127 lines
4.5 KiB
Python
127 lines
4.5 KiB
Python
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"""Newton-CG trust-region optimization."""
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import math
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import numpy as np
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import scipy.linalg
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from ._trustregion import (_minimize_trust_region, BaseQuadraticSubproblem)
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__all__ = []
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def _minimize_trust_ncg(fun, x0, args=(), jac=None, hess=None, hessp=None,
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**trust_region_options):
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"""
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Minimization of scalar function of one or more variables using
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the Newton conjugate gradient trust-region algorithm.
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Options
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-------
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initial_trust_radius : float
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Initial trust-region radius.
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max_trust_radius : float
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Maximum value of the trust-region radius. No steps that are longer
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than this value will be proposed.
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eta : float
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Trust region related acceptance stringency for proposed steps.
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gtol : float
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Gradient norm must be less than `gtol` before successful
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termination.
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"""
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if jac is None:
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raise ValueError('Jacobian is required for Newton-CG trust-region '
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'minimization')
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if hess is None and hessp is None:
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raise ValueError('Either the Hessian or the Hessian-vector product '
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'is required for Newton-CG trust-region minimization')
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return _minimize_trust_region(fun, x0, args=args, jac=jac, hess=hess,
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hessp=hessp, subproblem=CGSteihaugSubproblem,
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**trust_region_options)
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class CGSteihaugSubproblem(BaseQuadraticSubproblem):
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"""Quadratic subproblem solved by a conjugate gradient method"""
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def solve(self, trust_radius):
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"""
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Solve the subproblem using a conjugate gradient method.
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Parameters
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----------
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trust_radius : float
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We are allowed to wander only this far away from the origin.
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Returns
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-------
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p : ndarray
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The proposed step.
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hits_boundary : bool
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True if the proposed step is on the boundary of the trust region.
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Notes
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-----
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This is algorithm (7.2) of Nocedal and Wright 2nd edition.
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Only the function that computes the Hessian-vector product is required.
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The Hessian itself is not required, and the Hessian does
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not need to be positive semidefinite.
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"""
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# get the norm of jacobian and define the origin
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p_origin = np.zeros_like(self.jac)
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# define a default tolerance
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tolerance = min(0.5, math.sqrt(self.jac_mag)) * self.jac_mag
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# Stop the method if the search direction
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# is a direction of nonpositive curvature.
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if self.jac_mag < tolerance:
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hits_boundary = False
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return p_origin, hits_boundary
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# init the state for the first iteration
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z = p_origin
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r = self.jac
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d = -r
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# Search for the min of the approximation of the objective function.
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while True:
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# do an iteration
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Bd = self.hessp(d)
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dBd = np.dot(d, Bd)
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if dBd <= 0:
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# Look at the two boundary points.
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# Find both values of t to get the boundary points such that
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# ||z + t d|| == trust_radius
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# and then choose the one with the predicted min value.
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ta, tb = self.get_boundaries_intersections(z, d, trust_radius)
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pa = z + ta * d
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pb = z + tb * d
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if self(pa) < self(pb):
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p_boundary = pa
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else:
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p_boundary = pb
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hits_boundary = True
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return p_boundary, hits_boundary
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r_squared = np.dot(r, r)
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alpha = r_squared / dBd
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z_next = z + alpha * d
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if scipy.linalg.norm(z_next) >= trust_radius:
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# Find t >= 0 to get the boundary point such that
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# ||z + t d|| == trust_radius
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ta, tb = self.get_boundaries_intersections(z, d, trust_radius)
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p_boundary = z + tb * d
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hits_boundary = True
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return p_boundary, hits_boundary
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r_next = r + alpha * Bd
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r_next_squared = np.dot(r_next, r_next)
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if math.sqrt(r_next_squared) < tolerance:
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hits_boundary = False
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return z_next, hits_boundary
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beta_next = r_next_squared / r_squared
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d_next = -r_next + beta_next * d
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# update the state for the next iteration
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z = z_next
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r = r_next
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d = d_next
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