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431 lines
15 KiB
Python
431 lines
15 KiB
Python
"""Nearly exact trust-region optimization subproblem."""
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import numpy as np
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from scipy.linalg import (norm, get_lapack_funcs, solve_triangular,
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cho_solve)
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from ._trustregion import (_minimize_trust_region, BaseQuadraticSubproblem)
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__all__ = ['_minimize_trustregion_exact',
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'estimate_smallest_singular_value',
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'singular_leading_submatrix',
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'IterativeSubproblem']
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def _minimize_trustregion_exact(fun, x0, args=(), jac=None, hess=None,
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**trust_region_options):
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"""
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Minimization of scalar function of one or more variables using
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a nearly exact trust-region algorithm.
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Options
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-------
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initial_tr_radius : float
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Initial trust-region radius.
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max_tr_radius : float
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Maximum value of the trust-region radius. No steps that are longer
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than this value will be proposed.
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eta : float
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Trust region related acceptance stringency for proposed steps.
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gtol : float
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Gradient norm must be less than ``gtol`` before successful
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termination.
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"""
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if jac is None:
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raise ValueError('Jacobian is required for trust region '
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'exact minimization.')
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if hess is None:
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raise ValueError('Hessian matrix is required for trust region '
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'exact minimization.')
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return _minimize_trust_region(fun, x0, args=args, jac=jac, hess=hess,
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subproblem=IterativeSubproblem,
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**trust_region_options)
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def estimate_smallest_singular_value(U):
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"""Given upper triangular matrix ``U`` estimate the smallest singular
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value and the correspondent right singular vector in O(n**2) operations.
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Parameters
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----------
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U : ndarray
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Square upper triangular matrix.
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Returns
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-------
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s_min : float
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Estimated smallest singular value of the provided matrix.
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z_min : ndarray
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Estimatied right singular vector.
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Notes
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-----
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The procedure is based on [1]_ and is done in two steps. First, it finds
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a vector ``e`` with components selected from {+1, -1} such that the
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solution ``w`` from the system ``U.T w = e`` is as large as possible.
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Next it estimate ``U v = w``. The smallest singular value is close
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to ``norm(w)/norm(v)`` and the right singular vector is close
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to ``v/norm(v)``.
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The estimation will be better more ill-conditioned is the matrix.
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References
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----------
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.. [1] Cline, A. K., Moler, C. B., Stewart, G. W., Wilkinson, J. H.
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An estimate for the condition number of a matrix. 1979.
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SIAM Journal on Numerical Analysis, 16(2), 368-375.
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"""
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U = np.atleast_2d(U)
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m, n = U.shape
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if m != n:
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raise ValueError("A square triangular matrix should be provided.")
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# A vector `e` with components selected from {+1, -1}
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# is selected so that the solution `w` to the system
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# `U.T w = e` is as large as possible. Implementation
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# based on algorithm 3.5.1, p. 142, from reference [2]
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# adapted for lower triangular matrix.
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p = np.zeros(n)
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w = np.empty(n)
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# Implemented according to: Golub, G. H., Van Loan, C. F. (2013).
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# "Matrix computations". Forth Edition. JHU press. pp. 140-142.
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for k in range(n):
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wp = (1-p[k]) / U.T[k, k]
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wm = (-1-p[k]) / U.T[k, k]
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pp = p[k+1:] + U.T[k+1:, k]*wp
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pm = p[k+1:] + U.T[k+1:, k]*wm
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if abs(wp) + norm(pp, 1) >= abs(wm) + norm(pm, 1):
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w[k] = wp
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p[k+1:] = pp
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else:
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w[k] = wm
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p[k+1:] = pm
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# The system `U v = w` is solved using backward substitution.
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v = solve_triangular(U, w)
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v_norm = norm(v)
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w_norm = norm(w)
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# Smallest singular value
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s_min = w_norm / v_norm
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# Associated vector
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z_min = v / v_norm
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return s_min, z_min
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def gershgorin_bounds(H):
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"""
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Given a square matrix ``H`` compute upper
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and lower bounds for its eigenvalues (Gregoshgorin Bounds).
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Defined ref. [1].
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References
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----------
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.. [1] Conn, A. R., Gould, N. I., & Toint, P. L.
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Trust region methods. 2000. Siam. pp. 19.
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"""
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H_diag = np.diag(H)
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H_diag_abs = np.abs(H_diag)
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H_row_sums = np.sum(np.abs(H), axis=1)
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lb = np.min(H_diag + H_diag_abs - H_row_sums)
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ub = np.max(H_diag - H_diag_abs + H_row_sums)
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return lb, ub
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def singular_leading_submatrix(A, U, k):
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"""
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Compute term that makes the leading ``k`` by ``k``
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submatrix from ``A`` singular.
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Parameters
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----------
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A : ndarray
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Symmetric matrix that is not positive definite.
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U : ndarray
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Upper triangular matrix resulting of an incomplete
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Cholesky decomposition of matrix ``A``.
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k : int
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Positive integer such that the leading k by k submatrix from
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`A` is the first non-positive definite leading submatrix.
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Returns
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-------
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delta : float
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Amount that should be added to the element (k, k) of the
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leading k by k submatrix of ``A`` to make it singular.
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v : ndarray
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A vector such that ``v.T B v = 0``. Where B is the matrix A after
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``delta`` is added to its element (k, k).
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"""
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# Compute delta
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delta = np.sum(U[:k-1, k-1]**2) - A[k-1, k-1]
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n = len(A)
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# Inicialize v
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v = np.zeros(n)
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v[k-1] = 1
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# Compute the remaining values of v by solving a triangular system.
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if k != 1:
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v[:k-1] = solve_triangular(U[:k-1, :k-1], -U[:k-1, k-1])
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return delta, v
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class IterativeSubproblem(BaseQuadraticSubproblem):
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"""Quadratic subproblem solved by nearly exact iterative method.
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Notes
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-----
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This subproblem solver was based on [1]_, [2]_ and [3]_,
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which implement similar algorithms. The algorithm is basically
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that of [1]_ but ideas from [2]_ and [3]_ were also used.
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References
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----------
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.. [1] A.R. Conn, N.I. Gould, and P.L. Toint, "Trust region methods",
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Siam, pp. 169-200, 2000.
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.. [2] J. Nocedal and S. Wright, "Numerical optimization",
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Springer Science & Business Media. pp. 83-91, 2006.
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.. [3] J.J. More and D.C. Sorensen, "Computing a trust region step",
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SIAM Journal on Scientific and Statistical Computing, vol. 4(3),
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pp. 553-572, 1983.
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"""
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# UPDATE_COEFF appears in reference [1]_
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# in formula 7.3.14 (p. 190) named as "theta".
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# As recommended there it value is fixed in 0.01.
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UPDATE_COEFF = 0.01
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EPS = np.finfo(float).eps
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def __init__(self, x, fun, jac, hess, hessp=None,
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k_easy=0.1, k_hard=0.2):
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super(IterativeSubproblem, self).__init__(x, fun, jac, hess)
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# When the trust-region shrinks in two consecutive
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# calculations (``tr_radius < previous_tr_radius``)
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# the lower bound ``lambda_lb`` may be reused,
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# facilitating the convergence. To indicate no
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# previous value is known at first ``previous_tr_radius``
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# is set to -1 and ``lambda_lb`` to None.
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self.previous_tr_radius = -1
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self.lambda_lb = None
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self.niter = 0
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# ``k_easy`` and ``k_hard`` are parameters used
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# to determine the stop criteria to the iterative
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# subproblem solver. Take a look at pp. 194-197
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# from reference _[1] for a more detailed description.
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self.k_easy = k_easy
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self.k_hard = k_hard
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# Get Lapack function for cholesky decomposition.
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# The implemented SciPy wrapper does not return
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# the incomplete factorization needed by the method.
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self.cholesky, = get_lapack_funcs(('potrf',), (self.hess,))
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# Get info about Hessian
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self.dimension = len(self.hess)
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self.hess_gershgorin_lb,\
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self.hess_gershgorin_ub = gershgorin_bounds(self.hess)
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self.hess_inf = norm(self.hess, np.Inf)
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self.hess_fro = norm(self.hess, 'fro')
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# A constant such that for vectors smaler than that
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# backward substituition is not reliable. It was stabilished
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# based on Golub, G. H., Van Loan, C. F. (2013).
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# "Matrix computations". Forth Edition. JHU press., p.165.
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self.CLOSE_TO_ZERO = self.dimension * self.EPS * self.hess_inf
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def _initial_values(self, tr_radius):
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"""Given a trust radius, return a good initial guess for
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the damping factor, the lower bound and the upper bound.
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The values were chosen accordingly to the guidelines on
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section 7.3.8 (p. 192) from [1]_.
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"""
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# Upper bound for the damping factor
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lambda_ub = max(0, self.jac_mag/tr_radius + min(-self.hess_gershgorin_lb,
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self.hess_fro,
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self.hess_inf))
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# Lower bound for the damping factor
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lambda_lb = max(0, -min(self.hess.diagonal()),
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self.jac_mag/tr_radius - min(self.hess_gershgorin_ub,
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self.hess_fro,
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self.hess_inf))
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# Improve bounds with previous info
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if tr_radius < self.previous_tr_radius:
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lambda_lb = max(self.lambda_lb, lambda_lb)
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# Initial guess for the damping factor
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if lambda_lb == 0:
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lambda_initial = 0
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else:
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lambda_initial = max(np.sqrt(lambda_lb * lambda_ub),
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lambda_lb + self.UPDATE_COEFF*(lambda_ub-lambda_lb))
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return lambda_initial, lambda_lb, lambda_ub
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def solve(self, tr_radius):
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"""Solve quadratic subproblem"""
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lambda_current, lambda_lb, lambda_ub = self._initial_values(tr_radius)
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n = self.dimension
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hits_boundary = True
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already_factorized = False
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self.niter = 0
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while True:
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# Compute Cholesky factorization
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if already_factorized:
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already_factorized = False
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else:
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H = self.hess+lambda_current*np.eye(n)
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U, info = self.cholesky(H, lower=False,
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overwrite_a=False,
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clean=True)
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self.niter += 1
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# Check if factorization succeeded
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if info == 0 and self.jac_mag > self.CLOSE_TO_ZERO:
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# Successful factorization
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# Solve `U.T U p = s`
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p = cho_solve((U, False), -self.jac)
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p_norm = norm(p)
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# Check for interior convergence
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if p_norm <= tr_radius and lambda_current == 0:
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hits_boundary = False
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break
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# Solve `U.T w = p`
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w = solve_triangular(U, p, trans='T')
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w_norm = norm(w)
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# Compute Newton step accordingly to
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# formula (4.44) p.87 from ref [2]_.
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delta_lambda = (p_norm/w_norm)**2 * (p_norm-tr_radius)/tr_radius
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lambda_new = lambda_current + delta_lambda
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if p_norm < tr_radius: # Inside boundary
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s_min, z_min = estimate_smallest_singular_value(U)
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ta, tb = self.get_boundaries_intersections(p, z_min,
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tr_radius)
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# Choose `step_len` with the smallest magnitude.
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# The reason for this choice is explained at
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# ref [3]_, p. 6 (Immediately before the formula
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# for `tau`).
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step_len = min([ta, tb], key=abs)
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# Compute the quadratic term (p.T*H*p)
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quadratic_term = np.dot(p, np.dot(H, p))
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# Check stop criteria
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relative_error = (step_len**2 * s_min**2) / (quadratic_term + lambda_current*tr_radius**2)
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if relative_error <= self.k_hard:
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p += step_len * z_min
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break
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# Update uncertanty bounds
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lambda_ub = lambda_current
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lambda_lb = max(lambda_lb, lambda_current - s_min**2)
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# Compute Cholesky factorization
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H = self.hess + lambda_new*np.eye(n)
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c, info = self.cholesky(H, lower=False,
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overwrite_a=False,
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clean=True)
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# Check if the factorization have succeeded
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#
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if info == 0: # Successful factorization
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# Update damping factor
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lambda_current = lambda_new
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already_factorized = True
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else: # Unsuccessful factorization
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# Update uncertanty bounds
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lambda_lb = max(lambda_lb, lambda_new)
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# Update damping factor
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lambda_current = max(np.sqrt(lambda_lb * lambda_ub),
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lambda_lb + self.UPDATE_COEFF*(lambda_ub-lambda_lb))
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else: # Outside boundary
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# Check stop criteria
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relative_error = abs(p_norm - tr_radius) / tr_radius
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if relative_error <= self.k_easy:
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break
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# Update uncertanty bounds
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lambda_lb = lambda_current
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# Update damping factor
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lambda_current = lambda_new
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elif info == 0 and self.jac_mag <= self.CLOSE_TO_ZERO:
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# jac_mag very close to zero
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# Check for interior convergence
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if lambda_current == 0:
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p = np.zeros(n)
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hits_boundary = False
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break
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s_min, z_min = estimate_smallest_singular_value(U)
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step_len = tr_radius
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# Check stop criteria
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if step_len**2 * s_min**2 <= self.k_hard * lambda_current * tr_radius**2:
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p = step_len * z_min
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break
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# Update uncertanty bounds
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lambda_ub = lambda_current
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lambda_lb = max(lambda_lb, lambda_current - s_min**2)
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# Update damping factor
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lambda_current = max(np.sqrt(lambda_lb * lambda_ub),
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lambda_lb + self.UPDATE_COEFF*(lambda_ub-lambda_lb))
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else: # Unsuccessful factorization
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# Compute auxiliary terms
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delta, v = singular_leading_submatrix(H, U, info)
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v_norm = norm(v)
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# Update uncertanty interval
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lambda_lb = max(lambda_lb, lambda_current + delta/v_norm**2)
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# Update damping factor
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lambda_current = max(np.sqrt(lambda_lb * lambda_ub),
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lambda_lb + self.UPDATE_COEFF*(lambda_ub-lambda_lb))
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self.lambda_lb = lambda_lb
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self.lambda_current = lambda_current
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self.previous_tr_radius = tr_radius
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return p, hits_boundary
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