You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
250 lines
7.4 KiB
Python
250 lines
7.4 KiB
Python
"""The adaptation of Trust Region Reflective algorithm for a linear
|
|
least-squares problem."""
|
|
import numpy as np
|
|
from numpy.linalg import norm
|
|
from scipy.linalg import qr, solve_triangular
|
|
from scipy.sparse.linalg import lsmr
|
|
from scipy.optimize import OptimizeResult
|
|
|
|
from .givens_elimination import givens_elimination
|
|
from .common import (
|
|
EPS, step_size_to_bound, find_active_constraints, in_bounds,
|
|
make_strictly_feasible, build_quadratic_1d, evaluate_quadratic,
|
|
minimize_quadratic_1d, CL_scaling_vector, reflective_transformation,
|
|
print_header_linear, print_iteration_linear, compute_grad,
|
|
regularized_lsq_operator, right_multiplied_operator)
|
|
|
|
|
|
def regularized_lsq_with_qr(m, n, R, QTb, perm, diag, copy_R=True):
|
|
"""Solve regularized least squares using information from QR-decomposition.
|
|
|
|
The initial problem is to solve the following system in a least-squares
|
|
sense:
|
|
::
|
|
|
|
A x = b
|
|
D x = 0
|
|
|
|
where D is diagonal matrix. The method is based on QR decomposition
|
|
of the form A P = Q R, where P is a column permutation matrix, Q is an
|
|
orthogonal matrix and R is an upper triangular matrix.
|
|
|
|
Parameters
|
|
----------
|
|
m, n : int
|
|
Initial shape of A.
|
|
R : ndarray, shape (n, n)
|
|
Upper triangular matrix from QR decomposition of A.
|
|
QTb : ndarray, shape (n,)
|
|
First n components of Q^T b.
|
|
perm : ndarray, shape (n,)
|
|
Array defining column permutation of A, such that ith column of
|
|
P is perm[i]-th column of identity matrix.
|
|
diag : ndarray, shape (n,)
|
|
Array containing diagonal elements of D.
|
|
|
|
Returns
|
|
-------
|
|
x : ndarray, shape (n,)
|
|
Found least-squares solution.
|
|
"""
|
|
if copy_R:
|
|
R = R.copy()
|
|
v = QTb.copy()
|
|
|
|
givens_elimination(R, v, diag[perm])
|
|
|
|
abs_diag_R = np.abs(np.diag(R))
|
|
threshold = EPS * max(m, n) * np.max(abs_diag_R)
|
|
nns, = np.nonzero(abs_diag_R > threshold)
|
|
|
|
R = R[np.ix_(nns, nns)]
|
|
v = v[nns]
|
|
|
|
x = np.zeros(n)
|
|
x[perm[nns]] = solve_triangular(R, v)
|
|
|
|
return x
|
|
|
|
|
|
def backtracking(A, g, x, p, theta, p_dot_g, lb, ub):
|
|
"""Find an appropriate step size using backtracking line search."""
|
|
alpha = 1
|
|
while True:
|
|
x_new, _ = reflective_transformation(x + alpha * p, lb, ub)
|
|
step = x_new - x
|
|
cost_change = -evaluate_quadratic(A, g, step)
|
|
if cost_change > -0.1 * alpha * p_dot_g:
|
|
break
|
|
alpha *= 0.5
|
|
|
|
active = find_active_constraints(x_new, lb, ub)
|
|
if np.any(active != 0):
|
|
x_new, _ = reflective_transformation(x + theta * alpha * p, lb, ub)
|
|
x_new = make_strictly_feasible(x_new, lb, ub, rstep=0)
|
|
step = x_new - x
|
|
cost_change = -evaluate_quadratic(A, g, step)
|
|
|
|
return x, step, cost_change
|
|
|
|
|
|
def select_step(x, A_h, g_h, c_h, p, p_h, d, lb, ub, theta):
|
|
"""Select the best step according to Trust Region Reflective algorithm."""
|
|
if in_bounds(x + p, lb, ub):
|
|
return p
|
|
|
|
p_stride, hits = step_size_to_bound(x, p, lb, ub)
|
|
r_h = np.copy(p_h)
|
|
r_h[hits.astype(bool)] *= -1
|
|
r = d * r_h
|
|
|
|
# Restrict step, such that it hits the bound.
|
|
p *= p_stride
|
|
p_h *= p_stride
|
|
x_on_bound = x + p
|
|
|
|
# Find the step size along reflected direction.
|
|
r_stride_u, _ = step_size_to_bound(x_on_bound, r, lb, ub)
|
|
|
|
# Stay interior.
|
|
r_stride_l = (1 - theta) * r_stride_u
|
|
r_stride_u *= theta
|
|
|
|
if r_stride_u > 0:
|
|
a, b, c = build_quadratic_1d(A_h, g_h, r_h, s0=p_h, diag=c_h)
|
|
r_stride, r_value = minimize_quadratic_1d(
|
|
a, b, r_stride_l, r_stride_u, c=c)
|
|
r_h = p_h + r_h * r_stride
|
|
r = d * r_h
|
|
else:
|
|
r_value = np.inf
|
|
|
|
# Now correct p_h to make it strictly interior.
|
|
p_h *= theta
|
|
p *= theta
|
|
p_value = evaluate_quadratic(A_h, g_h, p_h, diag=c_h)
|
|
|
|
ag_h = -g_h
|
|
ag = d * ag_h
|
|
ag_stride_u, _ = step_size_to_bound(x, ag, lb, ub)
|
|
ag_stride_u *= theta
|
|
a, b = build_quadratic_1d(A_h, g_h, ag_h, diag=c_h)
|
|
ag_stride, ag_value = minimize_quadratic_1d(a, b, 0, ag_stride_u)
|
|
ag *= ag_stride
|
|
|
|
if p_value < r_value and p_value < ag_value:
|
|
return p
|
|
elif r_value < p_value and r_value < ag_value:
|
|
return r
|
|
else:
|
|
return ag
|
|
|
|
|
|
def trf_linear(A, b, x_lsq, lb, ub, tol, lsq_solver, lsmr_tol, max_iter,
|
|
verbose):
|
|
m, n = A.shape
|
|
x, _ = reflective_transformation(x_lsq, lb, ub)
|
|
x = make_strictly_feasible(x, lb, ub, rstep=0.1)
|
|
|
|
if lsq_solver == 'exact':
|
|
QT, R, perm = qr(A, mode='economic', pivoting=True)
|
|
QT = QT.T
|
|
|
|
if m < n:
|
|
R = np.vstack((R, np.zeros((n - m, n))))
|
|
|
|
QTr = np.zeros(n)
|
|
k = min(m, n)
|
|
elif lsq_solver == 'lsmr':
|
|
r_aug = np.zeros(m + n)
|
|
auto_lsmr_tol = False
|
|
if lsmr_tol is None:
|
|
lsmr_tol = 1e-2 * tol
|
|
elif lsmr_tol == 'auto':
|
|
auto_lsmr_tol = True
|
|
|
|
r = A.dot(x) - b
|
|
g = compute_grad(A, r)
|
|
cost = 0.5 * np.dot(r, r)
|
|
initial_cost = cost
|
|
|
|
termination_status = None
|
|
step_norm = None
|
|
cost_change = None
|
|
|
|
if max_iter is None:
|
|
max_iter = 100
|
|
|
|
if verbose == 2:
|
|
print_header_linear()
|
|
|
|
for iteration in range(max_iter):
|
|
v, dv = CL_scaling_vector(x, g, lb, ub)
|
|
g_scaled = g * v
|
|
g_norm = norm(g_scaled, ord=np.inf)
|
|
if g_norm < tol:
|
|
termination_status = 1
|
|
|
|
if verbose == 2:
|
|
print_iteration_linear(iteration, cost, cost_change,
|
|
step_norm, g_norm)
|
|
|
|
if termination_status is not None:
|
|
break
|
|
|
|
diag_h = g * dv
|
|
diag_root_h = diag_h ** 0.5
|
|
d = v ** 0.5
|
|
g_h = d * g
|
|
|
|
A_h = right_multiplied_operator(A, d)
|
|
if lsq_solver == 'exact':
|
|
QTr[:k] = QT.dot(r)
|
|
p_h = -regularized_lsq_with_qr(m, n, R * d[perm], QTr, perm,
|
|
diag_root_h, copy_R=False)
|
|
elif lsq_solver == 'lsmr':
|
|
lsmr_op = regularized_lsq_operator(A_h, diag_root_h)
|
|
r_aug[:m] = r
|
|
if auto_lsmr_tol:
|
|
eta = 1e-2 * min(0.5, g_norm)
|
|
lsmr_tol = max(EPS, min(0.1, eta * g_norm))
|
|
p_h = -lsmr(lsmr_op, r_aug, atol=lsmr_tol, btol=lsmr_tol)[0]
|
|
|
|
p = d * p_h
|
|
|
|
p_dot_g = np.dot(p, g)
|
|
if p_dot_g > 0:
|
|
termination_status = -1
|
|
|
|
theta = 1 - min(0.005, g_norm)
|
|
step = select_step(x, A_h, g_h, diag_h, p, p_h, d, lb, ub, theta)
|
|
cost_change = -evaluate_quadratic(A, g, step)
|
|
|
|
# Perhaps almost never executed, the idea is that `p` is descent
|
|
# direction thus we must find acceptable cost decrease using simple
|
|
# "backtracking", otherwise the algorithm's logic would break.
|
|
if cost_change < 0:
|
|
x, step, cost_change = backtracking(
|
|
A, g, x, p, theta, p_dot_g, lb, ub)
|
|
else:
|
|
x = make_strictly_feasible(x + step, lb, ub, rstep=0)
|
|
|
|
step_norm = norm(step)
|
|
r = A.dot(x) - b
|
|
g = compute_grad(A, r)
|
|
|
|
if cost_change < tol * cost:
|
|
termination_status = 2
|
|
|
|
cost = 0.5 * np.dot(r, r)
|
|
|
|
if termination_status is None:
|
|
termination_status = 0
|
|
|
|
active_mask = find_active_constraints(x, lb, ub, rtol=tol)
|
|
|
|
return OptimizeResult(
|
|
x=x, fun=r, cost=cost, optimality=g_norm, active_mask=active_mask,
|
|
nit=iteration + 1, status=termination_status,
|
|
initial_cost=initial_cost)
|