266 lines
9 KiB
Python
266 lines
9 KiB
Python
"""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 .optimize import (_check_unknown_options, wrap_function, _status_message,
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OptimizeResult, _prepare_scalar_function)
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__all__ = []
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class BaseQuadraticSubproblem(object):
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"""
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Base/abstract class defining the quadratic model for trust-region
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minimization. Child classes must implement the ``solve`` method.
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Values of the objective function, Jacobian and Hessian (if provided) at
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the current iterate ``x`` are evaluated on demand and then stored as
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attributes ``fun``, ``jac``, ``hess``.
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"""
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def __init__(self, x, fun, jac, hess=None, hessp=None):
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self._x = x
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self._f = None
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self._g = None
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self._h = None
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self._g_mag = None
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self._cauchy_point = None
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self._newton_point = None
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self._fun = fun
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self._jac = jac
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self._hess = hess
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self._hessp = hessp
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def __call__(self, p):
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return self.fun + np.dot(self.jac, p) + 0.5 * np.dot(p, self.hessp(p))
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@property
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def fun(self):
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"""Value of objective function at current iteration."""
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if self._f is None:
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self._f = self._fun(self._x)
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return self._f
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@property
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def jac(self):
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"""Value of Jacobian of objective function at current iteration."""
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if self._g is None:
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self._g = self._jac(self._x)
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return self._g
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@property
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def hess(self):
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"""Value of Hessian of objective function at current iteration."""
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if self._h is None:
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self._h = self._hess(self._x)
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return self._h
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def hessp(self, p):
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if self._hessp is not None:
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return self._hessp(self._x, p)
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else:
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return np.dot(self.hess, p)
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@property
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def jac_mag(self):
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"""Magnitude of jacobian of objective function at current iteration."""
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if self._g_mag is None:
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self._g_mag = scipy.linalg.norm(self.jac)
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return self._g_mag
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def get_boundaries_intersections(self, z, d, trust_radius):
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"""
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Solve the scalar quadratic equation ||z + t d|| == trust_radius.
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This is like a line-sphere intersection.
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Return the two values of t, sorted from low to high.
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"""
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a = np.dot(d, d)
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b = 2 * np.dot(z, d)
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c = np.dot(z, z) - trust_radius**2
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sqrt_discriminant = math.sqrt(b*b - 4*a*c)
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# The following calculation is mathematically
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# equivalent to:
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# ta = (-b - sqrt_discriminant) / (2*a)
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# tb = (-b + sqrt_discriminant) / (2*a)
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# but produce smaller round off errors.
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# Look at Matrix Computation p.97
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# for a better justification.
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aux = b + math.copysign(sqrt_discriminant, b)
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ta = -aux / (2*a)
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tb = -2*c / aux
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return sorted([ta, tb])
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def solve(self, trust_radius):
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raise NotImplementedError('The solve method should be implemented by '
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'the child class')
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def _minimize_trust_region(fun, x0, args=(), jac=None, hess=None, hessp=None,
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subproblem=None, initial_trust_radius=1.0,
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max_trust_radius=1000.0, eta=0.15, gtol=1e-4,
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maxiter=None, disp=False, return_all=False,
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callback=None, inexact=True, **unknown_options):
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"""
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Minimization of scalar function of one or more variables using a
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trust-region algorithm.
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Options for the trust-region algorithm are:
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initial_trust_radius : float
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Initial trust radius.
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max_trust_radius : float
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Never propose steps that are longer than this value.
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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`
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before successful termination.
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maxiter : int
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Maximum number of iterations to perform.
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disp : bool
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If True, print convergence message.
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inexact : bool
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Accuracy to solve subproblems. If True requires less nonlinear
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iterations, but more vector products. Only effective for method
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trust-krylov.
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This function is called by the `minimize` function.
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It is not supposed to be called directly.
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"""
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_check_unknown_options(unknown_options)
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if jac is None:
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raise ValueError('Jacobian is currently required for trust-region '
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'methods')
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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 currently required for trust-region methods')
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if subproblem is None:
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raise ValueError('A subproblem solving strategy is required for '
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'trust-region methods')
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if not (0 <= eta < 0.25):
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raise Exception('invalid acceptance stringency')
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if max_trust_radius <= 0:
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raise Exception('the max trust radius must be positive')
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if initial_trust_radius <= 0:
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raise ValueError('the initial trust radius must be positive')
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if initial_trust_radius >= max_trust_radius:
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raise ValueError('the initial trust radius must be less than the '
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'max trust radius')
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# force the initial guess into a nice format
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x0 = np.asarray(x0).flatten()
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# A ScalarFunction representing the problem. This caches calls to fun, jac,
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# hess.
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sf = _prepare_scalar_function(fun, x0, jac=jac, hess=hess, args=args)
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fun = sf.fun
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jac = sf.grad
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if hess is not None:
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hess = sf.hess
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# ScalarFunction doesn't represent hessp
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nhessp, hessp = wrap_function(hessp, args)
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# limit the number of iterations
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if maxiter is None:
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maxiter = len(x0)*200
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# init the search status
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warnflag = 0
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# initialize the search
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trust_radius = initial_trust_radius
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x = x0
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if return_all:
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allvecs = [x]
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m = subproblem(x, fun, jac, hess, hessp)
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k = 0
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# search for the function min
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# do not even start if the gradient is small enough
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while m.jac_mag >= gtol:
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# Solve the sub-problem.
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# This gives us the proposed step relative to the current position
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# and it tells us whether the proposed step
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# has reached the trust region boundary or not.
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try:
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p, hits_boundary = m.solve(trust_radius)
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except np.linalg.linalg.LinAlgError:
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warnflag = 3
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break
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# calculate the predicted value at the proposed point
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predicted_value = m(p)
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# define the local approximation at the proposed point
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x_proposed = x + p
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m_proposed = subproblem(x_proposed, fun, jac, hess, hessp)
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# evaluate the ratio defined in equation (4.4)
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actual_reduction = m.fun - m_proposed.fun
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predicted_reduction = m.fun - predicted_value
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if predicted_reduction <= 0:
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warnflag = 2
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break
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rho = actual_reduction / predicted_reduction
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# update the trust radius according to the actual/predicted ratio
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if rho < 0.25:
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trust_radius *= 0.25
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elif rho > 0.75 and hits_boundary:
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trust_radius = min(2*trust_radius, max_trust_radius)
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# if the ratio is high enough then accept the proposed step
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if rho > eta:
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x = x_proposed
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m = m_proposed
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# append the best guess, call back, increment the iteration count
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if return_all:
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allvecs.append(np.copy(x))
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if callback is not None:
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callback(np.copy(x))
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k += 1
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# check if the gradient is small enough to stop
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if m.jac_mag < gtol:
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warnflag = 0
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break
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# check if we have looked at enough iterations
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if k >= maxiter:
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warnflag = 1
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break
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# print some stuff if requested
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status_messages = (
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_status_message['success'],
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_status_message['maxiter'],
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'A bad approximation caused failure to predict improvement.',
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'A linalg error occurred, such as a non-psd Hessian.',
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)
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if disp:
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if warnflag == 0:
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print(status_messages[warnflag])
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else:
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print('Warning: ' + status_messages[warnflag])
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print(" Current function value: %f" % m.fun)
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print(" Iterations: %d" % k)
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print(" Function evaluations: %d" % sf.nfev)
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print(" Gradient evaluations: %d" % sf.ngev)
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print(" Hessian evaluations: %d" % (sf.nhev + nhessp[0]))
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result = OptimizeResult(x=x, success=(warnflag == 0), status=warnflag,
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fun=m.fun, jac=m.jac, nfev=sf.nfev, njev=sf.ngev,
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nhev=sf.nhev + nhessp[0], nit=k,
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message=status_messages[warnflag])
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if hess is not None:
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result['hess'] = m.hess
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if return_all:
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result['allvecs'] = allvecs
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return result
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