# Source code for gemseo.post.core.hessians

# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
# Contributors:
#    INITIAL AUTHORS - initial API and implementation and/or initial
#                        documentation
#        :author: Francois Gallard
#    OTHER AUTHORS   - MACROSCOPIC CHANGES
r"""Approximation of the Hessian matrix from an optimization history.

Notations:

- :math:f: the function of interest for which to approximate the Hessian matrix,
- :math:y: the output value of :math:f,
- :math:x\in\mathbb{R}^d: the :math:d input variables of :math:f,
- :math:k: the :math:k-th iteration of the optimization history,
- :math:K: the iteration of the optimization history
at which to approximate the Hessian matrix,
- :math:x_k: the input value at iteration :math:k,
- :math:\Delta x_k=x_{k+1}-x_k: the variation of :math:x
from iteration :math:k to iteration :math:k+1,
- :math:y_k: the output value at iteration :math:k,
- :math:\Delta y_k=y_{k+1}-y_k: the variation of the function output
from iteration :math:k to iteration :math:k+1,
- :math:g_k: the gradient of :math:f at :math:x_k,
- :math:\Delta g_k=g_{k+1}-g_k: the variation of the gradient
from iteration :math:k to iteration :math:k+1,
- :math:B_k: the approximation of the Hessian of :math:f at :math:x_k,
- :math:H_k: the inverse of :math:B_k.
"""
from __future__ import annotations

import logging
from typing import Generator

from numpy import array
from numpy import atleast_2d
from numpy import concatenate
from numpy import cumsum
from numpy import diag as np_diag
from numpy import dot
from numpy import eye
from numpy import inf
from numpy import ndarray
from numpy import sqrt
from numpy import trace
from numpy import zeros
from numpy.linalg import cholesky
from numpy.linalg import inv
from numpy.linalg import LinAlgError
from numpy.linalg import multi_dot
from numpy.linalg import norm
from numpy.matlib import repmat
from scipy.optimize import leastsq

from gemseo.algos.database import Database
from gemseo.algos.design_space import DesignSpace

LOGGER = logging.getLogger(__name__)

r"""Approximation of the Hessian matrix from an optimization history."""

history: Database
"""The optimization history
containing input values, output values and Jacobian values.
"""

x_ref: ndarray | None
"""The value :math:x_K
of the input variables :math:x
at the iteration :math:K of the optimization history;
this is the point at which
the Hessian matrix and its inverse are approximated."""

"""The value :math:g_K
of the gradient function :math:g of :math:f at :math:x_K."""

f_ref: ndarray | None
"""The value :math:y_K of the output of :math:f at :math:x_K."""

b_mat_history: list[ndarray]
r"""The history :math:B_0,B_1,\ldots,B_K
of the approximations of the Hessian matrix :math:B."""

h_mat_history: list[ndarray]
r"""The history :math:H_0,H_1,\ldots,H_K
of the approximations of the inverse Hessian matrix :math:H."""

def __init__(
self,
history: Database,
) -> None:  # noqa: E262, E261
"""
Args:
history: The optimization history
containing input values, output values and Jacobian values.
"""
self.history = history
self.x_ref = None
self.f_ref = None
self.b_mat_history = []
self.h_mat_history = []

self,
funcname: str,
first_iter: int = 0,
last_iter: int = 0,
at_most_niter: int = -1,
func_index: int | None = None,
normalize_design_space: bool = False,
design_space: DesignSpace | None = None,
) -> tuple[ndarray, ndarray, int, int]:
"""Return the histories of the inputs and gradient.

Args:
funcname: The name of the function for which to get the gradient.
first_iter: The first iteration of the history to be considered.
last_iter: The last iteration of the history to be considered.
If 0, consider all the iterations.
at_most_niter: The maximum number of iterations to be considered.
If -1, consider all the iterations.
func_index: The index of the output of interest
to be defined if the function has a multidimensional output.
If None and if the output is multidimensional, an error is raised.
normalize_design_space: Whether to scale the input values between 0 and 1
to work in a normalized input space.
design_space: The input space used to scale the input values
if normalize_design_space is True.

Returns:
* The history of the input variables.
* The history of the gradient.
* The length of the history.
* The dimension of the input space.

Raises:
ValueError: When either
the gradient history contains a single element,
func_index is None while the function output is a vector,
func_index is not an output index,
the shape of the history of the input variables
is not consistent with the shape of the history of the gradient
or the optimization history size is insufficient.
"""
# TODO: use None as default value for last_iter and at_most_iter
if normalize_design_space:
(
x_hist,

assert grad_hist_length == len(x_hist)  # TODO: remove it

raise ValueError(
"Cannot build approximation for function: {} "
"because its gradient history is too small : {}.".format(
)
)

x_hist = array(x_hist)
# TODO: add shapes in the exception message
raise ValueError(
"Inconsistent gradient and design variables optimization history."
)

# Function is a vector, Jacobian is a 2D matrix
if func_index is None:
raise ValueError(
"Function {} has a vector output "
"then function index of output "
"must be specified.".format(funcname)
)

if not 0 <= func_index < output_size:
raise ValueError(
"Function {} has a vector output of size {}, "
"function index {} is out of range.".format(
funcname, output_size, func_index
)
)

if last_iter == 0:
x_hist = x_hist[first_iter:, :]
else:
x_hist = x_hist[first_iter:last_iter, :]

n_iterations = x_hist.shape[0]
if 0 < at_most_niter < n_iterations:
x_hist = x_hist[n_iterations - at_most_niter :, :]

n_iterations, input_dimension = x_hist.shape
if n_iterations < 2 or input_dimension == 0:
# TODO: split into two tests
raise ValueError(
"Insufficient optimization history size, "
"niter={} nparam = {}.".format(n_iterations, input_dimension)
)

self.x_ref = x_hist[-1]
if last_iter == 0:
self.f_ref = array(self.history.get_func_history(funcname))[-1]
else:
self.f_ref = array(self.history.get_func_history(funcname))[:last_iter][-1]

@staticmethod
def _normalize_x_g(
x_hist: ndarray,
design_space: DesignSpace,
) -> tuple[ndarray, ndarray]:
"""Scale the design variables between 0 and 1 in the histories.

Args:
x_hist: The history of the input variables.
design_space: The input space used to scale the input variables.

Returns:
* The history of the scaled input variables.
* The history of the gradient.

Raises:
ValueError: When the input space is None.
"""
if design_space is None:
raise ValueError(
"Design space must be provided "
"when using a normalize_design_space option."
)

scaled_x_hist.append(design_space.normalize_vect(x_value))

[docs]    @staticmethod
def get_s_k_y_k(
x_hist: ndarray,
iteration: int,
) -> tuple[ndarray, ndarray]:
r"""Compute the variation of the input variables and gradient from an iteration.

The variations from the iteration :math:k are defined by:

- :math:\Delta x_k = x_{k+1}-x_k for the input variables,
- :math:\Delta g_k = g_{k+1} - g_k for the gradient.

Args:
x_hist: The history of the input variables.
iteration: The optimization iteration at which to compute the variations.

Returns:
* The difference between the input variables at iteration iteration+1
and the input variables at iteration iteration.
* The difference between the gradient at iteration iteration+1
and the gradient at iteration iteration.

Raises:
ValueError: When the iteration is not stored in the database.
"""
if iteration >= n_iterations:
raise ValueError(
"Iteration {} is higher than number of gradients "
"in the database : {}.".format(iteration, n_iterations)
)

input_diff = atleast_2d(x_hist[iteration + 1] - x_hist[iteration]).T

[docs]    @staticmethod
def iterate_s_k_y_k(
x_hist: ndarray,
) -> Generator[tuple[ndarray, ndarray]]:
r"""Compute the variations of the input variables and gradient.

The variations from the iteration :math:k are defined by:

- :math:\Delta x_k = x_{k+1}-x_k for the input variables,
- :math:\Delta g_k = g_{k+1} - g_k for the gradient.

Args:
x_hist: The history of the input variables.

Returns:
* The difference between the input variables at iteration iteration
and the input variables at iteration iteration+1.
* The difference between the gradient at iteration iteration
and the gradient at iteration iteration+1.
"""
for iteration in range(len(x_hist) - 1):
)

[docs]    def build_approximation(
self,
funcname: str,
save_diag: bool = False,
first_iter: int = 0,
last_iter: int = -1,
b_mat0: ndarray | None = None,
at_most_niter: int = -1,
func_index: int | None = None,
save_matrix: bool = False,
scaling: bool = False,
normalize_design_space: bool = False,
design_space: DesignSpace | None = None,
) -> tuple[ndarray, ndarray, ndarray | None, ndarray | None]:
# pylint: disable=W0221
"""Compute :math:B, the approximation of the Hessian matrix.

Args:
funcname: The name of the function
for which to approximate the Hessian matrix.
save_diag: Whether to return the approximations of the Hessian's diagonal.
first_iter: The first iteration of the history to be considered.
last_iter: The last iteration of the history to be considered.
b_mat0: The initial approximation of the Hessian matrix.
at_most_niter: The maximum number of iterations to be considered.
at the last iteration.
func_index: The index of the output of interest
to be defined if the function has a multidimensional output.
If None and if the output is multidimensional, an error is raised.
save_matrix: Whether to store the approximations of the Hessian
in :attr:.HessianApproximation.b_mat_history.
scaling: do scaling step
normalize_design_space: Whether to scale the input values between 0 and 1
to work in a normalized input space.
design_space: The input space used to scale the input values
if normalize_design_space is True.

Returns:
* :math:B, the approximation of the Hessian matrix.
* The diagonal of :math:B.
* The history of the input variables if return_x_grad is True.
* The history of the gradient if return_x_grad is True.
"""
funcname,
first_iter,
last_iter,
at_most_niter,
func_index,
normalize_design_space,
design_space,
)
if b_mat0 is None:
hessian = (1.0 / alpha) * eye(grad_hist.shape[1])
elif b_mat0.size == 0:
hessian = zeros((x_hist.shape[1],) * 2)
else:
hessian = b_mat0

hessian_diagonal = []

if save_diag:
hessian_diagonal.append(np_diag(hessian).copy())
if save_matrix:
self.b_mat_history.append(hessian.copy())

return hessian, hessian_diagonal, x_hist[-1, :], grad_hist[-1, :]

return hessian, hessian_diagonal, None, None

[docs]    @staticmethod
def compute_scaling(
hessk: ndarray,
hessk_dsk: ndarray,
dskt_hessk_dsk: ndarray,
dyk: ndarray,
dyt_dsk: ndarray,
) -> tuple[float, float]:
r"""Compute the scaling coefficients :math:c_1 and :math:c_2.

- :math:c_1=\frac{d-1}{\mathrm{Tr}(B_k)-\frac{\|B_k\Delta x_k\|_2^2}
{\Delta x_k^T B_k\Delta x_k}},
- :math:c_2=\frac{\Delta g_k^T\Delta x_k}{\|\Delta g_k\|_2^2}.

Args:
hessk: The approximation :math:B_k of the Hessian matrix
at iteration :math:k.
hessk_dsk: The product :math:B_k\Delta x_k.
dskt_hessk_dsk: The product :math:\Delta x_k^T B_k\Delta x_k.
dyk: The variation of the gradient :math:\Delta g_k.
dyt_dsk: The product
:math:\Delta g_k^T\Delta x_k.

Returns:
* coeff1: TODO
* coeff2: TODO
"""
coeff1 = (len(hessk_dsk) - 1) / (
trace(hessk) - norm(hessk_dsk) ** 2 / dskt_hessk_dsk
)
coeff2 = dyt_dsk / norm(dyk) ** 2
return coeff1, coeff2

[docs]    @staticmethod
def iterate_approximation(
hessk: ndarray,
dsk: ndarray,
dyk: ndarray,
scaling: bool = False,
) -> None:
r"""Update :math:B from iteration :math:k to iteration :math:k+1.

Based on an iteration of the BFGS algorithm:

:math:B_{k+1} =
B_k
- c_1\frac{B_k\Delta x_k\Delta x_k^TB_k}{\Delta x_k^TB_k\Delta x_k}
+ c_2\frac{\Delta g_k\Delta g_k^T}{\Delta g_k^T\Delta x_k}

where :math:c_1=c_2=1 if scaling is False, otherwise:

- :math:c_1=\frac{d-1}{\mathrm{Tr}(B_k)-\frac{\|B_k\Delta x_k\|_2^2}
{\Delta x_k^T B_k\Delta x_k}},
- :math:c_2=\frac{\Delta g_k^T\Delta x_k}{\|\Delta g_k\|_2^2}.

.. note::
hessk represents :math:B_k initially
before to be overwritten by :math:B_{k+1} when passed to this method.

.. seealso::
BFGS algorithm.
<https://en.wikipedia.org/wiki/Broyden-Fletcher-Goldfarb-Shanno_algorithm>_

Args:
hessk: The approximation :math:B_k of the Hessian matrix
at iteration :math:k.
dsk: The variation :math:\Delta x_k of the input variables.
dyk: The variation :math:\Delta g_k of the gradient.
scaling: Whether to use a scaling stage.
"""
dyt_dsk = dot(dyk.T, dsk)
hessk_dsk = dot(hessk, dsk)
dskt_hessk_dsk = multi_dot((dsk.T, hessk, dsk))
# Build the next approximation:
b_first_term = hessk - multi_dot((hessk, dsk, dsk.T, hessk)) / dskt_hessk_dsk
b_second_term = dot(dyk, dyk.T) / dyt_dsk
if not scaling:
hessk[:, :] = b_first_term + b_second_term
else:
c_1, c_2 = HessianApproximation.compute_scaling(
hessk, hessk_dsk, dskt_hessk_dsk, dyk, dyt_dsk
)
hessk[:, :] = c_1 * b_first_term + c_2 * b_second_term

[docs]    def build_inverse_approximation(
self,
funcname: str,
save_diag: int = False,
first_iter: int = 0,
last_iter: int = -1,
h_mat0: ndarray | None = None,
at_most_niter: int = -1,
func_index: int | None = None,
save_matrix: bool = False,
factorize: bool = False,
scaling: bool = False,
angle_tol: float = 1e-5,
step_tol: float = 1e10,
normalize_design_space: bool = False,
design_space: DesignSpace | None = None,
) -> tuple[ndarray, ndarray, ndarray | None, ndarray | None]:
r"""Compute :math:H, the approximation of the inverse of the Hessian matrix.

Args:
funcname: The name of the function
for which to approximate the inverse of the Hessian matrix.
save_diag: Whether to return the list of diagonal approximations.
first_iter: The first iteration of the history to be considered.
last_iter: The last iteration of the history to be considered.
h_mat0: The initial approximation of the inverse of the Hessian matrix.
If None,
use :math:H_0=\frac{\Delta g_k^T\Delta x_k}
{\Delta g_k^T\Delta g_k}I_d.
at_most_niter: The maximum number of iterations to take.
at the last iteration.
func_index: The output index of the function
to be provided if the function output is a vector.
save_matrix: Whether to store the approximations of the inverse Hessian
in :attr:.HessianApproximation.h_mat_history.
factorize: Whether to factorize the approximations of the Hessian matrix
and its inverse, as :math:A=A_{1/2}A_{1/2}^T for a matrix :math:A.
scaling: do scaling step
angle_tol: The significativity level for
:math:\Delta g_k^T\Delta x_k.
step_tol: The significativity level for
:math:\|\Delta g_k\|_{\infty}.
normalize_design_space: Whether to scale the input values between 0 and 1
to work in a normalized input space.
design_space: The input space used to scale the input values
if normalize_design_space is True.

Returns:
* :math:H, the approximation of the inverse of the Hessian matrix.
* The diagonal of :math:H.
* The history of the input variables if return_x_grad is True.
* The history of the gradient if return_x_grad is True.
* The matrix :math:H_{1/2} such that :math:H=H_{1/2}H_{1/2}^T
if factorize is True.
* :math:B, the approximation of the Hessian matrix.
* A matrix :math:B_{1/2} such that :math:B=B_{1/2}B_{1/2}^T
if factorize is True.

Raises:
LinAlgError: When either
the inversion of :math:H fails
or the Cholesky decomposition of :math:H or :math:B fails.
"""
funcname,
first_iter,
last_iter,
at_most_niter,
func_index,
normalize_design_space,
design_space,
)
h_factor = None  # to become a matrix G such that H = G*G', optionally
b_factor = None  # to become the inverse of the matrix G
if h_mat0 is None:
alpha = dot(y_k.T, s_k) / dot(y_k.T, y_k)
h_mat = alpha * eye(n_x)
b_mat = 1.0 / alpha * eye(n_x)
if factorize:
h_factor = sqrt(alpha) * eye(n_x)
b_factor = eye(n_x) / sqrt(alpha)

elif len(h_mat0) == 0:
n_x = len(x_hist[0])
h_mat = zeros((n_x, n_x))
b_mat = zeros((n_x, n_x))
if factorize:
h_factor = zeros((n_x, n_x))
b_factor = zeros((n_x, n_x))

else:
h_mat = h_mat0
try:
b_mat = inv(h_mat)
except LinAlgError:
raise LinAlgError("The inversion of h_mat failed.")

if factorize or scaling:
try:
h_factor = cholesky(h_mat)
b_factor = cholesky(b_mat).T
except LinAlgError:
raise LinAlgError(
"The Cholesky decomposition of h_factor or b_factor failed."
)

diag = []
count = 0
k = 0
for s_k, y_k in self.iterate_s_k_y_k(x_hist, grad_hist):
k = k + 1
if dot(s_k.T, y_k) > angle_tol and norm(y_k, inf) < step_tol:
count = count + 1
self.iterate_inverse_approximation(
h_mat,
s_k,
y_k,
h_factor,
b_mat,
b_factor,
factorize=factorize,
scaling=scaling,
)

if save_diag:
diag.append(np_diag(h_mat).copy())

if save_matrix:
self.h_mat_history.append(h_mat.copy())

return h_mat, diag, x_hist[-1, :], grad_hist[-1, :], None, None, None

return h_mat, diag, None, None, h_factor, b_mat, b_factor

[docs]    @staticmethod
def compute_corrections(
x_hist: ndarray,
) -> tuple[ndarray, ndarray]:
"""Compute the successive variations of both input variables and gradient.

These variations are called *corrections*.

Args:
x_hist: The history of the input variables.

Returns:
* The successive variations of the input variables.
* The successive variations of the gradient.
"""
n_iter = len(x_hist)
x_corr = x_hist[1:n_iter].T - x_hist[: n_iter - 1].T

[docs]    @staticmethod
def rebuild_history(
x_corr: ndarray,
x_0: ndarray,
g_0: ndarray,
) -> tuple[ndarray, ndarray]:
"""Compute the history from the corrections of input variables and gradient.

A *correction* is the variation of a quantity between two successive iterations.

Args:
x_corr: The corrections of the input variables.
x_0: The initial values of the input variables.
g_0: The initial value of the gradient.

Returns:
* The history of the input variables.
* The history of the gradient.
"""
# Rebuild the argument history:
x_hist = repmat(x_0, x_corr.shape[1], 1) + cumsum(x_corr.T, axis=0)
x_hist = concatenate((atleast_2d(x_0), x_hist), axis=0)

[docs]    @staticmethod
def iterate_inverse_approximation(
h_mat: ndarray,
s_k: ndarray,
y_k: ndarray,
h_factor: ndarray | None = None,
b_mat: ndarray | None = None,
b_factor: ndarray | None = None,
factorize: bool = False,
scaling: bool = False,
):
r"""Update :math:H and :math:B from step :math:k to step :math:k+1.

Use an iteration of the BFGS algorithm:

:math:B_{k+1} =
B_k
- c_1\frac{B_k\Delta x_k\Delta x_k^TB_k}{\Delta x_k^TB_k\Delta x_k}
+ c_2\frac{\Delta g_k\Delta g_k^T}{\Delta g_k^T\Delta x_k}

and

:math:H_{k+1}=c_1^{-1}\Pi_{k+1}H_k\Pi_{k+1}^T
+c_2^{-1}\frac{\Delta x_k\Delta x_k^T}{\Delta g_k^T\Delta x_k}

where:

:math:\Pi_{k+1}=I_d-\frac{\Delta x_k\Delta g_k^T}
{\Delta g_k^T\Delta x_k}

and where :math:c_1=c_2=1 if scaling is False, otherwise:

- :math:c_1=\frac{d-1}{\mathrm{Tr}(B_k)-\frac{\|B_k\Delta x_k\|_2^2}
{\Delta x_k^T B_k\Delta x_k}},
- :math:c_2=\frac{\Delta g_k^T\Delta x_k}{\|\Delta g_k\|_2^2}.

.. note::
h_mat and b_mat represent :math:H_k and :math:B_k initially
before to be overwritten by :math:H_{k+1} and :math:B_{k+1}
when passed to this method.

.. seealso::
BFGS algorithm.
<https://en.wikipedia.org/wiki/Broyden-Fletcher-Goldfarb-Shanno_algorithm>_

Args:
h_mat: The approximation :math:H_k of the inverse of the Hessian matrix
at iteration :math:k.
s_k: The variation :math:\Delta x_k of the input variables.
y_k: The variation :math:\Delta g_k of the gradient.
h_factor: The square root of the :math:H_k at iteration :math:k.
b_mat: The approximation :math:B_k of the Hessian matrix
at iteration :math:k if factorize is True.
b_factor: The square root of the :math:B_k at iteration :math:k
if factorize is True.
factorize: Whether to update the approximations of the Hessian matrix
and its inverse, as :math:A=A_{1/2}A_{1/2}^T for a matrix :math:A.
scaling: do scaling step
"""
# Compute the two terms of the non-scaled updated matrix:
yts = dot(y_k.T, s_k)
proj = eye(len(s_k)) - dot(s_k, y_k.T) / yts
h_first_term = multi_dot((proj, h_mat, proj.T))
h_second_term = dot(s_k, s_k.T) / yts
b_s = dot(b_mat, s_k)
st_b_s = dot(s_k.T, b_s)
# Compute the scaling coefficients:
if scaling:
coeff1, coeff2 = HessianApproximation.compute_scaling(
b_mat, b_s, st_b_s, y_k, yts
)
else:
coeff1, coeff2 = 1.0, 1.0

# Update the inverse approximation H and, optionally, the factor G:
h_mat[:, :] = h_first_term / coeff1 + h_second_term / coeff2
if factorize:
sst_b = dot(s_k, b_s.T)
left = proj / sqrt(coeff1) + sst_b / sqrt(coeff2 * yts * st_b_s)
h_factor[:, :] = dot(left, h_factor)
# b_factor[:, :] = dot(eye(len(s_k)) - sstB.T / stBs / sqrt(coeff1)
#                      + dot(y_k, s_k.T)
#                      / sqrt(coeff2 * stBs * yts),
#                      b_factor)
right = sqrt(coeff1) * (eye(len(s_k)) - sst_b / st_b_s)
right += sqrt(coeff2) * dot(s_k, y_k.T) / sqrt(st_b_s * yts)
b_factor[:, :] = dot(b_factor, right)

# Update the Hessian approximation:
b_first_term = b_mat - multi_dot((b_s, b_s.T)) / st_b_s
b_second_term = dot(y_k, y_k.T) / yts
b_mat[:, :] = coeff1 * b_first_term + coeff2 * b_second_term

#             b_mat[:, :] = multi_dot((proj.T, b_mat, proj)) \
#                 + dot(y_k, y_k.T) / yts

[docs]class BFGSApprox(HessianApproximation):
"""Hessian matrix approximation with the BFGS algorithm."""

[docs]    @staticmethod
def iterate_s_k_y_k(
x_hist: ndarray,
) -> Generator[tuple[ndarray, ndarray]]:
for iteration in range(len(x_hist) - 1):
)
# All pairs curvatures shall be positive
# if dot(s_k.T, y_k) > 0.:

[docs]class SR1Approx(HessianApproximation):
r"""Hessian matrix approximation with the Symmetric Rank One (SR1) algorithm.

The approximation at iteration :math:k+1 is:

.. math::
B_{k+1}=B_k +
\frac{(\Delta g_k-B_k\Delta x_k)(\Delta g_k-B_k\Delta x_k)^T}
{(\Delta g_k-B_k\Delta x_k)^T\Delta x_k}

This update from iteration :math:k to iteration :math:k+1 is applied only if
:math:|(\Delta g_k-B_k\Delta x_k)^T\Delta x_k|
\geq \varepsilon\|\Delta x_k\|\|\Delta g_k\|
where :math:\varepsilon is a small number, e.g. :math:10^{-8}.

.. seealso::

SR1 algorithm. <https://en.wikipedia.org/wiki/Symmetric_rank-one>_
"""

EPSILON = 1e-8

[docs]    @staticmethod
def iterate_approximation(
b_mat: ndarray,
s_k: ndarray,
y_k: ndarray,
scaling: bool = False,
):
residuals = y_k - multi_dot((b_mat, s_k))
denominator = multi_dot((residuals.T, s_k))
if abs(denominator) > SR1Approx.EPSILON * norm(s_k) * norm(residuals):
b_mat[:, :] = b_mat + multi_dot((residuals, residuals.T)) / denominator
else:
LOGGER.debug(
"Denominator of SR1 update is too small, update skipped %s.",
denominator,
)

[docs]class LSTSQApprox(HessianApproximation):
"""Least squares approximation of an Hessian matrix from an optimization history."""

[docs]    def build_approximation(
self,
funcname: str,
save_diag: bool = False,
first_iter: int = 0,
last_iter: int = -1,
b_mat0: ndarray | None = None,
at_most_niter: int = -1,
scaling: bool = False,
func_index: int = -1,
normalize_design_space: bool = False,
design_space: DesignSpace | None = None,
) -> tuple[ndarray, ndarray, ndarray | None, ndarray | None]:
funcname,
first_iter,
last_iter,
at_most_niter,
func_index=func_index,
normalize_design_space=normalize_design_space,
design_space=design_space,
)
assert len(grad_hist) == len(x_hist)  # TODO: replace with an if/raise

sec_dim = max(input_dimension, n_iterations)
hessian_diagonal = []

def y_to_b(
y_vars: ndarray,
) -> ndarray:
"""Reshape the approximation from vector to matrix.

Args:
y_vars: The vector approximation.

Returns:
The square matrix version of the passed vector.
"""
y_mat = y_vars.reshape((input_dimension, input_dimension))
return y_mat + y_mat.T

def compute_error(
y_vars: ndarray,
) -> ndarray:
"""Create the least square function.

Args:
y_vars: The current approximation vector.

Returns:
The estimated error vector.
"""
hessian = y_to_b(y_vars)
err = zeros((input_dimension, sec_dim))
for item, x_current in enumerate(x_hist):
err[:, item] = dot(hessian, x_current - self.x_ref) - grad_hist[item]

err = err.reshape(-1)
if n_iterations < input_dimension:
err += y_vars

return err

x_0 = zeros(input_dimension * input_dimension)
LOGGER.debug("Start least squares problem..")
x_opt, ier = leastsq(compute_error, x0=x_0)  # , cov_x, infodict, mesg, ier
LOGGER.debug("End least squares, msg=%s", str(ier))
hessian = y_to_b(x_opt)