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.
316 lines
12 KiB
Python
316 lines
12 KiB
Python
"""Linear least squares with bound constraints on independent variables."""
|
|
import numpy as np
|
|
from numpy.linalg import norm
|
|
from scipy.sparse import issparse, csr_matrix
|
|
from scipy.sparse.linalg import LinearOperator, lsmr
|
|
from scipy.optimize import OptimizeResult
|
|
|
|
from .common import in_bounds, compute_grad
|
|
from .trf_linear import trf_linear
|
|
from .bvls import bvls
|
|
|
|
|
|
def prepare_bounds(bounds, n):
|
|
lb, ub = [np.asarray(b, dtype=float) for b in bounds]
|
|
|
|
if lb.ndim == 0:
|
|
lb = np.resize(lb, n)
|
|
|
|
if ub.ndim == 0:
|
|
ub = np.resize(ub, n)
|
|
|
|
return lb, ub
|
|
|
|
|
|
TERMINATION_MESSAGES = {
|
|
-1: "The algorithm was not able to make progress on the last iteration.",
|
|
0: "The maximum number of iterations is exceeded.",
|
|
1: "The first-order optimality measure is less than `tol`.",
|
|
2: "The relative change of the cost function is less than `tol`.",
|
|
3: "The unconstrained solution is optimal."
|
|
}
|
|
|
|
|
|
def lsq_linear(A, b, bounds=(-np.inf, np.inf), method='trf', tol=1e-10,
|
|
lsq_solver=None, lsmr_tol=None, max_iter=None, verbose=0):
|
|
r"""Solve a linear least-squares problem with bounds on the variables.
|
|
|
|
Given a m-by-n design matrix A and a target vector b with m elements,
|
|
`lsq_linear` solves the following optimization problem::
|
|
|
|
minimize 0.5 * ||A x - b||**2
|
|
subject to lb <= x <= ub
|
|
|
|
This optimization problem is convex, hence a found minimum (if iterations
|
|
have converged) is guaranteed to be global.
|
|
|
|
Parameters
|
|
----------
|
|
A : array_like, sparse matrix of LinearOperator, shape (m, n)
|
|
Design matrix. Can be `scipy.sparse.linalg.LinearOperator`.
|
|
b : array_like, shape (m,)
|
|
Target vector.
|
|
bounds : 2-tuple of array_like, optional
|
|
Lower and upper bounds on independent variables. Defaults to no bounds.
|
|
Each array must have shape (n,) or be a scalar, in the latter
|
|
case a bound will be the same for all variables. Use ``np.inf`` with
|
|
an appropriate sign to disable bounds on all or some variables.
|
|
method : 'trf' or 'bvls', optional
|
|
Method to perform minimization.
|
|
|
|
* 'trf' : Trust Region Reflective algorithm adapted for a linear
|
|
least-squares problem. This is an interior-point-like method
|
|
and the required number of iterations is weakly correlated with
|
|
the number of variables.
|
|
* 'bvls' : Bounded-variable least-squares algorithm. This is
|
|
an active set method, which requires the number of iterations
|
|
comparable to the number of variables. Can't be used when `A` is
|
|
sparse or LinearOperator.
|
|
|
|
Default is 'trf'.
|
|
tol : float, optional
|
|
Tolerance parameter. The algorithm terminates if a relative change
|
|
of the cost function is less than `tol` on the last iteration.
|
|
Additionally, the first-order optimality measure is considered:
|
|
|
|
* ``method='trf'`` terminates if the uniform norm of the gradient,
|
|
scaled to account for the presence of the bounds, is less than
|
|
`tol`.
|
|
* ``method='bvls'`` terminates if Karush-Kuhn-Tucker conditions
|
|
are satisfied within `tol` tolerance.
|
|
|
|
lsq_solver : {None, 'exact', 'lsmr'}, optional
|
|
Method of solving unbounded least-squares problems throughout
|
|
iterations:
|
|
|
|
* 'exact' : Use dense QR or SVD decomposition approach. Can't be
|
|
used when `A` is sparse or LinearOperator.
|
|
* 'lsmr' : Use `scipy.sparse.linalg.lsmr` iterative procedure
|
|
which requires only matrix-vector product evaluations. Can't
|
|
be used with ``method='bvls'``.
|
|
|
|
If None (default), the solver is chosen based on type of `A`.
|
|
lsmr_tol : None, float or 'auto', optional
|
|
Tolerance parameters 'atol' and 'btol' for `scipy.sparse.linalg.lsmr`
|
|
If None (default), it is set to ``1e-2 * tol``. If 'auto', the
|
|
tolerance will be adjusted based on the optimality of the current
|
|
iterate, which can speed up the optimization process, but is not always
|
|
reliable.
|
|
max_iter : None or int, optional
|
|
Maximum number of iterations before termination. If None (default), it
|
|
is set to 100 for ``method='trf'`` or to the number of variables for
|
|
``method='bvls'`` (not counting iterations for 'bvls' initialization).
|
|
verbose : {0, 1, 2}, optional
|
|
Level of algorithm's verbosity:
|
|
|
|
* 0 : work silently (default).
|
|
* 1 : display a termination report.
|
|
* 2 : display progress during iterations.
|
|
|
|
Returns
|
|
-------
|
|
OptimizeResult with the following fields defined:
|
|
x : ndarray, shape (n,)
|
|
Solution found.
|
|
cost : float
|
|
Value of the cost function at the solution.
|
|
fun : ndarray, shape (m,)
|
|
Vector of residuals at the solution.
|
|
optimality : float
|
|
First-order optimality measure. The exact meaning depends on `method`,
|
|
refer to the description of `tol` parameter.
|
|
active_mask : ndarray of int, shape (n,)
|
|
Each component shows whether a corresponding constraint is active
|
|
(that is, whether a variable is at the bound):
|
|
|
|
* 0 : a constraint is not active.
|
|
* -1 : a lower bound is active.
|
|
* 1 : an upper bound is active.
|
|
|
|
Might be somewhat arbitrary for the `trf` method as it generates a
|
|
sequence of strictly feasible iterates and active_mask is determined
|
|
within a tolerance threshold.
|
|
nit : int
|
|
Number of iterations. Zero if the unconstrained solution is optimal.
|
|
status : int
|
|
Reason for algorithm termination:
|
|
|
|
* -1 : the algorithm was not able to make progress on the last
|
|
iteration.
|
|
* 0 : the maximum number of iterations is exceeded.
|
|
* 1 : the first-order optimality measure is less than `tol`.
|
|
* 2 : the relative change of the cost function is less than `tol`.
|
|
* 3 : the unconstrained solution is optimal.
|
|
|
|
message : str
|
|
Verbal description of the termination reason.
|
|
success : bool
|
|
True if one of the convergence criteria is satisfied (`status` > 0).
|
|
|
|
See Also
|
|
--------
|
|
nnls : Linear least squares with non-negativity constraint.
|
|
least_squares : Nonlinear least squares with bounds on the variables.
|
|
|
|
Notes
|
|
-----
|
|
The algorithm first computes the unconstrained least-squares solution by
|
|
`numpy.linalg.lstsq` or `scipy.sparse.linalg.lsmr` depending on
|
|
`lsq_solver`. This solution is returned as optimal if it lies within the
|
|
bounds.
|
|
|
|
Method 'trf' runs the adaptation of the algorithm described in [STIR]_ for
|
|
a linear least-squares problem. The iterations are essentially the same as
|
|
in the nonlinear least-squares algorithm, but as the quadratic function
|
|
model is always accurate, we don't need to track or modify the radius of
|
|
a trust region. The line search (backtracking) is used as a safety net
|
|
when a selected step does not decrease the cost function. Read more
|
|
detailed description of the algorithm in `scipy.optimize.least_squares`.
|
|
|
|
Method 'bvls' runs a Python implementation of the algorithm described in
|
|
[BVLS]_. The algorithm maintains active and free sets of variables, on
|
|
each iteration chooses a new variable to move from the active set to the
|
|
free set and then solves the unconstrained least-squares problem on free
|
|
variables. This algorithm is guaranteed to give an accurate solution
|
|
eventually, but may require up to n iterations for a problem with n
|
|
variables. Additionally, an ad-hoc initialization procedure is
|
|
implemented, that determines which variables to set free or active
|
|
initially. It takes some number of iterations before actual BVLS starts,
|
|
but can significantly reduce the number of further iterations.
|
|
|
|
References
|
|
----------
|
|
.. [STIR] M. A. Branch, T. F. Coleman, and Y. Li, "A Subspace, Interior,
|
|
and Conjugate Gradient Method for Large-Scale Bound-Constrained
|
|
Minimization Problems," SIAM Journal on Scientific Computing,
|
|
Vol. 21, Number 1, pp 1-23, 1999.
|
|
.. [BVLS] P. B. Start and R. L. Parker, "Bounded-Variable Least-Squares:
|
|
an Algorithm and Applications", Computational Statistics, 10,
|
|
129-141, 1995.
|
|
|
|
Examples
|
|
--------
|
|
In this example, a problem with a large sparse matrix and bounds on the
|
|
variables is solved.
|
|
|
|
>>> from scipy.sparse import rand
|
|
>>> from scipy.optimize import lsq_linear
|
|
...
|
|
>>> np.random.seed(0)
|
|
...
|
|
>>> m = 20000
|
|
>>> n = 10000
|
|
...
|
|
>>> A = rand(m, n, density=1e-4)
|
|
>>> b = np.random.randn(m)
|
|
...
|
|
>>> lb = np.random.randn(n)
|
|
>>> ub = lb + 1
|
|
...
|
|
>>> res = lsq_linear(A, b, bounds=(lb, ub), lsmr_tol='auto', verbose=1)
|
|
# may vary
|
|
The relative change of the cost function is less than `tol`.
|
|
Number of iterations 16, initial cost 1.5039e+04, final cost 1.1112e+04,
|
|
first-order optimality 4.66e-08.
|
|
"""
|
|
if method not in ['trf', 'bvls']:
|
|
raise ValueError("`method` must be 'trf' or 'bvls'")
|
|
|
|
if lsq_solver not in [None, 'exact', 'lsmr']:
|
|
raise ValueError("`solver` must be None, 'exact' or 'lsmr'.")
|
|
|
|
if verbose not in [0, 1, 2]:
|
|
raise ValueError("`verbose` must be in [0, 1, 2].")
|
|
|
|
if issparse(A):
|
|
A = csr_matrix(A)
|
|
elif not isinstance(A, LinearOperator):
|
|
A = np.atleast_2d(np.asarray(A))
|
|
|
|
if method == 'bvls':
|
|
if lsq_solver == 'lsmr':
|
|
raise ValueError("method='bvls' can't be used with "
|
|
"lsq_solver='lsmr'")
|
|
|
|
if not isinstance(A, np.ndarray):
|
|
raise ValueError("method='bvls' can't be used with `A` being "
|
|
"sparse or LinearOperator.")
|
|
|
|
if lsq_solver is None:
|
|
if isinstance(A, np.ndarray):
|
|
lsq_solver = 'exact'
|
|
else:
|
|
lsq_solver = 'lsmr'
|
|
elif lsq_solver == 'exact' and not isinstance(A, np.ndarray):
|
|
raise ValueError("`exact` solver can't be used when `A` is "
|
|
"sparse or LinearOperator.")
|
|
|
|
if len(A.shape) != 2: # No ndim for LinearOperator.
|
|
raise ValueError("`A` must have at most 2 dimensions.")
|
|
|
|
if len(bounds) != 2:
|
|
raise ValueError("`bounds` must contain 2 elements.")
|
|
|
|
if max_iter is not None and max_iter <= 0:
|
|
raise ValueError("`max_iter` must be None or positive integer.")
|
|
|
|
m, n = A.shape
|
|
|
|
b = np.atleast_1d(b)
|
|
if b.ndim != 1:
|
|
raise ValueError("`b` must have at most 1 dimension.")
|
|
|
|
if b.size != m:
|
|
raise ValueError("Inconsistent shapes between `A` and `b`.")
|
|
|
|
lb, ub = prepare_bounds(bounds, n)
|
|
|
|
if lb.shape != (n,) and ub.shape != (n,):
|
|
raise ValueError("Bounds have wrong shape.")
|
|
|
|
if np.any(lb >= ub):
|
|
raise ValueError("Each lower bound must be strictly less than each "
|
|
"upper bound.")
|
|
|
|
if lsq_solver == 'exact':
|
|
x_lsq = np.linalg.lstsq(A, b, rcond=-1)[0]
|
|
elif lsq_solver == 'lsmr':
|
|
x_lsq = lsmr(A, b, atol=tol, btol=tol)[0]
|
|
|
|
if in_bounds(x_lsq, lb, ub):
|
|
r = A @ x_lsq - b
|
|
cost = 0.5 * np.dot(r, r)
|
|
termination_status = 3
|
|
termination_message = TERMINATION_MESSAGES[termination_status]
|
|
g = compute_grad(A, r)
|
|
g_norm = norm(g, ord=np.inf)
|
|
|
|
if verbose > 0:
|
|
print(termination_message)
|
|
print("Final cost {0:.4e}, first-order optimality {1:.2e}"
|
|
.format(cost, g_norm))
|
|
|
|
return OptimizeResult(
|
|
x=x_lsq, fun=r, cost=cost, optimality=g_norm,
|
|
active_mask=np.zeros(n), nit=0, status=termination_status,
|
|
message=termination_message, success=True)
|
|
|
|
if method == 'trf':
|
|
res = trf_linear(A, b, x_lsq, lb, ub, tol, lsq_solver, lsmr_tol,
|
|
max_iter, verbose)
|
|
elif method == 'bvls':
|
|
res = bvls(A, b, x_lsq, lb, ub, tol, max_iter, verbose)
|
|
|
|
res.message = TERMINATION_MESSAGES[res.status]
|
|
res.success = res.status > 0
|
|
|
|
if verbose > 0:
|
|
print(res.message)
|
|
print("Number of iterations {0}, initial cost {1:.4e}, "
|
|
"final cost {2:.4e}, first-order optimality {3:.2e}."
|
|
.format(res.nit, res.initial_cost, res.cost, res.optimality))
|
|
|
|
del res.initial_cost
|
|
|
|
return res
|