scipy.optimize.nnls#
- scipy.optimize.nnls(A, b, maxiter=None, *, atol=None)[source]#
Solve
argmin_x || Ax - b ||_2
forx>=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 asmax(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:
- xndarray
Solution vector.
- rnormfloat
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)