Non-negative least squares

In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares problem where the coefficients are not allowed to become negative. That is, given a matrix and a (column) vector of response variables , the goal is to find
Here means that each component of the vector should be non-negative, and denotes the Euclidean norm.
Non-negative least squares problems turn up as subproblems in matrix decomposition, e.g. in algorithms for PARAFAC and non-negative matrix/tensor factorization. The latter can be considered a generalization of NNLS.
Another generalization of NNLS is bounded-variable least squares (BVLS), with simultaneous upper and lower bounds .
Quadratic programming version.
The NNLS problem is equivalent to a quadratic programming problem
where = and = . This problem is convex, as is positive semidefinite and the non-negativity constraints form a convex feasible set.
Algorithms.
The first widely used algorithm for solving this problem is an active-set method published by Lawson and Hanson in their 1974 book "Solving Least Squares Problems". In pseudocode, this algorithm looks as follows:
This algorithm takes a finite number of steps to reach a solution and smoothly improves its candidate solution as it goes (so it can find good approximate solutions when cut off at a reasonable number of iterations), but is very slow in practice, owing largely to the computation of the pseudoinverse . Variants of this algorithm are available in MATLAB as the routine and in SciPy as .
Many improved algorithms have been suggested since 1974. Fast NNLS (FNNLS) is an optimized version of the Lawson—Hanson algorithm. Other algorithms include variants of Landweber's gradient descent method and coordinate-wise optimization based on the quadratic programming problem above.