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

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.

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.


Solution vector.


The 2-norm of the residual, || Ax-b ||_2.

See also


Linear least squares with bounds on the variables


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.



: Lawson C., Hanson R.J., “Solving Least Squares Problems”, SIAM, 1995, DOI:10.1137/1.9781611971217


: 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


>>> 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)