125 lines
4.3 KiB
Python
125 lines
4.3 KiB
Python
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"""Dog-leg trust-region optimization."""
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from __future__ import division, print_function, absolute_import
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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_dogleg(fun, x0, args=(), jac=None, hess=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 dog-leg 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 dogleg minimization')
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if hess is None:
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raise ValueError('Hessian is required for dogleg minimization')
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return _minimize_trust_region(fun, x0, args=args, jac=jac, hess=hess,
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subproblem=DoglegSubproblem,
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**trust_region_options)
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class DoglegSubproblem(BaseQuadraticSubproblem):
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"""Quadratic subproblem solved by the dogleg method"""
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def cauchy_point(self):
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"""
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The Cauchy point is minimal along the direction of steepest descent.
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"""
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if self._cauchy_point is None:
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g = self.jac
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Bg = self.hessp(g)
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self._cauchy_point = -(np.dot(g, g) / np.dot(g, Bg)) * g
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return self._cauchy_point
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def newton_point(self):
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"""
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The Newton point is a global minimum of the approximate function.
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"""
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if self._newton_point is None:
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g = self.jac
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B = self.hess
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cho_info = scipy.linalg.cho_factor(B)
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self._newton_point = -scipy.linalg.cho_solve(cho_info, g)
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return self._newton_point
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def solve(self, trust_radius):
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"""
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Minimize a function using the dog-leg trust-region algorithm.
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This algorithm requires function values and first and second derivatives.
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It also performs a costly Hessian decomposition for most iterations,
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and the Hessian is required to be positive definite.
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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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The Hessian is required to be positive definite.
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References
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----------
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.. [1] Jorge Nocedal and Stephen Wright,
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Numerical Optimization, second edition,
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Springer-Verlag, 2006, page 73.
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"""
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# Compute the Newton point.
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# This is the optimum for the quadratic model function.
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# If it is inside the trust radius then return this point.
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p_best = self.newton_point()
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if scipy.linalg.norm(p_best) < trust_radius:
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hits_boundary = False
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return p_best, hits_boundary
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# Compute the Cauchy point.
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# This is the predicted optimum along the direction of steepest descent.
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p_u = self.cauchy_point()
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# If the Cauchy point is outside the trust region,
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# then return the point where the path intersects the boundary.
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p_u_norm = scipy.linalg.norm(p_u)
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if p_u_norm >= trust_radius:
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p_boundary = p_u * (trust_radius / p_u_norm)
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hits_boundary = True
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return p_boundary, hits_boundary
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# Compute the intersection of the trust region boundary
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# and the line segment connecting the Cauchy and Newton points.
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# This requires solving a quadratic equation.
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# ||p_u + t*(p_best - p_u)||**2 == trust_radius**2
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# Solve this for positive time t using the quadratic formula.
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_, tb = self.get_boundaries_intersections(p_u, p_best - p_u,
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trust_radius)
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p_boundary = p_u + tb * (p_best - p_u)
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hits_boundary = True
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return p_boundary, hits_boundary
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