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333 lines
11 KiB
Python
333 lines
11 KiB
Python
"""
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dogleg algorithm with rectangular trust regions for least-squares minimization.
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The description of the algorithm can be found in [Voglis]_. The algorithm does
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trust-region iterations, but the shape of trust regions is rectangular as
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opposed to conventional elliptical. The intersection of a trust region and
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an initial feasible region is again some rectangle. Thus on each iteration a
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bound-constrained quadratic optimization problem is solved.
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A quadratic problem is solved by well-known dogleg approach, where the
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function is minimized along piecewise-linear "dogleg" path [NumOpt]_,
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Chapter 4. If Jacobian is not rank-deficient then the function is decreasing
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along this path, and optimization amounts to simply following along this
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path as long as a point stays within the bounds. A constrained Cauchy step
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(along the anti-gradient) is considered for safety in rank deficient cases,
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in this situations the convergence might be slow.
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If during iterations some variable hit the initial bound and the component
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of anti-gradient points outside the feasible region, then a next dogleg step
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won't make any progress. At this state such variables satisfy first-order
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optimality conditions and they are excluded before computing a next dogleg
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step.
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Gauss-Newton step can be computed exactly by `numpy.linalg.lstsq` (for dense
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Jacobian matrices) or by iterative procedure `scipy.sparse.linalg.lsmr` (for
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dense and sparse matrices, or Jacobian being LinearOperator). The second
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option allows to solve very large problems (up to couple of millions of
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residuals on a regular PC), provided the Jacobian matrix is sufficiently
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sparse. But note that dogbox is not very good for solving problems with
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large number of constraints, because of variables exclusion-inclusion on each
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iteration (a required number of function evaluations might be high or accuracy
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of a solution will be poor), thus its large-scale usage is probably limited
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to unconstrained problems.
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References
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----------
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.. [Voglis] C. Voglis and I. E. Lagaris, "A Rectangular Trust Region Dogleg
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Approach for Unconstrained and Bound Constrained Nonlinear
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Optimization", WSEAS International Conference on Applied
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Mathematics, Corfu, Greece, 2004.
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.. [NumOpt] J. Nocedal and S. J. Wright, "Numerical optimization, 2nd edition".
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"""
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from __future__ import division, print_function, absolute_import
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import numpy as np
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from numpy.linalg import lstsq, norm
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from scipy.sparse.linalg import LinearOperator, aslinearoperator, lsmr
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from scipy.optimize import OptimizeResult
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from scipy._lib.six import string_types
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from .common import (
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step_size_to_bound, in_bounds, update_tr_radius, evaluate_quadratic,
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build_quadratic_1d, minimize_quadratic_1d, compute_grad,
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compute_jac_scale, check_termination, scale_for_robust_loss_function,
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print_header_nonlinear, print_iteration_nonlinear)
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def lsmr_operator(Jop, d, active_set):
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"""Compute LinearOperator to use in LSMR by dogbox algorithm.
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`active_set` mask is used to excluded active variables from computations
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of matrix-vector products.
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"""
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m, n = Jop.shape
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def matvec(x):
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x_free = x.ravel().copy()
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x_free[active_set] = 0
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return Jop.matvec(x * d)
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def rmatvec(x):
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r = d * Jop.rmatvec(x)
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r[active_set] = 0
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return r
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return LinearOperator((m, n), matvec=matvec, rmatvec=rmatvec, dtype=float)
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def find_intersection(x, tr_bounds, lb, ub):
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"""Find intersection of trust-region bounds and initial bounds.
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Returns
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-------
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lb_total, ub_total : ndarray with shape of x
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Lower and upper bounds of the intersection region.
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orig_l, orig_u : ndarray of bool with shape of x
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True means that an original bound is taken as a corresponding bound
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in the intersection region.
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tr_l, tr_u : ndarray of bool with shape of x
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True means that a trust-region bound is taken as a corresponding bound
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in the intersection region.
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"""
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lb_centered = lb - x
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ub_centered = ub - x
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lb_total = np.maximum(lb_centered, -tr_bounds)
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ub_total = np.minimum(ub_centered, tr_bounds)
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orig_l = np.equal(lb_total, lb_centered)
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orig_u = np.equal(ub_total, ub_centered)
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tr_l = np.equal(lb_total, -tr_bounds)
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tr_u = np.equal(ub_total, tr_bounds)
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return lb_total, ub_total, orig_l, orig_u, tr_l, tr_u
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def dogleg_step(x, newton_step, g, a, b, tr_bounds, lb, ub):
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"""Find dogleg step in a rectangular region.
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Returns
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-------
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step : ndarray, shape (n,)
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Computed dogleg step.
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bound_hits : ndarray of int, shape (n,)
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Each component shows whether a corresponding variable hits the
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initial bound after the step is taken:
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* 0 - a variable doesn't hit the bound.
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* -1 - lower bound is hit.
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* 1 - upper bound is hit.
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tr_hit : bool
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Whether the step hit the boundary of the trust-region.
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"""
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lb_total, ub_total, orig_l, orig_u, tr_l, tr_u = find_intersection(
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x, tr_bounds, lb, ub
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)
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bound_hits = np.zeros_like(x, dtype=int)
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if in_bounds(newton_step, lb_total, ub_total):
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return newton_step, bound_hits, False
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to_bounds, _ = step_size_to_bound(np.zeros_like(x), -g, lb_total, ub_total)
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# The classical dogleg algorithm would check if Cauchy step fits into
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# the bounds, and just return it constrained version if not. But in a
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# rectangular trust region it makes sense to try to improve constrained
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# Cauchy step too. Thus we don't distinguish these two cases.
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cauchy_step = -minimize_quadratic_1d(a, b, 0, to_bounds)[0] * g
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step_diff = newton_step - cauchy_step
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step_size, hits = step_size_to_bound(cauchy_step, step_diff,
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lb_total, ub_total)
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bound_hits[(hits < 0) & orig_l] = -1
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bound_hits[(hits > 0) & orig_u] = 1
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tr_hit = np.any((hits < 0) & tr_l | (hits > 0) & tr_u)
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return cauchy_step + step_size * step_diff, bound_hits, tr_hit
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def dogbox(fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev, x_scale,
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loss_function, tr_solver, tr_options, verbose):
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f = f0
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f_true = f.copy()
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nfev = 1
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J = J0
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njev = 1
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if loss_function is not None:
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rho = loss_function(f)
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cost = 0.5 * np.sum(rho[0])
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J, f = scale_for_robust_loss_function(J, f, rho)
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else:
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cost = 0.5 * np.dot(f, f)
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g = compute_grad(J, f)
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jac_scale = isinstance(x_scale, string_types) and x_scale == 'jac'
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if jac_scale:
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scale, scale_inv = compute_jac_scale(J)
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else:
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scale, scale_inv = x_scale, 1 / x_scale
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Delta = norm(x0 * scale_inv, ord=np.inf)
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if Delta == 0:
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Delta = 1.0
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on_bound = np.zeros_like(x0, dtype=int)
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on_bound[np.equal(x0, lb)] = -1
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on_bound[np.equal(x0, ub)] = 1
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x = x0
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step = np.empty_like(x0)
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if max_nfev is None:
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max_nfev = x0.size * 100
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termination_status = None
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iteration = 0
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step_norm = None
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actual_reduction = None
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if verbose == 2:
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print_header_nonlinear()
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while True:
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active_set = on_bound * g < 0
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free_set = ~active_set
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g_free = g[free_set]
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g_full = g.copy()
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g[active_set] = 0
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g_norm = norm(g, ord=np.inf)
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if g_norm < gtol:
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termination_status = 1
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if verbose == 2:
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print_iteration_nonlinear(iteration, nfev, cost, actual_reduction,
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step_norm, g_norm)
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if termination_status is not None or nfev == max_nfev:
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break
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x_free = x[free_set]
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lb_free = lb[free_set]
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ub_free = ub[free_set]
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scale_free = scale[free_set]
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# Compute (Gauss-)Newton and build quadratic model for Cauchy step.
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if tr_solver == 'exact':
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J_free = J[:, free_set]
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newton_step = lstsq(J_free, -f, rcond=-1)[0]
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# Coefficients for the quadratic model along the anti-gradient.
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a, b = build_quadratic_1d(J_free, g_free, -g_free)
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elif tr_solver == 'lsmr':
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Jop = aslinearoperator(J)
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# We compute lsmr step in scaled variables and then
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# transform back to normal variables, if lsmr would give exact lsq
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# solution this would be equivalent to not doing any
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# transformations, but from experience it's better this way.
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# We pass active_set to make computations as if we selected
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# the free subset of J columns, but without actually doing any
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# slicing, which is expensive for sparse matrices and impossible
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# for LinearOperator.
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lsmr_op = lsmr_operator(Jop, scale, active_set)
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newton_step = -lsmr(lsmr_op, f, **tr_options)[0][free_set]
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newton_step *= scale_free
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# Components of g for active variables were zeroed, so this call
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# is correct and equivalent to using J_free and g_free.
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a, b = build_quadratic_1d(Jop, g, -g)
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actual_reduction = -1.0
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while actual_reduction <= 0 and nfev < max_nfev:
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tr_bounds = Delta * scale_free
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step_free, on_bound_free, tr_hit = dogleg_step(
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x_free, newton_step, g_free, a, b, tr_bounds, lb_free, ub_free)
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step.fill(0.0)
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step[free_set] = step_free
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if tr_solver == 'exact':
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predicted_reduction = -evaluate_quadratic(J_free, g_free,
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step_free)
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elif tr_solver == 'lsmr':
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predicted_reduction = -evaluate_quadratic(Jop, g, step)
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x_new = x + step
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f_new = fun(x_new)
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nfev += 1
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step_h_norm = norm(step * scale_inv, ord=np.inf)
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if not np.all(np.isfinite(f_new)):
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Delta = 0.25 * step_h_norm
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continue
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# Usual trust-region step quality estimation.
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if loss_function is not None:
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cost_new = loss_function(f_new, cost_only=True)
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else:
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cost_new = 0.5 * np.dot(f_new, f_new)
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actual_reduction = cost - cost_new
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Delta, ratio = update_tr_radius(
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Delta, actual_reduction, predicted_reduction,
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step_h_norm, tr_hit
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)
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step_norm = norm(step)
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termination_status = check_termination(
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actual_reduction, cost, step_norm, norm(x), ratio, ftol, xtol)
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if termination_status is not None:
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break
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if actual_reduction > 0:
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on_bound[free_set] = on_bound_free
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x = x_new
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# Set variables exactly at the boundary.
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mask = on_bound == -1
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x[mask] = lb[mask]
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mask = on_bound == 1
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x[mask] = ub[mask]
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f = f_new
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f_true = f.copy()
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cost = cost_new
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J = jac(x, f)
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njev += 1
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if loss_function is not None:
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rho = loss_function(f)
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J, f = scale_for_robust_loss_function(J, f, rho)
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g = compute_grad(J, f)
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if jac_scale:
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scale, scale_inv = compute_jac_scale(J, scale_inv)
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else:
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step_norm = 0
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actual_reduction = 0
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iteration += 1
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if termination_status is None:
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termination_status = 0
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return OptimizeResult(
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x=x, cost=cost, fun=f_true, jac=J, grad=g_full, optimality=g_norm,
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active_mask=on_bound, nfev=nfev, njev=njev, status=termination_status)
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