scipy.optimize.

nnls#

scipy.optimize.nnls(A, b, *, maxiter=None)[source]#

Solve argmin_x || Ax - b ||_2^2 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.
maxiterint, optional: Maximum number of iterations, optional. Default value is 3 * n.

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 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

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)