123 lines
4.3 KiB
Python
123 lines
4.3 KiB
Python
|
"""Dog-leg trust-region optimization."""
|
||
|
|
import numpy as np
|
||
|
|
import scipy.linalg
|
||
|
|
from ._trustregion import (_minimize_trust_region, BaseQuadraticSubproblem)
|
||
|
|
|
||
|
|
__all__ = []
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_dogleg(fun, x0, args=(), jac=None, hess=None,
|
||
|
|
**trust_region_options):
|
||
|
|
"""
|
||
|
|
Minimization of scalar function of one or more variables using
|
||
|
|
the dog-leg trust-region algorithm.
|
||
|
|
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
initial_trust_radius : float
|
||
|
|
Initial trust-region radius.
|
||
|
|
max_trust_radius : float
|
||
|
|
Maximum value of the trust-region radius. No steps that are longer
|
||
|
|
than this value will be proposed.
|
||
|
|
eta : float
|
||
|
|
Trust region related acceptance stringency for proposed steps.
|
||
|
|
gtol : float
|
||
|
|
Gradient norm must be less than `gtol` before successful
|
||
|
|
termination.
|
||
|
|
|
||
|
|
"""
|
||
|
|
if jac is None:
|
||
|
|
raise ValueError('Jacobian is required for dogleg minimization')
|
||
|
|
if hess is None:
|
||
|
|
raise ValueError('Hessian is required for dogleg minimization')
|
||
|
|
return _minimize_trust_region(fun, x0, args=args, jac=jac, hess=hess,
|
||
|
|
subproblem=DoglegSubproblem,
|
||
|
|
**trust_region_options)
|
||
|
|
|
||
|
|
|
||
|
|
class DoglegSubproblem(BaseQuadraticSubproblem):
|
||
|
|
"""Quadratic subproblem solved by the dogleg method"""
|
||
|
|
|
||
|
|
def cauchy_point(self):
|
||
|
|
"""
|
||
|
|
The Cauchy point is minimal along the direction of steepest descent.
|
||
|
|
"""
|
||
|
|
if self._cauchy_point is None:
|
||
|
|
g = self.jac
|
||
|
|
Bg = self.hessp(g)
|
||
|
|
self._cauchy_point = -(np.dot(g, g) / np.dot(g, Bg)) * g
|
||
|
|
return self._cauchy_point
|
||
|
|
|
||
|
|
def newton_point(self):
|
||
|
|
"""
|
||
|
|
The Newton point is a global minimum of the approximate function.
|
||
|
|
"""
|
||
|
|
if self._newton_point is None:
|
||
|
|
g = self.jac
|
||
|
|
B = self.hess
|
||
|
|
cho_info = scipy.linalg.cho_factor(B)
|
||
|
|
self._newton_point = -scipy.linalg.cho_solve(cho_info, g)
|
||
|
|
return self._newton_point
|
||
|
|
|
||
|
|
def solve(self, trust_radius):
|
||
|
|
"""
|
||
|
|
Minimize a function using the dog-leg trust-region algorithm.
|
||
|
|
|
||
|
|
This algorithm requires function values and first and second derivatives.
|
||
|
|
It also performs a costly Hessian decomposition for most iterations,
|
||
|
|
and the Hessian is required to be positive definite.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
trust_radius : float
|
||
|
|
We are allowed to wander only this far away from the origin.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
p : ndarray
|
||
|
|
The proposed step.
|
||
|
|
hits_boundary : bool
|
||
|
|
True if the proposed step is on the boundary of the trust region.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The Hessian is required to be positive definite.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Jorge Nocedal and Stephen Wright,
|
||
|
|
Numerical Optimization, second edition,
|
||
|
|
Springer-Verlag, 2006, page 73.
|
||
|
|
"""
|
||
|
|
|
||
|
|
# Compute the Newton point.
|
||
|
|
# This is the optimum for the quadratic model function.
|
||
|
|
# If it is inside the trust radius then return this point.
|
||
|
|
p_best = self.newton_point()
|
||
|
|
if scipy.linalg.norm(p_best) < trust_radius:
|
||
|
|
hits_boundary = False
|
||
|
|
return p_best, hits_boundary
|
||
|
|
|
||
|
|
# Compute the Cauchy point.
|
||
|
|
# This is the predicted optimum along the direction of steepest descent.
|
||
|
|
p_u = self.cauchy_point()
|
||
|
|
|
||
|
|
# If the Cauchy point is outside the trust region,
|
||
|
|
# then return the point where the path intersects the boundary.
|
||
|
|
p_u_norm = scipy.linalg.norm(p_u)
|
||
|
|
if p_u_norm >= trust_radius:
|
||
|
|
p_boundary = p_u * (trust_radius / p_u_norm)
|
||
|
|
hits_boundary = True
|
||
|
|
return p_boundary, hits_boundary
|
||
|
|
|
||
|
|
# Compute the intersection of the trust region boundary
|
||
|
|
# and the line segment connecting the Cauchy and Newton points.
|
||
|
|
# This requires solving a quadratic equation.
|
||
|
|
# ||p_u + t*(p_best - p_u)||**2 == trust_radius**2
|
||
|
|
# Solve this for positive time t using the quadratic formula.
|
||
|
|
_, tb = self.get_boundaries_intersections(p_u, p_best - p_u,
|
||
|
|
trust_radius)
|
||
|
|
p_boundary = p_u + tb * (p_best - p_u)
|
||
|
|
hits_boundary = True
|
||
|
|
return p_boundary, hits_boundary
|