EVOLUTION-MANAGER

Edit File: _trustregion_dogleg.py

"""Dog-leg trust-region optimization."""
from __future__ import division, print_function, absolute_import

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