def check_for_interior_convergence_and_update(
x_candidate,
hessian_info,
lambdas,
stopping_criteria,
converged,
):
"""Check for interior convergence, update candidate vector and lambdas."""
if lambdas.candidate == 0:
x_candidate = np.zeros_like(x_candidate)
converged = True
s_min, z_min = estimate_smallest_singular_value(
hessian_info.upper_triangular)
step_len = 2
if step_len**2 * s_min**2 <= stopping_criteria["k_hard"] * lambdas.current:
x_candidate = step_len * z_min
converged = True
lambda_lower_bound = max(lambdas.lower_bound,
lambdas.upper_bound - s_min**2)
lambda_new_candidate = _get_new_lambda_candidate(
lower_bound=lambda_lower_bound, upper_bound=lambdas.candidate)
lambdas_new = lambdas._replace(
candidate=lambda_new_candidate,
lower_bound=lambda_lower_bound,
upper_bound=lambdas.candidate,
)
return x_candidate, lambdas_new, converged
def _update_candidate_and_parameters_when_candidate_within_trustregion(
x_candidate,
main_model,
hessian_info,
lambdas,
newton_step,
stopping_criteria,
converged,
):
"""Update candidate vector, hessian, and lambdas when x outside trust-region."""
n = len(x_candidate)
s_min, z_min = estimate_smallest_singular_value(
hessian_info.upper_triangular)
step_len = _compute_smallest_step_len_for_candidate_vector(
x_candidate, z_min)
quadratic_term = x_candidate.T @ hessian_info.hessian_plus_lambda @ x_candidate
relative_error = (step_len**2 * s_min**2) / (quadratic_term +
lambdas.candidate)
if relative_error <= stopping_criteria["k_hard"]:
x_candidate = x_candidate + step_len * z_min
converged = True
lambda_new_lower_bound = max(lambdas.lower_bound,
lambdas.candidate - s_min**2)
hessian_plus_lambda = main_model.square_terms + newton_step * np.eye(n)
_, factorization_unsuccessful = compute_cholesky_factorization(
hessian_plus_lambda,
lower=False,
overwrite_a=False,
clean=True,
)
if factorization_unsuccessful == 0:
hessian_already_factorized = True
lambda_new_candidate = newton_step
else:
hessian_already_factorized = hessian_info.already_factorized
lambda_new_lower_bound = max(lambda_new_lower_bound, newton_step)
lambda_new_candidate = _get_new_lambda_candidate(
lower_bound=lambda_new_lower_bound, upper_bound=lambdas.candidate)
hessian_info_new = hessian_info._replace(
hessian_plus_lambda=hessian_plus_lambda,
already_factorized=hessian_already_factorized,
)
lambdas_new = lambdas._replace(
candidate=lambda_new_candidate,
lower_bound=lambda_new_lower_bound,
upper_bound=lambdas.candidate,
)
return x_candidate, hessian_info_new, lambdas_new, converged
def test_for_ill_condiotioned_matrix(self):
# Ill-conditioned triangular matrix
C = np.array([[1, 2, 3, 4], [0, 0.05, 60, 7], [0, 0, 0.8, 9],
[0, 0, 0, 10]])
# Get svd decomposition
U, s, Vt = svd(C)
# Get smallest singular value and correspondent right singular vector.
smin_svd = s[-1]
zmin_svd = Vt[-1, :]
# Estimate smallest singular value
smin, zmin = estimate_smallest_singular_value(C)
# Check the estimation
assert_array_almost_equal(smin, smin_svd, decimal=8)
assert_array_almost_equal(abs(zmin), abs(zmin_svd), decimal=8)
def test_for_ill_condiotioned_matrix(self):
# Ill-conditioned triangular matrix
C = np.array([[1, 2, 3, 4],
[0, 0.05, 60, 7],
[0, 0, 0.8, 9],
[0, 0, 0, 10]])
# Get svd decomposition
U, s, Vt = svd(C)
# Get smallest singular value and correspondent right singular vector.
smin_svd = s[-1]
zmin_svd = Vt[-1, :]
# Estimate smallest singular value
smin, zmin = estimate_smallest_singular_value(C)
# Check the estimation
assert_array_almost_equal(smin, smin_svd, decimal=8)
assert_array_almost_equal(abs(zmin), abs(zmin_svd), decimal=8)