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import numpy as np
from ._cython_nnls import _nnls
__all__ = ['nnls']
def nnls(A, b, maxiter=None, *, atol=None):
"""
Solve ``argmin_x || Ax - b ||_2`` for ``x>=0``.
This problem, often called as NonNegative Least Squares, is a convex
optimization problem with convex constraints. It typically arises when
the ``x`` models quantities for which only nonnegative values are
attainable; weight of ingredients, component costs and so on.
Parameters
----------
A : (m, n) ndarray
Coefficient array
b : (m,) ndarray, float
Right-hand side vector.
maxiter: int, optional
Maximum number of iterations, optional. Default value is ``3 * n``.
atol: float
Tolerance value used in the algorithm to assess closeness to zero in
the projected residual ``(A.T @ (A x - b)`` entries. Increasing this
value relaxes the solution constraints. A typical relaxation value can
be selected as ``max(m, n) * np.linalg.norm(a, 1) * np.spacing(1.)``.
This value is not set as default since the norm operation becomes
expensive for large problems hence can be used only when necessary.
Returns
-------
x : ndarray
Solution vector.
rnorm : float
The 2-norm of the residual, ``|| Ax-b ||_2``.
See Also
--------
lsq_linear : Linear least squares with bounds on the variables
Notes
-----
The code is based on [2]_ which is an improved version of the classical
algorithm of [1]_. It utilizes an active set method and solves the KKT
(Karush-Kuhn-Tucker) conditions for the non-negative least squares problem.
References
----------
.. [1] : Lawson C., Hanson R.J., "Solving Least Squares Problems", SIAM,
1995, :doi:`10.1137/1.9781611971217`
.. [2] : Bro, Rasmus and de Jong, Sijmen, "A Fast Non-Negativity-
Constrained Least Squares Algorithm", Journal Of Chemometrics, 1997,
:doi:`10.1002/(SICI)1099-128X(199709/10)11:5<393::AID-CEM483>3.0.CO;2-L`
Examples
--------
>>> import numpy as np
>>> from scipy.optimize import nnls
...
>>> A = np.array([[1, 0], [1, 0], [0, 1]])
>>> b = np.array([2, 1, 1])
>>> nnls(A, b)
(array([1.5, 1. ]), 0.7071067811865475)
>>> b = np.array([-1, -1, -1])
>>> nnls(A, b)
(array([0., 0.]), 1.7320508075688772)
"""
A = np.asarray_chkfinite(A, dtype=np.float64, order='C')
b = np.asarray_chkfinite(b, dtype=np.float64)
if len(A.shape) != 2:
raise ValueError("Expected a two-dimensional array (matrix)" +
f", but the shape of A is {A.shape}")
if len(b.shape) != 1:
raise ValueError("Expected a one-dimensional array (vector)" +
f", but the shape of b is {b.shape}")
m, n = A.shape
if m != b.shape[0]:
raise ValueError(
"Incompatible dimensions. The first dimension of " +
f"A is {m}, while the shape of b is {(b.shape[0], )}")
if not maxiter:
maxiter = 3*n
x, rnorm, info = _nnls(A, b, maxiter)
if info == -1:
raise RuntimeError("Maximum number of iterations reached.")
return x, rnorm
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