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"""
Unit tests for trust-region iterative subproblem.
To run it in its simplest form::
nosetests test_optimize.py
"""
from __future__ import division, print_function, absolute_import
import numpy as np
from scipy.optimize._trustregion_exact import (
estimate_smallest_singular_value,
singular_leading_submatrix,
IterativeSubproblem)
from scipy.linalg import (svd, get_lapack_funcs, det,
cho_factor, cho_solve, qr,
eigvalsh, eig, norm)
from numpy.testing import (assert_, assert_array_equal,
assert_equal, assert_array_almost_equal,
assert_array_less)
def random_entry(n, min_eig, max_eig, case):
# Generate random matrix
rand = np.random.uniform(-1, 1, (n, n))
# QR decomposition
Q, _, _ = qr(rand, pivoting='True')
# Generate random eigenvalues
eigvalues = np.random.uniform(min_eig, max_eig, n)
eigvalues = np.sort(eigvalues)[::-1]
# Generate matrix
Qaux = np.multiply(eigvalues, Q)
A = np.dot(Qaux, Q.T)
# Generate gradient vector accordingly
# to the case is being tested.
if case == 'hard':
g = np.zeros(n)
g[:-1] = np.random.uniform(-1, 1, n-1)
g = np.dot(Q, g)
elif case == 'jac_equal_zero':
g = np.zeros(n)
else:
g = np.random.uniform(-1, 1, n)
return A, g
class TestEstimateSmallestSingularValue(object):
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)
class TestSingularLeadingSubmatrix(object):
def test_for_already_singular_leading_submatrix(self):
# Define test matrix A.
# Note that the leading 2x2 submatrix is singular.
A = np.array([[1, 2, 3],
[2, 4, 5],
[3, 5, 6]])
# Get Cholesky from lapack functions
cholesky, = get_lapack_funcs(('potrf',), (A,))
# Compute Cholesky Decomposition
c, k = cholesky(A, lower=False, overwrite_a=False, clean=True)
delta, v = singular_leading_submatrix(A, c, k)
A[k-1, k-1] += delta
# Check if the leading submatrix is singular.
assert_array_almost_equal(det(A[:k, :k]), 0)
# Check if `v` fullfil the specified properties
quadratic_term = np.dot(v, np.dot(A, v))
assert_array_almost_equal(quadratic_term, 0)
def test_for_simetric_indefinite_matrix(self):
# Define test matrix A.
# Note that the leading 5x5 submatrix is indefinite.
A = np.asarray([[1, 2, 3, 7, 8],
[2, 5, 5, 9, 0],
[3, 5, 11, 1, 2],
[7, 9, 1, 7, 5],
[8, 0, 2, 5, 8]])
# Get Cholesky from lapack functions
cholesky, = get_lapack_funcs(('potrf',), (A,))
# Compute Cholesky Decomposition
c, k = cholesky(A, lower=False, overwrite_a=False, clean=True)
delta, v = singular_leading_submatrix(A, c, k)
A[k-1, k-1] += delta
# Check if the leading submatrix is singular.
assert_array_almost_equal(det(A[:k, :k]), 0)
# Check if `v` fullfil the specified properties
quadratic_term = np.dot(v, np.dot(A, v))
assert_array_almost_equal(quadratic_term, 0)
def test_for_first_element_equal_to_zero(self):
# Define test matrix A.
# Note that the leading 2x2 submatrix is singular.
A = np.array([[0, 3, 11],
[3, 12, 5],
[11, 5, 6]])
# Get Cholesky from lapack functions
cholesky, = get_lapack_funcs(('potrf',), (A,))
# Compute Cholesky Decomposition
c, k = cholesky(A, lower=False, overwrite_a=False, clean=True)
delta, v = singular_leading_submatrix(A, c, k)
A[k-1, k-1] += delta
# Check if the leading submatrix is singular
assert_array_almost_equal(det(A[:k, :k]), 0)
# Check if `v` fullfil the specified properties
quadratic_term = np.dot(v, np.dot(A, v))
assert_array_almost_equal(quadratic_term, 0)
class TestIterativeSubproblem(object):
def test_for_the_easy_case(self):
# `H` is chosen such that `g` is not orthogonal to the
# eigenvector associated with the smallest eigenvalue `s`.
H = [[10, 2, 3, 4],
[2, 1, 7, 1],
[3, 7, 1, 7],
[4, 1, 7, 2]]
g = [1, 1, 1, 1]
# Trust Radius
trust_radius = 1
# Solve Subproblem
subprob = IterativeSubproblem(x=0,
fun=lambda x: 0,
jac=lambda x: np.array(g),
hess=lambda x: np.array(H),
k_easy=1e-10,
k_hard=1e-10)
p, hits_boundary = subprob.solve(trust_radius)
assert_array_almost_equal(p, [0.00393332, -0.55260862,
0.67065477, -0.49480341])
assert_array_almost_equal(hits_boundary, True)
def test_for_the_hard_case(self):
# `H` is chosen such that `g` is orthogonal to the
# eigenvector associated with the smallest eigenvalue `s`.
H = [[10, 2, 3, 4],
[2, 1, 7, 1],
[3, 7, 1, 7],
[4, 1, 7, 2]]
g = [6.4852641521327437, 1, 1, 1]
s = -8.2151519874416614
# Trust Radius
trust_radius = 1
# Solve Subproblem
subprob = IterativeSubproblem(x=0,
fun=lambda x: 0,
jac=lambda x: np.array(g),
hess=lambda x: np.array(H),
k_easy=1e-10,
k_hard=1e-10)
p, hits_boundary = subprob.solve(trust_radius)
assert_array_almost_equal(-s, subprob.lambda_current)
def test_for_interior_convergence(self):
H = [[1.812159, 0.82687265, 0.21838879, -0.52487006, 0.25436988],
[0.82687265, 2.66380283, 0.31508988, -0.40144163, 0.08811588],
[0.21838879, 0.31508988, 2.38020726, -0.3166346, 0.27363867],
[-0.52487006, -0.40144163, -0.3166346, 1.61927182, -0.42140166],
[0.25436988, 0.08811588, 0.27363867, -0.42140166, 1.33243101]]
g = [0.75798952, 0.01421945, 0.33847612, 0.83725004, -0.47909534]
# Solve Subproblem
subprob = IterativeSubproblem(x=0,
fun=lambda x: 0,
jac=lambda x: np.array(g),
hess=lambda x: np.array(H))
p, hits_boundary = subprob.solve(1.1)
assert_array_almost_equal(p, [-0.68585435, 0.1222621, -0.22090999,
-0.67005053, 0.31586769])
assert_array_almost_equal(hits_boundary, False)
assert_array_almost_equal(subprob.lambda_current, 0)
assert_array_almost_equal(subprob.niter, 1)
def test_for_jac_equal_zero(self):
H = [[0.88547534, 2.90692271, 0.98440885, -0.78911503, -0.28035809],
[2.90692271, -0.04618819, 0.32867263, -0.83737945, 0.17116396],
[0.98440885, 0.32867263, -0.87355957, -0.06521957, -1.43030957],
[-0.78911503, -0.83737945, -0.06521957, -1.645709, -0.33887298],
[-0.28035809, 0.17116396, -1.43030957, -0.33887298, -1.68586978]]
g = [0, 0, 0, 0, 0]
# Solve Subproblem
subprob = IterativeSubproblem(x=0,
fun=lambda x: 0,
jac=lambda x: np.array(g),
hess=lambda x: np.array(H),
k_easy=1e-10,
k_hard=1e-10)
p, hits_boundary = subprob.solve(1.1)
assert_array_almost_equal(p, [0.06910534, -0.01432721,
-0.65311947, -0.23815972,
-0.84954934])
assert_array_almost_equal(hits_boundary, True)
def test_for_jac_very_close_to_zero(self):
H = [[0.88547534, 2.90692271, 0.98440885, -0.78911503, -0.28035809],
[2.90692271, -0.04618819, 0.32867263, -0.83737945, 0.17116396],
[0.98440885, 0.32867263, -0.87355957, -0.06521957, -1.43030957],
[-0.78911503, -0.83737945, -0.06521957, -1.645709, -0.33887298],
[-0.28035809, 0.17116396, -1.43030957, -0.33887298, -1.68586978]]
g = [0, 0, 0, 0, 1e-15]
# Solve Subproblem
subprob = IterativeSubproblem(x=0,
fun=lambda x: 0,
jac=lambda x: np.array(g),
hess=lambda x: np.array(H),
k_easy=1e-10,
k_hard=1e-10)
p, hits_boundary = subprob.solve(1.1)
assert_array_almost_equal(p, [0.06910534, -0.01432721,
-0.65311947, -0.23815972,
-0.84954934])
assert_array_almost_equal(hits_boundary, True)
def test_for_random_entries(self):
# Seed
np.random.seed(1)
# Dimension
n = 5
for case in ('easy', 'hard', 'jac_equal_zero'):
eig_limits = [(-20, -15),
(-10, -5),
(-10, 0),
(-5, 5),
(-10, 10),
(0, 10),
(5, 10),
(15, 20)]
for min_eig, max_eig in eig_limits:
# Generate random symmetric matrix H with
# eigenvalues between min_eig and max_eig.
H, g = random_entry(n, min_eig, max_eig, case)
# Trust radius
trust_radius_list = [0.1, 0.3, 0.6, 0.8, 1, 1.2, 3.3, 5.5, 10]
for trust_radius in trust_radius_list:
# Solve subproblem with very high accuracy
subprob_ac = IterativeSubproblem(0,
lambda x: 0,
lambda x: g,
lambda x: H,
k_easy=1e-10,
k_hard=1e-10)
p_ac, hits_boundary_ac = subprob_ac.solve(trust_radius)
# Compute objective function value
J_ac = 1/2*np.dot(p_ac, np.dot(H, p_ac))+np.dot(g, p_ac)
stop_criteria = [(0.1, 2),
(0.5, 1.1),
(0.9, 1.01)]
for k_opt, k_trf in stop_criteria:
# k_easy and k_hard computed in function
# of k_opt and k_trf accordingly to
# Conn, A. R., Gould, N. I., & Toint, P. L. (2000).
# "Trust region methods". Siam. p. 197.
k_easy = min(k_trf-1,
1-np.sqrt(k_opt))
k_hard = 1-k_opt
# Solve subproblem
subprob = IterativeSubproblem(0,
lambda x: 0,
lambda x: g,
lambda x: H,
k_easy=k_easy,
k_hard=k_hard)
p, hits_boundary = subprob.solve(trust_radius)
# Compute objective function value
J = 1/2*np.dot(p, np.dot(H, p))+np.dot(g, p)
# Check if it respect k_trf
if hits_boundary:
assert_array_equal(np.abs(norm(p)-trust_radius) <=
(k_trf-1)*trust_radius, True)
else:
assert_equal(norm(p) <= trust_radius, True)
# Check if it respect k_opt
assert_equal(J <= k_opt*J_ac, True)
|
