""" Unit tests for trust-region iterative subproblem. To run it in its simplest form:: nosetests test_optimize.py """ 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, qr, norm) from numpy.testing import (assert_array_equal, assert_equal, assert_array_almost_equal) 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: 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: 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: 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)