manyi/venv/Lib/site-packages/scipy/optimize/tests/test_trustregion_exact.py

"""
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)