
||
|
|
self.args = args
|
||
|
|
self.tol = tol
|
||
|
|
self.maxiter = maxiter
|
||
|
|
self._mintol = 1.0e-11
|
||
|
|
self._cg = 0.3819660
|
||
|
|
self.xmin = None
|
||
|
|
self.fval = None
|
||
|
|
self.iter = 0
|
||
|
|
self.funcalls = 0
|
||
|
|
|
||
|
|
# need to rethink design of set_bracket (new options, etc)
|
||
|
|
def set_bracket(self, brack=None):
|
||
|
|
self.brack = brack
|
||
|
|
|
||
|
|
def get_bracket_info(self):
|
||
|
|
#set up
|
||
|
|
func = self.func
|
||
|
|
args = self.args
|
||
|
|
brack = self.brack
|
||
|
|
### BEGIN core bracket_info code ###
|
||
|
|
### carefully DOCUMENT any CHANGES in core ##
|
||
|
|
if brack is None:
|
||
|
|
xa, xb, xc, fa, fb, fc, funcalls = bracket(func, args=args)
|
||
|
|
elif len(brack) == 2:
|
||
|
|
xa, xb, xc, fa, fb, fc, funcalls = bracket(func, xa=brack[0],
|
||
|
|
xb=brack[1], args=args)
|
||
|
|
elif len(brack) == 3:
|
||
|
|
xa, xb, xc = brack
|
||
|
|
if (xa > xc): # swap so xa < xc can be assumed
|
||
|
|
xc, xa = xa, xc
|
||
|
|
if not ((xa < xb) and (xb < xc)):
|
||
|
|
raise ValueError("Not a bracketing interval.")
|
||
|
|
fa = func(*((xa,) + args))
|
||
|
|
fb = func(*((xb,) + args))
|
||
|
|
fc = func(*((xc,) + args))
|
||
|
|
if not ((fb < fa) and (fb < fc)):
|
||
|
|
raise ValueError("Not a bracketing interval.")
|
||
|
|
funcalls = 3
|
||
|
|
else:
|
||
|
|
raise ValueError("Bracketing interval must be "
|
||
|
|
"length 2 or 3 sequence.")
|
||
|
|
### END core bracket_info code ###
|
||
|
|
|
||
|
|
return xa, xb, xc, fa, fb, fc, funcalls
|
||
|
|
|
||
|
|
def optimize(self):
|
||
|
|
# set up for optimization
|
||
|
|
func = self.func
|
||
|
|
xa, xb, xc, fa, fb, fc, funcalls = self.get_bracket_info()
|
||
|
|
_mintol = self._mintol
|
||
|
|
_cg = self._cg
|
||
|
|
#################################
|
||
|
|
#BEGIN CORE ALGORITHM
|
||
|
|
#################################
|
||
|
|
x = w = v = xb
|
||
|
|
fw = fv = fx = func(*((x,) + self.args))
|
||
|
|
if (xa < xc):
|
||
|
|
a = xa
|
||
|
|
b = xc
|
||
|
|
else:
|
||
|
|
a = xc
|
||
|
|
b = xa
|
||
|
|
deltax = 0.0
|
||
|
|
funcalls += 1
|
||
|
|
iter = 0
|
||
|
|
while (iter < self.maxiter):
|
||
|
|
tol1 = self.tol * numpy.abs(x) + _mintol
|
||
|
|
tol2 = 2.0 * tol1
|
||
|
|
xmid = 0.5 * (a + b)
|
||
|
|
# check for convergence
|
||
|
|
if numpy.abs(x - xmid) < (tol2 - 0.5 * (b - a)):
|
||
|
|
break
|
||
|
|
# XXX In the first iteration, rat is only bound in the true case
|
||
|
|
# of this conditional. This used to cause an UnboundLocalError
|
||
|
|
# (gh-4140). It should be set before the if (but to what?).
|
||
|
|
if (numpy.abs(deltax) <= tol1):
|
||
|
|
if (x >= xmid):
|
||
|
|
deltax = a - x # do a golden section step
|
||
|
|
else:
|
||
|
|
deltax = b - x
|
||
|
|
rat = _cg * deltax
|
||
|
|
else: # do a parabolic step
|
||
|
|
tmp1 = (x - w) * (fx - fv)
|
||
|
|
tmp2 = (x - v) * (fx - fw)
|
||
|
|
p = (x - v) * tmp2 - (x - w) * tmp1
|
||
|
|
tmp2 = 2.0 * (tmp2 - tmp1)
|
||
|
|
if (tmp2 > 0.0):
|
||
|
|
p = -p
|
||
|
|
tmp2 = numpy.abs(tmp2)
|
||
|
|
dx_temp = deltax
|
||
|
|
deltax = rat
|
||
|
|
# check parabolic fit
|
||
|
|
if ((p > tmp2 * (a - x)) and (p < tmp2 * (b - x)) and
|
||
|
|
(numpy.abs(p) < numpy.abs(0.5 * tmp2 * dx_temp))):
|
||
|
|
rat = p * 1.0 / tmp2 # if parabolic step is useful.
|
||
|
|
u = x + rat
|
||
|
|
if ((u - a) < tol2 or (b - u) < tol2):
|
||
|
|
if xmid - x >= 0:
|
||
|
|
rat = tol1
|
||
|
|
else:
|
||
|
|
rat = -tol1
|
||
|
|
else:
|
||
|
|
if (x >= xmid):
|
||
|
|
deltax = a - x # if it's not do a golden section step
|
||
|
|
else:
|
||
|
|
deltax = b - x
|
||
|
|
rat = _cg * deltax
|
||
|
|
|
||
|
|
if (numpy.abs(rat) < tol1): # update by at least tol1
|
||
|
|
if rat >= 0:
|
||
|
|
u = x + tol1
|
||
|
|
else:
|
||
|
|
u = x - tol1
|
||
|
|
else:
|
||
|
|
u = x + rat
|
||
|
|
fu = func(*((u,) + self.args)) # calculate new output value
|
||
|
|
funcalls += 1
|
||
|
|
|
||
|
|
if (fu > fx): # if it's bigger than current
|
||
|
|
if (u < x):
|
||
|
|
a = u
|
||
|
|
else:
|
||
|
|
b = u
|
||
|
|
if (fu <= fw) or (w == x):
|
||
|
|
v = w
|
||
|
|
w = u
|
||
|
|
fv = fw
|
||
|
|
fw = fu
|
||
|
|
elif (fu <= fv) or (v == x) or (v == w):
|
||
|
|
v = u
|
||
|
|
fv = fu
|
||
|
|
else:
|
||
|
|
if (u >= x):
|
||
|
|
a = x
|
||
|
|
else:
|
||
|
|
b = x
|
||
|
|
v = w
|
||
|
|
w = x
|
||
|
|
x = u
|
||
|
|
fv = fw
|
||
|
|
fw = fx
|
||
|
|
fx = fu
|
||
|
|
|
||
|
|
iter += 1
|
||
|
|
#################################
|
||
|
|
#END CORE ALGORITHM
|
||
|
|
#################################
|
||
|
|
|
||
|
|
self.xmin = x
|
||
|
|
self.fval = fx
|
||
|
|
self.iter = iter
|
||
|
|
self.funcalls = funcalls
|
||
|
|
|
||
|
|
def get_result(self, full_output=False):
|
||
|
|
if full_output:
|
||
|
|
return self.xmin, self.fval, self.iter, self.funcalls
|
||
|
|
else:
|
||
|
|
return self.xmin
|
||
|
|
|
||
|
|
|
||
|
|
def brent(func, args=(), brack=None, tol=1.48e-8, full_output=0, maxiter=500):
|
||
|
|
"""
|
||
|
|
Given a function of one-variable and a possible bracket, return
|
||
|
|
the local minimum of the function isolated to a fractional precision
|
||
|
|
of tol.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable f(x,*args)
|
||
|
|
Objective function.
|
||
|
|
args : tuple, optional
|
||
|
|
Additional arguments (if present).
|
||
|
|
brack : tuple, optional
|
||
|
|
Either a triple (xa,xb,xc) where xa<xb<xc and func(xb) <
|
||
|
|
func(xa), func(xc) or a pair (xa,xb) which are used as a
|
||
|
|
starting interval for a downhill bracket search (see
|
||
|
|
`bracket`). Providing the pair (xa,xb) does not always mean
|
||
|
|
the obtained solution will satisfy xa<=x<=xb.
|
||
|
|
tol : float, optional
|
||
|
|
Stop if between iteration change is less than `tol`.
|
||
|
|
full_output : bool, optional
|
||
|
|
If True, return all output args (xmin, fval, iter,
|
||
|
|
funcalls).
|
||
|
|
maxiter : int, optional
|
||
|
|
Maximum number of iterations in solution.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
xmin : ndarray
|
||
|
|
Optimum point.
|
||
|
|
fval : float
|
||
|
|
Optimum value.
|
||
|
|
iter : int
|
||
|
|
Number of iterations.
|
||
|
|
funcalls : int
|
||
|
|
Number of objective function evaluations made.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
minimize_scalar: Interface to minimization algorithms for scalar
|
||
|
|
univariate functions. See the 'Brent' `method` in particular.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Uses inverse parabolic interpolation when possible to speed up
|
||
|
|
convergence of golden section method.
|
||
|
|
|
||
|
|
Does not ensure that the minimum lies in the range specified by
|
||
|
|
`brack`. See `fminbound`.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
We illustrate the behaviour of the function when `brack` is of
|
||
|
|
size 2 and 3 respectively. In the case where `brack` is of the
|
||
|
|
form (xa,xb), we can see for the given values, the output need
|
||
|
|
not necessarily lie in the range (xa,xb).
|
||
|
|
|
||
|
|
>>> def f(x):
|
||
|
|
... return x**2
|
||
|
|
|
||
|
|
>>> from scipy import optimize
|
||
|
|
|
||
|
|
>>> minimum = optimize.brent(f,brack=(1,2))
|
||
|
|
>>> minimum
|
||
|
|
0.0
|
||
|
|
>>> minimum = optimize.brent(f,brack=(-1,0.5,2))
|
||
|
|
>>> minimum
|
||
|
|
-2.7755575615628914e-17
|
||
|
|
|
||
|
|
"""
|
||
|
|
options = {'xtol': tol,
|
||
|
|
'maxiter': maxiter}
|
||
|
|
res = _minimize_scalar_brent(func, brack, args, **options)
|
||
|
|
if full_output:
|
||
|
|
return res['x'], res['fun'], res['nit'], res['nfev']
|
||
|
|
else:
|
||
|
|
return res['x']
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_scalar_brent(func, brack=None, args=(),
|
||
|
|
xtol=1.48e-8, maxiter=500,
|
||
|
|
**unknown_options):
|
||
|
|
"""
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
maxiter : int
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
xtol : float
|
||
|
|
Relative error in solution `xopt` acceptable for convergence.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Uses inverse parabolic interpolation when possible to speed up
|
||
|
|
convergence of golden section method.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
tol = xtol
|
||
|
|
if tol < 0:
|
||
|
|
raise ValueError('tolerance should be >= 0, got %r' % tol)
|
||
|
|
|
||
|
|
brent = Brent(func=func, args=args, tol=tol,
|
||
|
|
full_output=True, maxiter=maxiter)
|
||
|
|
brent.set_bracket(brack)
|
||
|
|
brent.optimize()
|
||
|
|
x, fval, nit, nfev = brent.get_result(full_output=True)
|
||
|
|
return OptimizeResult(fun=fval, x=x, nit=nit, nfev=nfev,
|
||
|
|
success=nit < maxiter)
|
||
|
|
|
||
|
|
|
||
|
|
def golden(func, args=(), brack=None, tol=_epsilon,
|
||
|
|
full_output=0, maxiter=5000):
|
||
|
|
"""
|
||
|
|
Return the minimum of a function of one variable using golden section
|
||
|
|
method.
|
||
|
|
|
||
|
|
Given a function of one variable and a possible bracketing interval,
|
||
|
|
return the minimum of the function isolated to a fractional precision of
|
||
|
|
tol.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable func(x,*args)
|
||
|
|
Objective function to minimize.
|
||
|
|
args : tuple, optional
|
||
|
|
Additional arguments (if present), passed to func.
|
||
|
|
brack : tuple, optional
|
||
|
|
Triple (a,b,c), where (a<b<c) and func(b) <
|
||
|
|
func(a),func(c). If bracket consists of two numbers (a,
|
||
|
|
c), then they are assumed to be a starting interval for a
|
||
|
|
downhill bracket search (see `bracket`); it doesn't always
|
||
|
|
mean that obtained solution will satisfy a<=x<=c.
|
||
|
|
tol : float, optional
|
||
|
|
x tolerance stop criterion
|
||
|
|
full_output : bool, optional
|
||
|
|
If True, return optional outputs.
|
||
|
|
maxiter : int
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
minimize_scalar: Interface to minimization algorithms for scalar
|
||
|
|
univariate functions. See the 'Golden' `method` in particular.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Uses analog of bisection method to decrease the bracketed
|
||
|
|
interval.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
We illustrate the behaviour of the function when `brack` is of
|
||
|
|
size 2 and 3 respectively. In the case where `brack` is of the
|
||
|
|
form (xa,xb), we can see for the given values, the output need
|
||
|
|
not necessarily lie in the range ``(xa, xb)``.
|
||
|
|
|
||
|
|
>>> def f(x):
|
||
|
|
... return x**2
|
||
|
|
|
||
|
|
>>> from scipy import optimize
|
||
|
|
|
||
|
|
>>> minimum = optimize.golden(f, brack=(1, 2))
|
||
|
|
>>> minimum
|
||
|
|
1.5717277788484873e-162
|
||
|
|
>>> minimum = optimize.golden(f, brack=(-1, 0.5, 2))
|
||
|
|
>>> minimum
|
||
|
|
-1.5717277788484873e-162
|
||
|
|
|
||
|
|
"""
|
||
|
|
options = {'xtol': tol, 'maxiter': maxiter}
|
||
|
|
res = _minimize_scalar_golden(func, brack, args, **options)
|
||
|
|
if full_output:
|
||
|
|
return res['x'], res['fun'], res['nfev']
|
||
|
|
else:
|
||
|
|
return res['x']
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_scalar_golden(func, brack=None, args=(),
|
||
|
|
xtol=_epsilon, maxiter=5000, **unknown_options):
|
||
|
|
"""
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
maxiter : int
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
xtol : float
|
||
|
|
Relative error in solution `xopt` acceptable for convergence.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
tol = xtol
|
||
|
|
if brack is None:
|
||
|
|
xa, xb, xc, fa, fb, fc, funcalls = bracket(func, args=args)
|
||
|
|
elif len(brack) == 2:
|
||
|
|
xa, xb, xc, fa, fb, fc, funcalls = bracket(func, xa=brack[0],
|
||
|
|
xb=brack[1], args=args)
|
||
|
|
elif len(brack) == 3:
|
||
|
|
xa, xb, xc = brack
|
||
|
|
if (xa > xc): # swap so xa < xc can be assumed
|
||
|
|
xc, xa = xa, xc
|
||
|
|
if not ((xa < xb) and (xb < xc)):
|
||
|
|
raise ValueError("Not a bracketing interval.")
|
||
|
|
fa = func(*((xa,) + args))
|
||
|
|
fb = func(*((xb,) + args))
|
||
|
|
fc = func(*((xc,) + args))
|
||
|
|
if not ((fb < fa) and (fb < fc)):
|
||
|
|
raise ValueError("Not a bracketing interval.")
|
||
|
|
funcalls = 3
|
||
|
|
else:
|
||
|
|
raise ValueError("Bracketing interval must be length 2 or 3 sequence.")
|
||
|
|
|
||
|
|
_gR = 0.61803399 # golden ratio conjugate: 2.0/(1.0+sqrt(5.0))
|
||
|
|
_gC = 1.0 - _gR
|
||
|
|
x3 = xc
|
||
|
|
x0 = xa
|
||
|
|
if (numpy.abs(xc - xb) > numpy.abs(xb - xa)):
|
||
|
|
x1 = xb
|
||
|
|
x2 = xb + _gC * (xc - xb)
|
||
|
|
else:
|
||
|
|
x2 = xb
|
||
|
|
x1 = xb - _gC * (xb - xa)
|
||
|
|
f1 = func(*((x1,) + args))
|
||
|
|
f2 = func(*((x2,) + args))
|
||
|
|
funcalls += 2
|
||
|
|
nit = 0
|
||
|
|
for i in xrange(maxiter):
|
||
|
|
if numpy.abs(x3 - x0) <= tol * (numpy.abs(x1) + numpy.abs(x2)):
|
||
|
|
break
|
||
|
|
if (f2 < f1):
|
||
|
|
x0 = x1
|
||
|
|
x1 = x2
|
||
|
|
x2 = _gR * x1 + _gC * x3
|
||
|
|
f1 = f2
|
||
|
|
f2 = func(*((x2,) + args))
|
||
|
|
else:
|
||
|
|
x3 = x2
|
||
|
|
x2 = x1
|
||
|
|
x1 = _gR * x2 + _gC * x0
|
||
|
|
f2 = f1
|
||
|
|
f1 = func(*((x1,) + args))
|
||
|
|
funcalls += 1
|
||
|
|
nit += 1
|
||
|
|
if (f1 < f2):
|
||
|
|
xmin = x1
|
||
|
|
fval = f1
|
||
|
|
else:
|
||
|
|
xmin = x2
|
||
|
|
fval = f2
|
||
|
|
|
||
|
|
return OptimizeResult(fun=fval, nfev=funcalls, x=xmin, nit=nit,
|
||
|
|
success=nit < maxiter)
|
||
|
|
|
||
|
|
|
||
|
|
def bracket(func, xa=0.0, xb=1.0, args=(), grow_limit=110.0, maxiter=1000):
|
||
|
|
"""
|
||
|
|
Bracket the minimum of the function.
|
||
|
|
|
||
|
|
Given a function and distinct initial points, search in the
|
||
|
|
downhill direction (as defined by the initital points) and return
|
||
|
|
new points xa, xb, xc that bracket the minimum of the function
|
||
|
|
f(xa) > f(xb) < f(xc). It doesn't always mean that obtained
|
||
|
|
solution will satisfy xa<=x<=xb
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable f(x,*args)
|
||
|
|
Objective function to minimize.
|
||
|
|
xa, xb : float, optional
|
||
|
|
Bracketing interval. Defaults `xa` to 0.0, and `xb` to 1.0.
|
||
|
|
args : tuple, optional
|
||
|
|
Additional arguments (if present), passed to `func`.
|
||
|
|
grow_limit : float, optional
|
||
|
|
Maximum grow limit. Defaults to 110.0
|
||
|
|
maxiter : int, optional
|
||
|
|
Maximum number of iterations to perform. Defaults to 1000.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
xa, xb, xc : float
|
||
|
|
Bracket.
|
||
|
|
fa, fb, fc : float
|
||
|
|
Objective function values in bracket.
|
||
|
|
funcalls : int
|
||
|
|
Number of function evaluations made.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_gold = 1.618034 # golden ratio: (1.0+sqrt(5.0))/2.0
|
||
|
|
_verysmall_num = 1e-21
|
||
|
|
fa = func(*(xa,) + args)
|
||
|
|
fb = func(*(xb,) + args)
|
||
|
|
if (fa < fb): # Switch so fa > fb
|
||
|
|
xa, xb = xb, xa
|
||
|
|
fa, fb = fb, fa
|
||
|
|
xc = xb + _gold * (xb - xa)
|
||
|
|
fc = func(*((xc,) + args))
|
||
|
|
funcalls = 3
|
||
|
|
iter = 0
|
||
|
|
while (fc < fb):
|
||
|
|
tmp1 = (xb - xa) * (fb - fc)
|
||
|
|
tmp2 = (xb - xc) * (fb - fa)
|
||
|
|
val = tmp2 - tmp1
|
||
|
|
if numpy.abs(val) < _verysmall_num:
|
||
|
|
denom = 2.0 * _verysmall_num
|
||
|
|
else:
|
||
|
|
denom = 2.0 * val
|
||
|
|
w = xb - ((xb - xc) * tmp2 - (xb - xa) * tmp1) / denom
|
||
|
|
wlim = xb + grow_limit * (xc - xb)
|
||
|
|
if iter > maxiter:
|
||
|
|
raise RuntimeError("Too many iterations.")
|
||
|
|
iter += 1
|
||
|
|
if (w - xc) * (xb - w) > 0.0:
|
||
|
|
fw = func(*((w,) + args))
|
||
|
|
funcalls += 1
|
||
|
|
if (fw < fc):
|
||
|
|
xa = xb
|
||
|
|
xb = w
|
||
|
|
fa = fb
|
||
|
|
fb = fw
|
||
|
|
return xa, xb, xc, fa, fb, fc, funcalls
|
||
|
|
elif (fw > fb):
|
||
|
|
xc = w
|
||
|
|
fc = fw
|
||
|
|
return xa, xb, xc, fa, fb, fc, funcalls
|
||
|
|
w = xc + _gold * (xc - xb)
|
||
|
|
fw = func(*((w,) + args))
|
||
|
|
funcalls += 1
|
||
|
|
elif (w - wlim)*(wlim - xc) >= 0.0:
|
||
|
|
w = wlim
|
||
|
|
fw = func(*((w,) + args))
|
||
|
|
funcalls += 1
|
||
|
|
elif (w - wlim)*(xc - w) > 0.0:
|
||
|
|
fw = func(*((w,) + args))
|
||
|
|
funcalls += 1
|
||
|
|
if (fw < fc):
|
||
|
|
xb = xc
|
||
|
|
xc = w
|
||
|
|
w = xc + _gold * (xc - xb)
|
||
|
|
fb = fc
|
||
|
|
fc = fw
|
||
|
|
fw = func(*((w,) + args))
|
||
|
|
funcalls += 1
|
||
|
|
else:
|
||
|
|
w = xc + _gold * (xc - xb)
|
||
|
|
fw = func(*((w,) + args))
|
||
|
|
funcalls += 1
|
||
|
|
xa = xb
|
||
|
|
xb = xc
|
||
|
|
xc = w
|
||
|
|
fa = fb
|
||
|
|
fb = fc
|
||
|
|
fc = fw
|
||
|
|
return xa, xb, xc, fa, fb, fc, funcalls
|
||
|
|
|
||
|
|
|
||
|
|
def _linesearch_powell(func, p, xi, tol=1e-3):
|
||
|
|
"""Line-search algorithm using fminbound.
|
||
|
|
|
||
|
|
Find the minimium of the function ``func(x0+ alpha*direc)``.
|
||
|
|
|
||
|
|
"""
|
||
|
|
def myfunc(alpha):
|
||
|
|
return func(p + alpha*xi)
|
||
|
|
alpha_min, fret, iter, num = brent(myfunc, full_output=1, tol=tol)
|
||
|
|
xi = alpha_min*xi
|
||
|
|
return squeeze(fret), p + xi, xi
|
||
|
|
|
||
|
|
|
||
|
|
def fmin_powell(func, x0, args=(), xtol=1e-4, ftol=1e-4, maxiter=None,
|
||
|
|
maxfun=None, full_output=0, disp=1, retall=0, callback=None,
|
||
|
|
direc=None):
|
||
|
|
"""
|
||
|
|
Minimize a function using modified Powell's method. This method
|
||
|
|
only uses function values, not derivatives.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable f(x,*args)
|
||
|
|
Objective function to be minimized.
|
||
|
|
x0 : ndarray
|
||
|
|
Initial guess.
|
||
|
|
args : tuple, optional
|
||
|
|
Extra arguments passed to func.
|
||
|
|
callback : callable, optional
|
||
|
|
An optional user-supplied function, called after each
|
||
|
|
iteration. Called as ``callback(xk)``, where ``xk`` is the
|
||
|
|
current parameter vector.
|
||
|
|
direc : ndarray, optional
|
||
|
|
Initial direction set.
|
||
|
|
xtol : float, optional
|
||
|
|
Line-search error tolerance.
|
||
|
|
ftol : float, optional
|
||
|
|
Relative error in ``func(xopt)`` acceptable for convergence.
|
||
|
|
maxiter : int, optional
|
||
|
|
Maximum number of iterations to perform.
|
||
|
|
maxfun : int, optional
|
||
|
|
Maximum number of function evaluations to make.
|
||
|
|
full_output : bool, optional
|
||
|
|
If True, fopt, xi, direc, iter, funcalls, and
|
||
|
|
warnflag are returned.
|
||
|
|
disp : bool, optional
|
||
|
|
If True, print convergence messages.
|
||
|
|
retall : bool, optional
|
||
|
|
If True, return a list of the solution at each iteration.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
xopt : ndarray
|
||
|
|
Parameter which minimizes `func`.
|
||
|
|
fopt : number
|
||
|
|
Value of function at minimum: ``fopt = func(xopt)``.
|
||
|
|
direc : ndarray
|
||
|
|
Current direction set.
|
||
|
|
iter : int
|
||
|
|
Number of iterations.
|
||
|
|
funcalls : int
|
||
|
|
Number of function calls made.
|
||
|
|
warnflag : int
|
||
|
|
Integer warning flag:
|
||
|
|
1 : Maximum number of function evaluations.
|
||
|
|
2 : Maximum number of iterations.
|
||
|
|
allvecs : list
|
||
|
|
List of solutions at each iteration.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
minimize: Interface to unconstrained minimization algorithms for
|
||
|
|
multivariate functions. See the 'Powell' `method` in particular.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Uses a modification of Powell's method to find the minimum of
|
||
|
|
a function of N variables. Powell's method is a conjugate
|
||
|
|
direction method.
|
||
|
|
|
||
|
|
The algorithm has two loops. The outer loop
|
||
|
|
merely iterates over the inner loop. The inner loop minimizes
|
||
|
|
over each current direction in the direction set. At the end
|
||
|
|
of the inner loop, if certain conditions are met, the direction
|
||
|
|
that gave the largest decrease is dropped and replaced with
|
||
|
|
the difference between the current estimated x and the estimated
|
||
|
|
x from the beginning of the inner-loop.
|
||
|
|
|
||
|
|
The technical conditions for replacing the direction of greatest
|
||
|
|
increase amount to checking that
|
||
|
|
|
||
|
|
1. No further gain can be made along the direction of greatest increase
|
||
|
|
from that iteration.
|
||
|
|
2. The direction of greatest increase accounted for a large sufficient
|
||
|
|
fraction of the decrease in the function value from that iteration of
|
||
|
|
the inner loop.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> def f(x):
|
||
|
|
... return x**2
|
||
|
|
|
||
|
|
>>> from scipy import optimize
|
||
|
|
|
||
|
|
>>> minimum = optimize.fmin_powell(f, -1)
|
||
|
|
Optimization terminated successfully.
|
||
|
|
Current function value: 0.000000
|
||
|
|
Iterations: 2
|
||
|
|
Function evaluations: 18
|
||
|
|
>>> minimum
|
||
|
|
array(0.0)
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
Powell M.J.D. (1964) An efficient method for finding the minimum of a
|
||
|
|
function of several variables without calculating derivatives,
|
||
|
|
Computer Journal, 7 (2):155-162.
|
||
|
|
|
||
|
|
Press W., Teukolsky S.A., Vetterling W.T., and Flannery B.P.:
|
||
|
|
Numerical Recipes (any edition), Cambridge University Press
|
||
|
|
|
||
|
|
"""
|
||
|
|
opts = {'xtol': xtol,
|
||
|
|
'ftol': ftol,
|
||
|
|
'maxiter': maxiter,
|
||
|
|
'maxfev': maxfun,
|
||
|
|
'disp': disp,
|
||
|
|
'direc': direc,
|
||
|
|
'return_all': retall}
|
||
|
|
|
||
|
|
res = _minimize_powell(func, x0, args, callback=callback, **opts)
|
||
|
|
|
||
|
|
if full_output:
|
||
|
|
retlist = (res['x'], res['fun'], res['direc'], res['nit'],
|
||
|
|
res['nfev'], res['status'])
|
||
|
|
if retall:
|
||
|
|
retlist += (res['allvecs'], )
|
||
|
|
return retlist
|
||
|
|
else:
|
||
|
|
if retall:
|
||
|
|
return res['x'], res['allvecs']
|
||
|
|
else:
|
||
|
|
return res['x']
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_powell(func, x0, args=(), callback=None,
|
||
|
|
xtol=1e-4, ftol=1e-4, maxiter=None, maxfev=None,
|
||
|
|
disp=False, direc=None, return_all=False,
|
||
|
|
**unknown_options):
|
||
|
|
"""
|
||
|
|
Minimization of scalar function of one or more variables using the
|
||
|
|
modified Powell algorithm.
|
||
|
|
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
disp : bool
|
||
|
|
Set to True to print convergence messages.
|
||
|
|
xtol : float
|
||
|
|
Relative error in solution `xopt` acceptable for convergence.
|
||
|
|
ftol : float
|
||
|
|
Relative error in ``fun(xopt)`` acceptable for convergence.
|
||
|
|
maxiter, maxfev : int
|
||
|
|
Maximum allowed number of iterations and function evaluations.
|
||
|
|
Will default to ``N*1000``, where ``N`` is the number of
|
||
|
|
variables, if neither `maxiter` or `maxfev` is set. If both
|
||
|
|
`maxiter` and `maxfev` are set, minimization will stop at the
|
||
|
|
first reached.
|
||
|
|
direc : ndarray
|
||
|
|
Initial set of direction vectors for the Powell method.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
maxfun = maxfev
|
||
|
|
retall = return_all
|
||
|
|
# we need to use a mutable object here that we can update in the
|
||
|
|
# wrapper function
|
||
|
|
fcalls, func = wrap_function(func, args)
|
||
|
|
x = asarray(x0).flatten()
|
||
|
|
if retall:
|
||
|
|
allvecs = [x]
|
||
|
|
N = len(x)
|
||
|
|
# If neither are set, then set both to default
|
||
|
|
if maxiter is None and maxfun is None:
|
||
|
|
maxiter = N * 1000
|
||
|
|
maxfun = N * 1000
|
||
|
|
elif maxiter is None:
|
||
|
|
# Convert remaining Nones, to np.inf, unless the other is np.inf, in
|
||
|
|
# which case use the default to avoid unbounded iteration
|
||
|
|
if maxfun == np.inf:
|
||
|
|
maxiter = N * 1000
|
||
|
|
else:
|
||
|
|
maxiter = np.inf
|
||
|
|
elif maxfun is None:
|
||
|
|
if maxiter == np.inf:
|
||
|
|
maxfun = N * 1000
|
||
|
|
else:
|
||
|
|
maxfun = np.inf
|
||
|
|
|
||
|
|
if direc is None:
|
||
|
|
direc = eye(N, dtype=float)
|
||
|
|
else:
|
||
|
|
direc = asarray(direc, dtype=float)
|
||
|
|
|
||
|
|
fval = squeeze(func(x))
|
||
|
|
x1 = x.copy()
|
||
|
|
iter = 0
|
||
|
|
ilist = list(range(N))
|
||
|
|
while True:
|
||
|
|
fx = fval
|
||
|
|
bigind = 0
|
||
|
|
delta = 0.0
|
||
|
|
for i in ilist:
|
||
|
|
direc1 = direc[i]
|
||
|
|
fx2 = fval
|
||
|
|
fval, x, direc1 = _linesearch_powell(func, x, direc1,
|
||
|
|
tol=xtol * 100)
|
||
|
|
if (fx2 - fval) > delta:
|
||
|
|
delta = fx2 - fval
|
||
|
|
bigind = i
|
||
|
|
iter += 1
|
||
|
|
if callback is not None:
|
||
|
|
callback(x)
|
||
|
|
if retall:
|
||
|
|
allvecs.append(x)
|
||
|
|
bnd = ftol * (numpy.abs(fx) + numpy.abs(fval)) + 1e-20
|
||
|
|
if 2.0 * (fx - fval) <= bnd:
|
||
|
|
break
|
||
|
|
if fcalls[0] >= maxfun:
|
||
|
|
break
|
||
|
|
if iter >= maxiter:
|
||
|
|
break
|
||
|
|
|
||
|
|
# Construct the extrapolated point
|
||
|
|
direc1 = x - x1
|
||
|
|
x2 = 2*x - x1
|
||
|
|
x1 = x.copy()
|
||
|
|
fx2 = squeeze(func(x2))
|
||
|
|
|
||
|
|
if (fx > fx2):
|
||
|
|
t = 2.0*(fx + fx2 - 2.0*fval)
|
||
|
|
temp = (fx - fval - delta)
|
||
|
|
t *= temp*temp
|
||
|
|
temp = fx - fx2
|
||
|
|
t -= delta*temp*temp
|
||
|
|
if t < 0.0:
|
||
|
|
fval, x, direc1 = _linesearch_powell(func, x, direc1,
|
||
|
|
tol=xtol*100)
|
||
|
|
direc[bigind] = direc[-1]
|
||
|
|
direc[-1] = direc1
|
||
|
|
|
||
|
|
warnflag = 0
|
||
|
|
if fcalls[0] >= maxfun:
|
||
|
|
warnflag = 1
|
||
|
|
msg = _status_message['maxfev']
|
||
|
|
if disp:
|
||
|
|
print("Warning: " + msg)
|
||
|
|
elif iter >= maxiter:
|
||
|
|
warnflag = 2
|
||
|
|
msg = _status_message['maxiter']
|
||
|
|
if disp:
|
||
|
|
print("Warning: " + msg)
|
||
|
|
else:
|
||
|
|
msg = _status_message['success']
|
||
|
|
if disp:
|
||
|
|
print(msg)
|
||
|
|
print(" Current function value: %f" % fval)
|
||
|
|
print(" Iterations: %d" % iter)
|
||
|
|
print(" Function evaluations: %d" % fcalls[0])
|
||
|
|
|
||
|
|
x = squeeze(x)
|
||
|
|
|
||
|
|
result = OptimizeResult(fun=fval, direc=direc, nit=iter, nfev=fcalls[0],
|
||
|
|
status=warnflag, success=(warnflag == 0),
|
||
|
|
message=msg, x=x)
|
||
|
|
if retall:
|
||
|
|
result['allvecs'] = allvecs
|
||
|
|
return result
|
||
|
|
|
||
|
|
|
||
|
|
def _endprint(x, flag, fval, maxfun, xtol, disp):
|
||
|
|
if flag == 0:
|
||
|
|
if disp > 1:
|
||
|
|
print("\nOptimization terminated successfully;\n"
|
||
|
|
"The returned value satisfies the termination criteria\n"
|
||
|
|
"(using xtol = ", xtol, ")")
|
||
|
|
if flag == 1:
|
||
|
|
if disp:
|
||
|
|
print("\nMaximum number of function evaluations exceeded --- "
|
||
|
|
"increase maxfun argument.\n")
|
||
|
|
return
|
||
|
|
|
||
|
|
|
||
|
|
def brute(func, ranges, args=(), Ns=20, full_output=0, finish=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 ``vectorize`` NumPy
|
||
|
|
function. The value and type of the function evaluation returned when
|
||
|
|
``full_output=True`` are affected in addition by the ``finish`` argument
|
||
|
|
(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.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable
|
||
|
|
The objective function to be minimized. Must be in the
|
||
|
|
form ``f(x, *args)``, where ``x`` is the argument in
|
||
|
|
the form of a 1-D array and ``args`` is 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.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
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.)
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
basinhopping, differential_evolution
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
*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 `brute` function'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.
|
||
|
|
|
||
|
|
`brute` assumes that the `finish` function returns either an
|
||
|
|
`OptimizeResult` object or a tuple in the form:
|
||
|
|
``(xmin, Jmin, ... , statuscode)``, where ``xmin`` is the minimizing
|
||
|
|
value of the argument, ``Jmin`` is the minimum value of the objective
|
||
|
|
function, "..." may be some other returned values (which are not used
|
||
|
|
by `brute`), and ``statuscode`` is 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 `brute` confines 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 `numpy.mgrid` object.
|
||
|
|
For `brute` the `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, `brute` uses it directly. If the component is a two-tuple
|
||
|
|
range, `brute` internally converts it to a slice object that interpolates
|
||
|
|
`Ns` points from its low-value to its high-value, inclusive.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
We illustrate the use of `brute` to 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 ``params`` and 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 `fmin` to 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[0] # global minimum
|
||
|
|
array([-1.05665192, 1.80834843])
|
||
|
|
>>> resbrute[1] # 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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
N = len(ranges)
|
||
|
|
if N > 40:
|
||
|
|
raise ValueError("Brute Force not possible with more "
|
||
|
|
"than 40 variables.")
|
||
|
|
lrange = list(ranges)
|
||
|
|
for k in range(N):
|
||
|
|
if type(lrange[k]) is not type(slice(None)):
|
||
|
|
if len(lrange[k]) < 3:
|
||
|
|
lrange[k] = tuple(lrange[k]) + (complex(Ns),)
|
||
|
|
lrange[k] = slice(*lrange[k])
|
||
|
|
if (N == 1):
|
||
|
|
lrange = lrange[0]
|
||
|
|
|
||
|
|
def _scalarfunc(*params):
|
||
|
|
params = asarray(params).flatten()
|
||
|
|
return func(params, *args)
|
||
|
|
|
||
|
|
vecfunc = vectorize(_scalarfunc)
|
||
|
|
grid = mgrid[lrange]
|
||
|
|
if (N == 1):
|
||
|
|
grid = (grid,)
|
||
|
|
Jout = vecfunc(*grid)
|
||
|
|
Nshape = shape(Jout)
|
||
|
|
indx = argmin(Jout.ravel(), axis=-1)
|
||
|
|
Nindx = zeros(N, int)
|
||
|
|
xmin = zeros(N, float)
|
||
|
|
for k in range(N - 1, -1, -1):
|
||
|
|
thisN = Nshape[k]
|
||
|
|
Nindx[k] = indx % Nshape[k]
|
||
|
|
indx = indx // thisN
|
||
|
|
for k in range(N):
|
||
|
|
xmin[k] = grid[k][tuple(Nindx)]
|
||
|
|
|
||
|
|
Jmin = Jout[tuple(Nindx)]
|
||
|
|
if (N == 1):
|
||
|
|
grid = grid[0]
|
||
|
|
xmin = xmin[0]
|
||
|
|
if callable(finish):
|
||
|
|
# set up kwargs for `finish` function
|
||
|
|
finish_args = _getargspec(finish).args
|
||
|
|
finish_kwargs = dict()
|
||
|
|
if 'full_output' in finish_args:
|
||
|
|
finish_kwargs['full_output'] = 1
|
||
|
|
if 'disp' in finish_args:
|
||
|
|
finish_kwargs['disp'] = disp
|
||
|
|
elif 'options' in finish_args:
|
||
|
|
# pass 'disp' as `options`
|
||
|
|
# (e.g. if `finish` is `minimize`)
|
||
|
|
finish_kwargs['options'] = {'disp': disp}
|
||
|
|
|
||
|
|
# run minimizer
|
||
|
|
res = finish(func, xmin, args=args, **finish_kwargs)
|
||
|
|
|
||
|
|
if isinstance(res, OptimizeResult):
|
||
|
|
xmin = res.x
|
||
|
|
Jmin = res.fun
|
||
|
|
success = res.success
|
||
|
|
else:
|
||
|
|
xmin = res[0]
|
||
|
|
Jmin = res[1]
|
||
|
|
success = res[-1] == 0
|
||
|
|
if not success:
|
||
|
|
if disp:
|
||
|
|
print("Warning: Either final optimization did not succeed "
|
||
|
|
"or `finish` does not return `statuscode` as its last "
|
||
|
|
"argument.")
|
||
|
|
|
||
|
|
if full_output:
|
||
|
|
return xmin, Jmin, grid, Jout
|
||
|
|
else:
|
||
|
|
return xmin
|
||
|
|
|
||
|
|
|
||
|
|
def show_options(solver=None, method=None, disp=True):
|
||
|
|
"""
|
||
|
|
Show documentation for additional options of optimization solvers.
|
||
|
|
|
||
|
|
These are method-specific options that can be supplied through the
|
||
|
|
``options`` dict.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
solver : str
|
||
|
|
Type of optimization solver. One of 'minimize', 'minimize_scalar',
|
||
|
|
'root', or 'linprog'.
|
||
|
|
method : str, optional
|
||
|
|
If not given, shows all methods of the specified solver. Otherwise,
|
||
|
|
show only the options for the specified method. Valid values
|
||
|
|
corresponds to methods' names of respective solver (e.g. 'BFGS' for
|
||
|
|
'minimize').
|
||
|
|
disp : bool, optional
|
||
|
|
Whether to print the result rather than returning it.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
text
|
||
|
|
Either None (for disp=False) or the text string (disp=True)
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The solver-specific methods are:
|
||
|
|
|
||
|
|
`scipy.optimize.minimize`
|
||
|
|
|
||
|
|
- :ref:`Nelder-Mead <optimize.minimize-neldermead>`
|
||
|
|
- :ref:`Powell <optimize.minimize-powell>`
|
||
|
|
- :ref:`CG <optimize.minimize-cg>`
|
||
|
|
- :ref:`BFGS <optimize.minimize-bfgs>`
|
||
|
|
- :ref:`Newton-CG <optimize.minimize-newtoncg>`
|
||
|
|
- :ref:`L-BFGS-B <optimize.minimize-lbfgsb>`
|
||
|
|
- :ref:`TNC <optimize.minimize-tnc>`
|
||
|
|
- :ref:`COBYLA <optimize.minimize-cobyla>`
|
||
|
|
- :ref:`SLSQP <optimize.minimize-slsqp>`
|
||
|
|
- :ref:`dogleg <optimize.minimize-dogleg>`
|
||
|
|
- :ref:`trust-ncg <optimize.minimize-trustncg>`
|
||
|
|
|
||
|
|
`scipy.optimize.root`
|
||
|
|
|
||
|
|
- :ref:`hybr <optimize.root-hybr>`
|
||
|
|
- :ref:`lm <optimize.root-lm>`
|
||
|
|
- :ref:`broyden1 <optimize.root-broyden1>`
|
||
|
|
- :ref:`broyden2 <optimize.root-broyden2>`
|
||
|
|
- :ref:`anderson <optimize.root-anderson>`
|
||
|
|
- :ref:`linearmixing <optimize.root-linearmixing>`
|
||
|
|
- :ref:`diagbroyden <optimize.root-diagbroyden>`
|
||
|
|
- :ref:`excitingmixing <optimize.root-excitingmixing>`
|
||
|
|
- :ref:`krylov <optimize.root-krylov>`
|
||
|
|
- :ref:`df-sane <optimize.root-dfsane>`
|
||
|
|
|
||
|
|
`scipy.optimize.minimize_scalar`
|
||
|
|
|
||
|
|
- :ref:`brent <optimize.minimize_scalar-brent>`
|
||
|
|
- :ref:`golden <optimize.minimize_scalar-golden>`
|
||
|
|
- :ref:`bounded <optimize.minimize_scalar-bounded>`
|
||
|
|
|
||
|
|
`scipy.optimize.linprog`
|
||
|
|
|
||
|
|
- :ref:`simplex <optimize.linprog-simplex>`
|
||
|
|
- :ref:`interior-point <optimize.linprog-interior-point>`
|
||
|
|
|
||
|
|
"""
|
||
|
|
import textwrap
|
||
|
|
|
||
|
|
doc_routines = {
|
||
|
|
'minimize': (
|
||
|
|
('bfgs', 'scipy.optimize.optimize._minimize_bfgs'),
|
||
|
|
('cg', 'scipy.optimize.optimize._minimize_cg'),
|
||
|
|
('cobyla', 'scipy.optimize.cobyla._minimize_cobyla'),
|
||
|
|
('dogleg', 'scipy.optimize._trustregion_dogleg._minimize_dogleg'),
|
||
|
|
('l-bfgs-b', 'scipy.optimize.lbfgsb._minimize_lbfgsb'),
|
||
|
|
('nelder-mead', 'scipy.optimize.optimize._minimize_neldermead'),
|
||
|
|
('newton-cg', 'scipy.optimize.optimize._minimize_newtoncg'),
|
||
|
|
('powell', 'scipy.optimize.optimize._minimize_powell'),
|
||
|
|
('slsqp', 'scipy.optimize.slsqp._minimize_slsqp'),
|
||
|
|
('tnc', 'scipy.optimize.tnc._minimize_tnc'),
|
||
|
|
('trust-ncg', 'scipy.optimize._trustregion_ncg._minimize_trust_ncg'),
|
||
|
|
),
|
||
|
|
'root': (
|
||
|
|
('hybr', 'scipy.optimize.minpack._root_hybr'),
|
||
|
|
('lm', 'scipy.optimize._root._root_leastsq'),
|
||
|
|
('broyden1', 'scipy.optimize._root._root_broyden1_doc'),
|
||
|
|
('broyden2', 'scipy.optimize._root._root_broyden2_doc'),
|
||
|
|
('anderson', 'scipy.optimize._root._root_anderson_doc'),
|
||
|
|
('diagbroyden', 'scipy.optimize._root._root_diagbroyden_doc'),
|
||
|
|
('excitingmixing', 'scipy.optimize._root._root_excitingmixing_doc'),
|
||
|
|
('linearmixing', 'scipy.optimize._root._root_linearmixing_doc'),
|
||
|
|
('krylov', 'scipy.optimize._root._root_krylov_doc'),
|
||
|
|
('df-sane', 'scipy.optimize._spectral._root_df_sane'),
|
||
|
|
),
|
||
|
|
'root_scalar': (
|
||
|
|
('bisect', 'scipy.optimize._root_scalar._root_scalar_bisect_doc'),
|
||
|
|
('brentq', 'scipy.optimize._root_scalar._root_scalar_brentq_doc'),
|
||
|
|
('brenth', 'scipy.optimize._root_scalar._root_scalar_brenth_doc'),
|
||
|
|
('ridder', 'scipy.optimize._root_scalar._root_scalar_ridder_doc'),
|
||
|
|
('toms748', 'scipy.optimize._root_scalar._root_scalar_toms748_doc'),
|
||
|
|
('secant', 'scipy.optimize._root_scalar._root_scalar_secant_doc'),
|
||
|
|
('newton', 'scipy.optimize._root_scalar._root_scalar_newton_doc'),
|
||
|
|
('halley', 'scipy.optimize._root_scalar._root_scalar_halley_doc'),
|
||
|
|
),
|
||
|
|
'linprog': (
|
||
|
|
('simplex', 'scipy.optimize._linprog._linprog_simplex'),
|
||
|
|
('interior-point', 'scipy.optimize._linprog._linprog_ip'),
|
||
|
|
),
|
||
|
|
'minimize_scalar': (
|
||
|
|
('brent', 'scipy.optimize.optimize._minimize_scalar_brent'),
|
||
|
|
('bounded', 'scipy.optimize.optimize._minimize_scalar_bounded'),
|
||
|
|
('golden', 'scipy.optimize.optimize._minimize_scalar_golden'),
|
||
|
|
),
|
||
|
|
}
|
||
|
|
|
||
|
|
if solver is None:
|
||
|
|
text = ["\n\n\n========\n", "minimize\n", "========\n"]
|
||
|
|
text.append(show_options('minimize', disp=False))
|
||
|
|
text.extend(["\n\n===============\n", "minimize_scalar\n",
|
||
|
|
"===============\n"])
|
||
|
|
text.append(show_options('minimize_scalar', disp=False))
|
||
|
|
text.extend(["\n\n\n====\n", "root\n",
|
||
|
|
"====\n"])
|
||
|
|
text.append(show_options('root', disp=False))
|
||
|
|
text.extend(['\n\n\n=======\n', 'linprog\n',
|
||
|
|
'=======\n'])
|
||
|
|
text.append(show_options('linprog', disp=False))
|
||
|
|
text = "".join(text)
|
||
|
|
else:
|
||
|
|
solver = solver.lower()
|
||
|
|
if solver not in doc_routines:
|
||
|
|
raise ValueError('Unknown solver %r' % (solver,))
|
||
|
|
|
||
|
|
if method is None:
|
||
|
|
text = []
|
||
|
|
for name, _ in doc_routines[solver]:
|
||
|
|
text.extend(["\n\n" + name, "\n" + "="*len(name) + "\n\n"])
|
||
|
|
text.append(show_options(solver, name, disp=False))
|
||
|
|
text = "".join(text)
|
||
|
|
else:
|
||
|
|
method = method.lower()
|
||
|
|
methods = dict(doc_routines[solver])
|
||
|
|
if method not in methods:
|
||
|
|
raise ValueError("Unknown method %r" % (method,))
|
||
|
|
name = methods[method]
|
||
|
|
|
||
|
|
# Import function object
|
||
|
|
parts = name.split('.')
|
||
|
|
mod_name = ".".join(parts[:-1])
|
||
|
|
__import__(mod_name)
|
||
|
|
obj = getattr(sys.modules[mod_name], parts[-1])
|
||
|
|
|
||
|
|
# Get doc
|
||
|
|
doc = obj.__doc__
|
||
|
|
if doc is not None:
|
||
|
|
text = textwrap.dedent(doc).strip()
|
||
|
|
else:
|
||
|
|
text = ""
|
||
|
|
|
||
|
|
if disp:
|
||
|
|
print(text)
|
||
|
|
return
|
||
|
|
else:
|
||
|
|
return text
|
||
|
|
|
||
|
|
|
||
|
|
def main():
|
||
|
|
import time
|
||
|
|
|
||
|
|
times = []
|
||
|
|
algor = []
|
||
|
|
x0 = [0.8, 1.2, 0.7]
|
||
|
|
print("Nelder-Mead Simplex")
|
||
|
|
print("===================")
|
||
|
|
start = time.time()
|
||
|
|
x = fmin(rosen, x0)
|
||
|
|
print(x)
|
||
|
|
times.append(time.time() - start)
|
||
|
|
algor.append('Nelder-Mead Simplex\t')
|
||
|
|
|
||
|
|
print()
|
||
|
|
print("Powell Direction Set Method")
|
||
|
|
print("===========================")
|
||
|
|
start = time.time()
|
||
|
|
x = fmin_powell(rosen, x0)
|
||
|
|
print(x)
|
||
|
|
times.append(time.time() - start)
|
||
|
|
algor.append('Powell Direction Set Method.')
|
||
|
|
|
||
|
|
print()
|
||
|
|
print("Nonlinear CG")
|
||
|
|
print("============")
|
||
|
|
start = time.time()
|
||
|
|
x = fmin_cg(rosen, x0, fprime=rosen_der, maxiter=200)
|
||
|
|
print(x)
|
||
|
|
times.append(time.time() - start)
|
||
|
|
algor.append('Nonlinear CG \t')
|
||
|
|
|
||
|
|
print()
|
||
|
|
print("BFGS Quasi-Newton")
|
||
|
|
print("=================")
|
||
|
|
start = time.time()
|
||
|
|
x = fmin_bfgs(rosen, x0, fprime=rosen_der, maxiter=80)
|
||
|
|
print(x)
|
||
|
|
times.append(time.time() - start)
|
||
|
|
algor.append('BFGS Quasi-Newton\t')
|
||
|
|
|
||
|
|
print()
|
||
|
|
print("BFGS approximate gradient")
|
||
|
|
print("=========================")
|
||
|
|
start = time.time()
|
||
|
|
x = fmin_bfgs(rosen, x0, gtol=1e-4, maxiter=100)
|
||
|
|
print(x)
|
||
|
|
times.append(time.time() - start)
|
||
|
|
algor.append('BFGS without gradient\t')
|
||
|
|
|
||
|
|
print()
|
||
|
|
print("Newton-CG with Hessian product")
|
||
|
|
print("==============================")
|
||
|
|
start = time.time()
|
||
|
|
x = fmin_ncg(rosen, x0, rosen_der, fhess_p=rosen_hess_prod, maxiter=80)
|
||
|
|
print(x)
|
||
|
|
times.append(time.time() - start)
|
||
|
|
algor.append('Newton-CG with hessian product')
|
||
|
|
|
||
|
|
print()
|
||
|
|
print("Newton-CG with full Hessian")
|
||
|
|
print("===========================")
|
||
|
|
start = time.time()
|
||
|
|
x = fmin_ncg(rosen, x0, rosen_der, fhess=rosen_hess, maxiter=80)
|
||
|
|
print(x)
|
||
|
|
times.append(time.time() - start)
|
||
|
|
algor.append('Newton-CG with full hessian')
|
||
|
|
|
||
|
|
print()
|
||
|
|
print("\nMinimizing the Rosenbrock function of order 3\n")
|
||
|
|
print(" Algorithm \t\t\t Seconds")
|
||
|
|
print("===========\t\t\t =========")
|
||
|
|
for k in range(len(algor)):
|
||
|
|
print(algor[k], "\t -- ", times[k])
|
||
|
|
|
||
|
|
|
||
|
|
if __name__ == "__main__":
|
||
|
|
main()
|