617 lines
22 KiB
Python
617 lines
22 KiB
Python
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import numpy as np
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import scipy.sparse as sps
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from ._numdiff import approx_derivative, group_columns
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from ._hessian_update_strategy import HessianUpdateStrategy
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from scipy.sparse.linalg import LinearOperator
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FD_METHODS = ('2-point', '3-point', 'cs')
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class ScalarFunction:
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"""Scalar function and its derivatives.
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This class defines a scalar function F: R^n->R and methods for
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computing or approximating its first and second derivatives.
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Parameters
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----------
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fun : callable
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evaluates the scalar function. Must be of the form ``fun(x, *args)``,
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where ``x`` is the argument in the form of a 1-D array and ``args`` is
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a tuple of any additional fixed parameters needed to completely specify
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the function. Should return a scalar.
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x0 : array-like
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Provides an initial set of variables for evaluating fun. Array of real
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elements of size (n,), where 'n' is the number of independent
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variables.
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args : tuple, optional
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Any additional fixed parameters needed to completely specify the scalar
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function.
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grad : {callable, '2-point', '3-point', 'cs'}
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Method for computing the gradient vector.
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If it is a callable, it should be a function that returns the gradient
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vector:
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``grad(x, *args) -> array_like, shape (n,)``
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where ``x`` is an array with shape (n,) and ``args`` is a tuple with
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the fixed parameters.
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Alternatively, the keywords {'2-point', '3-point', 'cs'} can be used
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to select a finite difference scheme for numerical estimation of the
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gradient with a relative step size. These finite difference schemes
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obey any specified `bounds`.
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hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy}
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Method for computing the Hessian matrix. If it is callable, it should
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return the Hessian matrix:
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``hess(x, *args) -> {LinearOperator, spmatrix, array}, (n, n)``
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where x is a (n,) ndarray and `args` is a tuple with the fixed
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parameters. Alternatively, the keywords {'2-point', '3-point', 'cs'}
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select a finite difference scheme for numerical estimation. Or, objects
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implementing `HessianUpdateStrategy` interface can be used to
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approximate the Hessian.
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Whenever the gradient is estimated via finite-differences, the Hessian
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cannot be estimated with options {'2-point', '3-point', 'cs'} and needs
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to be estimated using one of the quasi-Newton strategies.
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finite_diff_rel_step : None or array_like
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Relative step size to use. The absolute step size is computed as
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``h = finite_diff_rel_step * sign(x0) * max(1, abs(x0))``, possibly
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adjusted to fit into the bounds. For ``method='3-point'`` the sign
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of `h` is ignored. If None then finite_diff_rel_step is selected
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automatically,
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finite_diff_bounds : tuple of array_like
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Lower and upper bounds on independent variables. Defaults to no bounds,
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(-np.inf, np.inf). Each bound must match the size of `x0` or be a
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scalar, in the latter case the bound will be the same for all
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variables. Use it to limit the range of function evaluation.
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epsilon : None or array_like, optional
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Absolute step size to use, possibly adjusted to fit into the bounds.
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For ``method='3-point'`` the sign of `epsilon` is ignored. By default
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relative steps are used, only if ``epsilon is not None`` are absolute
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steps used.
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Notes
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-----
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This class implements a memoization logic. There are methods `fun`,
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`grad`, hess` and corresponding attributes `f`, `g` and `H`. The following
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things should be considered:
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1. Use only public methods `fun`, `grad` and `hess`.
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2. After one of the methods is called, the corresponding attribute
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will be set. However, a subsequent call with a different argument
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of *any* of the methods may overwrite the attribute.
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"""
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def __init__(self, fun, x0, args, grad, hess, finite_diff_rel_step,
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finite_diff_bounds, epsilon=None):
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if not callable(grad) and grad not in FD_METHODS:
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raise ValueError(
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f"`grad` must be either callable or one of {FD_METHODS}."
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)
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if not (callable(hess) or hess in FD_METHODS
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or isinstance(hess, HessianUpdateStrategy)):
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raise ValueError(
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f"`hess` must be either callable, HessianUpdateStrategy"
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f" or one of {FD_METHODS}."
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)
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if grad in FD_METHODS and hess in FD_METHODS:
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raise ValueError("Whenever the gradient is estimated via "
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"finite-differences, we require the Hessian "
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"to be estimated using one of the "
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"quasi-Newton strategies.")
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# the astype call ensures that self.x is a copy of x0
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self.x = np.atleast_1d(x0).astype(float)
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self.n = self.x.size
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self.nfev = 0
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self.ngev = 0
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self.nhev = 0
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self.f_updated = False
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self.g_updated = False
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self.H_updated = False
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self._lowest_x = None
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self._lowest_f = np.inf
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finite_diff_options = {}
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if grad in FD_METHODS:
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finite_diff_options["method"] = grad
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finite_diff_options["rel_step"] = finite_diff_rel_step
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finite_diff_options["abs_step"] = epsilon
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finite_diff_options["bounds"] = finite_diff_bounds
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if hess in FD_METHODS:
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finite_diff_options["method"] = hess
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finite_diff_options["rel_step"] = finite_diff_rel_step
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finite_diff_options["abs_step"] = epsilon
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finite_diff_options["as_linear_operator"] = True
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# Function evaluation
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def fun_wrapped(x):
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self.nfev += 1
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# Send a copy because the user may overwrite it.
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# Overwriting results in undefined behaviour because
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# fun(self.x) will change self.x, with the two no longer linked.
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fx = fun(np.copy(x), *args)
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# Make sure the function returns a true scalar
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if not np.isscalar(fx):
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try:
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fx = np.asarray(fx).item()
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except (TypeError, ValueError) as e:
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raise ValueError(
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"The user-provided objective function "
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"must return a scalar value."
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) from e
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if fx < self._lowest_f:
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self._lowest_x = x
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self._lowest_f = fx
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return fx
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def update_fun():
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self.f = fun_wrapped(self.x)
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self._update_fun_impl = update_fun
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self._update_fun()
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# Gradient evaluation
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if callable(grad):
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def grad_wrapped(x):
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self.ngev += 1
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return np.atleast_1d(grad(np.copy(x), *args))
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def update_grad():
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self.g = grad_wrapped(self.x)
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elif grad in FD_METHODS:
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def update_grad():
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self._update_fun()
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self.ngev += 1
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self.g = approx_derivative(fun_wrapped, self.x, f0=self.f,
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**finite_diff_options)
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self._update_grad_impl = update_grad
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self._update_grad()
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# Hessian Evaluation
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if callable(hess):
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self.H = hess(np.copy(x0), *args)
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self.H_updated = True
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self.nhev += 1
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if sps.issparse(self.H):
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def hess_wrapped(x):
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self.nhev += 1
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return sps.csr_matrix(hess(np.copy(x), *args))
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self.H = sps.csr_matrix(self.H)
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elif isinstance(self.H, LinearOperator):
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def hess_wrapped(x):
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self.nhev += 1
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return hess(np.copy(x), *args)
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else:
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def hess_wrapped(x):
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self.nhev += 1
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return np.atleast_2d(np.asarray(hess(np.copy(x), *args)))
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self.H = np.atleast_2d(np.asarray(self.H))
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def update_hess():
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self.H = hess_wrapped(self.x)
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elif hess in FD_METHODS:
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def update_hess():
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self._update_grad()
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self.H = approx_derivative(grad_wrapped, self.x, f0=self.g,
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**finite_diff_options)
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return self.H
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update_hess()
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self.H_updated = True
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elif isinstance(hess, HessianUpdateStrategy):
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self.H = hess
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self.H.initialize(self.n, 'hess')
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self.H_updated = True
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self.x_prev = None
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self.g_prev = None
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def update_hess():
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self._update_grad()
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self.H.update(self.x - self.x_prev, self.g - self.g_prev)
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self._update_hess_impl = update_hess
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if isinstance(hess, HessianUpdateStrategy):
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def update_x(x):
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self._update_grad()
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self.x_prev = self.x
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self.g_prev = self.g
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# ensure that self.x is a copy of x. Don't store a reference
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# otherwise the memoization doesn't work properly.
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self.x = np.atleast_1d(x).astype(float)
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self.f_updated = False
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self.g_updated = False
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self.H_updated = False
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self._update_hess()
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else:
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def update_x(x):
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# ensure that self.x is a copy of x. Don't store a reference
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# otherwise the memoization doesn't work properly.
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self.x = np.atleast_1d(x).astype(float)
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self.f_updated = False
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self.g_updated = False
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self.H_updated = False
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self._update_x_impl = update_x
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def _update_fun(self):
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if not self.f_updated:
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self._update_fun_impl()
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self.f_updated = True
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def _update_grad(self):
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if not self.g_updated:
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self._update_grad_impl()
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self.g_updated = True
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def _update_hess(self):
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if not self.H_updated:
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self._update_hess_impl()
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self.H_updated = True
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def fun(self, x):
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if not np.array_equal(x, self.x):
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self._update_x_impl(x)
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self._update_fun()
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return self.f
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def grad(self, x):
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if not np.array_equal(x, self.x):
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self._update_x_impl(x)
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self._update_grad()
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return self.g
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def hess(self, x):
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if not np.array_equal(x, self.x):
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self._update_x_impl(x)
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self._update_hess()
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return self.H
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def fun_and_grad(self, x):
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if not np.array_equal(x, self.x):
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self._update_x_impl(x)
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self._update_fun()
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self._update_grad()
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return self.f, self.g
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class VectorFunction:
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"""Vector function and its derivatives.
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This class defines a vector function F: R^n->R^m and methods for
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computing or approximating its first and second derivatives.
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Notes
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-----
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This class implements a memoization logic. There are methods `fun`,
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`jac`, hess` and corresponding attributes `f`, `J` and `H`. The following
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things should be considered:
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1. Use only public methods `fun`, `jac` and `hess`.
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2. After one of the methods is called, the corresponding attribute
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will be set. However, a subsequent call with a different argument
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of *any* of the methods may overwrite the attribute.
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"""
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def __init__(self, fun, x0, jac, hess,
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finite_diff_rel_step, finite_diff_jac_sparsity,
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finite_diff_bounds, sparse_jacobian):
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if not callable(jac) and jac not in FD_METHODS:
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raise ValueError("`jac` must be either callable or one of {}."
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.format(FD_METHODS))
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if not (callable(hess) or hess in FD_METHODS
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or isinstance(hess, HessianUpdateStrategy)):
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raise ValueError("`hess` must be either callable,"
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"HessianUpdateStrategy or one of {}."
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.format(FD_METHODS))
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if jac in FD_METHODS and hess in FD_METHODS:
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raise ValueError("Whenever the Jacobian is estimated via "
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"finite-differences, we require the Hessian to "
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"be estimated using one of the quasi-Newton "
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"strategies.")
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self.x = np.atleast_1d(x0).astype(float)
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self.n = self.x.size
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self.nfev = 0
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self.njev = 0
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self.nhev = 0
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self.f_updated = False
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self.J_updated = False
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self.H_updated = False
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finite_diff_options = {}
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if jac in FD_METHODS:
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finite_diff_options["method"] = jac
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finite_diff_options["rel_step"] = finite_diff_rel_step
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if finite_diff_jac_sparsity is not None:
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sparsity_groups = group_columns(finite_diff_jac_sparsity)
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finite_diff_options["sparsity"] = (finite_diff_jac_sparsity,
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sparsity_groups)
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finite_diff_options["bounds"] = finite_diff_bounds
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self.x_diff = np.copy(self.x)
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if hess in FD_METHODS:
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finite_diff_options["method"] = hess
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finite_diff_options["rel_step"] = finite_diff_rel_step
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finite_diff_options["as_linear_operator"] = True
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self.x_diff = np.copy(self.x)
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if jac in FD_METHODS and hess in FD_METHODS:
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raise ValueError("Whenever the Jacobian is estimated via "
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"finite-differences, we require the Hessian to "
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"be estimated using one of the quasi-Newton "
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"strategies.")
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# Function evaluation
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def fun_wrapped(x):
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self.nfev += 1
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return np.atleast_1d(fun(x))
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def update_fun():
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|
|
self.f = fun_wrapped(self.x)
|
||
|
|
|
||
|
|
self._update_fun_impl = update_fun
|
||
|
|
update_fun()
|
||
|
|
|
||
|
|
self.v = np.zeros_like(self.f)
|
||
|
|
self.m = self.v.size
|
||
|
|
|
||
|
|
# Jacobian Evaluation
|
||
|
|
if callable(jac):
|
||
|
|
self.J = jac(self.x)
|
||
|
|
self.J_updated = True
|
||
|
|
self.njev += 1
|
||
|
|
|
||
|
|
if (sparse_jacobian or
|
||
|
|
sparse_jacobian is None and sps.issparse(self.J)):
|
||
|
|
def jac_wrapped(x):
|
||
|
|
self.njev += 1
|
||
|
|
return sps.csr_matrix(jac(x))
|
||
|
|
self.J = sps.csr_matrix(self.J)
|
||
|
|
self.sparse_jacobian = True
|
||
|
|
|
||
|
|
elif sps.issparse(self.J):
|
||
|
|
def jac_wrapped(x):
|
||
|
|
self.njev += 1
|
||
|
|
return jac(x).toarray()
|
||
|
|
self.J = self.J.toarray()
|
||
|
|
self.sparse_jacobian = False
|
||
|
|
|
||
|
|
else:
|
||
|
|
def jac_wrapped(x):
|
||
|
|
self.njev += 1
|
||
|
|
return np.atleast_2d(jac(x))
|
||
|
|
self.J = np.atleast_2d(self.J)
|
||
|
|
self.sparse_jacobian = False
|
||
|
|
|
||
|
|
def update_jac():
|
||
|
|
self.J = jac_wrapped(self.x)
|
||
|
|
|
||
|
|
elif jac in FD_METHODS:
|
||
|
|
self.J = approx_derivative(fun_wrapped, self.x, f0=self.f,
|
||
|
|
**finite_diff_options)
|
||
|
|
self.J_updated = True
|
||
|
|
|
||
|
|
if (sparse_jacobian or
|
||
|
|
sparse_jacobian is None and sps.issparse(self.J)):
|
||
|
|
def update_jac():
|
||
|
|
self._update_fun()
|
||
|
|
self.J = sps.csr_matrix(
|
||
|
|
approx_derivative(fun_wrapped, self.x, f0=self.f,
|
||
|
|
**finite_diff_options))
|
||
|
|
self.J = sps.csr_matrix(self.J)
|
||
|
|
self.sparse_jacobian = True
|
||
|
|
|
||
|
|
elif sps.issparse(self.J):
|
||
|
|
def update_jac():
|
||
|
|
self._update_fun()
|
||
|
|
self.J = approx_derivative(fun_wrapped, self.x, f0=self.f,
|
||
|
|
**finite_diff_options).toarray()
|
||
|
|
self.J = self.J.toarray()
|
||
|
|
self.sparse_jacobian = False
|
||
|
|
|
||
|
|
else:
|
||
|
|
def update_jac():
|
||
|
|
self._update_fun()
|
||
|
|
self.J = np.atleast_2d(
|
||
|
|
approx_derivative(fun_wrapped, self.x, f0=self.f,
|
||
|
|
**finite_diff_options))
|
||
|
|
self.J = np.atleast_2d(self.J)
|
||
|
|
self.sparse_jacobian = False
|
||
|
|
|
||
|
|
self._update_jac_impl = update_jac
|
||
|
|
|
||
|
|
# Define Hessian
|
||
|
|
if callable(hess):
|
||
|
|
self.H = hess(self.x, self.v)
|
||
|
|
self.H_updated = True
|
||
|
|
self.nhev += 1
|
||
|
|
|
||
|
|
if sps.issparse(self.H):
|
||
|
|
def hess_wrapped(x, v):
|
||
|
|
self.nhev += 1
|
||
|
|
return sps.csr_matrix(hess(x, v))
|
||
|
|
self.H = sps.csr_matrix(self.H)
|
||
|
|
|
||
|
|
elif isinstance(self.H, LinearOperator):
|
||
|
|
def hess_wrapped(x, v):
|
||
|
|
self.nhev += 1
|
||
|
|
return hess(x, v)
|
||
|
|
|
||
|
|
else:
|
||
|
|
def hess_wrapped(x, v):
|
||
|
|
self.nhev += 1
|
||
|
|
return np.atleast_2d(np.asarray(hess(x, v)))
|
||
|
|
self.H = np.atleast_2d(np.asarray(self.H))
|
||
|
|
|
||
|
|
def update_hess():
|
||
|
|
self.H = hess_wrapped(self.x, self.v)
|
||
|
|
elif hess in FD_METHODS:
|
||
|
|
def jac_dot_v(x, v):
|
||
|
|
return jac_wrapped(x).T.dot(v)
|
||
|
|
|
||
|
|
def update_hess():
|
||
|
|
self._update_jac()
|
||
|
|
self.H = approx_derivative(jac_dot_v, self.x,
|
||
|
|
f0=self.J.T.dot(self.v),
|
||
|
|
args=(self.v,),
|
||
|
|
**finite_diff_options)
|
||
|
|
update_hess()
|
||
|
|
self.H_updated = True
|
||
|
|
elif isinstance(hess, HessianUpdateStrategy):
|
||
|
|
self.H = hess
|
||
|
|
self.H.initialize(self.n, 'hess')
|
||
|
|
self.H_updated = True
|
||
|
|
self.x_prev = None
|
||
|
|
self.J_prev = None
|
||
|
|
|
||
|
|
def update_hess():
|
||
|
|
self._update_jac()
|
||
|
|
# When v is updated before x was updated, then x_prev and
|
||
|
|
# J_prev are None and we need this check.
|
||
|
|
if self.x_prev is not None and self.J_prev is not None:
|
||
|
|
delta_x = self.x - self.x_prev
|
||
|
|
delta_g = self.J.T.dot(self.v) - self.J_prev.T.dot(self.v)
|
||
|
|
self.H.update(delta_x, delta_g)
|
||
|
|
|
||
|
|
self._update_hess_impl = update_hess
|
||
|
|
|
||
|
|
if isinstance(hess, HessianUpdateStrategy):
|
||
|
|
def update_x(x):
|
||
|
|
self._update_jac()
|
||
|
|
self.x_prev = self.x
|
||
|
|
self.J_prev = self.J
|
||
|
|
self.x = np.atleast_1d(x).astype(float)
|
||
|
|
self.f_updated = False
|
||
|
|
self.J_updated = False
|
||
|
|
self.H_updated = False
|
||
|
|
self._update_hess()
|
||
|
|
else:
|
||
|
|
def update_x(x):
|
||
|
|
self.x = np.atleast_1d(x).astype(float)
|
||
|
|
self.f_updated = False
|
||
|
|
self.J_updated = False
|
||
|
|
self.H_updated = False
|
||
|
|
|
||
|
|
self._update_x_impl = update_x
|
||
|
|
|
||
|
|
def _update_v(self, v):
|
||
|
|
if not np.array_equal(v, self.v):
|
||
|
|
self.v = v
|
||
|
|
self.H_updated = False
|
||
|
|
|
||
|
|
def _update_x(self, x):
|
||
|
|
if not np.array_equal(x, self.x):
|
||
|
|
self._update_x_impl(x)
|
||
|
|
|
||
|
|
def _update_fun(self):
|
||
|
|
if not self.f_updated:
|
||
|
|
self._update_fun_impl()
|
||
|
|
self.f_updated = True
|
||
|
|
|
||
|
|
def _update_jac(self):
|
||
|
|
if not self.J_updated:
|
||
|
|
self._update_jac_impl()
|
||
|
|
self.J_updated = True
|
||
|
|
|
||
|
|
def _update_hess(self):
|
||
|
|
if not self.H_updated:
|
||
|
|
self._update_hess_impl()
|
||
|
|
self.H_updated = True
|
||
|
|
|
||
|
|
def fun(self, x):
|
||
|
|
self._update_x(x)
|
||
|
|
self._update_fun()
|
||
|
|
return self.f
|
||
|
|
|
||
|
|
def jac(self, x):
|
||
|
|
self._update_x(x)
|
||
|
|
self._update_jac()
|
||
|
|
return self.J
|
||
|
|
|
||
|
|
def hess(self, x, v):
|
||
|
|
# v should be updated before x.
|
||
|
|
self._update_v(v)
|
||
|
|
self._update_x(x)
|
||
|
|
self._update_hess()
|
||
|
|
return self.H
|
||
|
|
|
||
|
|
|
||
|
|
class LinearVectorFunction:
|
||
|
|
"""Linear vector function and its derivatives.
|
||
|
|
|
||
|
|
Defines a linear function F = A x, where x is N-D vector and
|
||
|
|
A is m-by-n matrix. The Jacobian is constant and equals to A. The Hessian
|
||
|
|
is identically zero and it is returned as a csr matrix.
|
||
|
|
"""
|
||
|
|
def __init__(self, A, x0, sparse_jacobian):
|
||
|
|
if sparse_jacobian or sparse_jacobian is None and sps.issparse(A):
|
||
|
|
self.J = sps.csr_matrix(A)
|
||
|
|
self.sparse_jacobian = True
|
||
|
|
elif sps.issparse(A):
|
||
|
|
self.J = A.toarray()
|
||
|
|
self.sparse_jacobian = False
|
||
|
|
else:
|
||
|
|
# np.asarray makes sure A is ndarray and not matrix
|
||
|
|
self.J = np.atleast_2d(np.asarray(A))
|
||
|
|
self.sparse_jacobian = False
|
||
|
|
|
||
|
|
self.m, self.n = self.J.shape
|
||
|
|
|
||
|
|
self.x = np.atleast_1d(x0).astype(float)
|
||
|
|
self.f = self.J.dot(self.x)
|
||
|
|
self.f_updated = True
|
||
|
|
|
||
|
|
self.v = np.zeros(self.m, dtype=float)
|
||
|
|
self.H = sps.csr_matrix((self.n, self.n))
|
||
|
|
|
||
|
|
def _update_x(self, x):
|
||
|
|
if not np.array_equal(x, self.x):
|
||
|
|
self.x = np.atleast_1d(x).astype(float)
|
||
|
|
self.f_updated = False
|
||
|
|
|
||
|
|
def fun(self, x):
|
||
|
|
self._update_x(x)
|
||
|
|
if not self.f_updated:
|
||
|
|
self.f = self.J.dot(x)
|
||
|
|
self.f_updated = True
|
||
|
|
return self.f
|
||
|
|
|
||
|
|
def jac(self, x):
|
||
|
|
self._update_x(x)
|
||
|
|
return self.J
|
||
|
|
|
||
|
|
def hess(self, x, v):
|
||
|
|
self._update_x(x)
|
||
|
|
self.v = v
|
||
|
|
return self.H
|
||
|
|
|
||
|
|
|
||
|
|
class IdentityVectorFunction(LinearVectorFunction):
|
||
|
|
"""Identity vector function and its derivatives.
|
||
|
|
|
||
|
|
The Jacobian is the identity matrix, returned as a dense array when
|
||
|
|
`sparse_jacobian=False` and as a csr matrix otherwise. The Hessian is
|
||
|
|
identically zero and it is returned as a csr matrix.
|
||
|
|
"""
|
||
|
|
def __init__(self, x0, sparse_jacobian):
|
||
|
|
n = len(x0)
|
||
|
|
if sparse_jacobian or sparse_jacobian is None:
|
||
|
|
A = sps.eye(n, format='csr')
|
||
|
|
sparse_jacobian = True
|
||
|
|
else:
|
||
|
|
A = np.eye(n)
|
||
|
|
sparse_jacobian = False
|
||
|
|
super().__init__(A, x0, sparse_jacobian)
|