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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 A and a (column) vector of response variables y, the goal is to find[1]
Here x ≥ 0 means that each component of the vector x should be non-negative, and ‖·‖2 denotes the Euclidean norm.
Non-negative least squares problems turn up as subproblems in matrix decomposition, e.g. in algorithms for PARAFAC[2] and non-negative matrix/tensor factorization.[3][4] The latter can be considered a generalization of NNLS.[1]
Another generalization of NNLS is bounded-variable least squares (BVLS), with simultaneous upper and lower bounds αi ≤ xi ≤ βi.[5]: 291 [6]
The NNLS problem is equivalent to a quadratic programming problem
where Q = ATA and c = −AT y. This problem is convex, as Q is positive semidefinite and the non-negativity constraints form a convex feasible set.[7]
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.[5]: 291 In pseudocode, this algorithm looks as follows:[1][2]
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 ((AP)T AP)−1.[1] Variants of this algorithm are available in MATLAB as the routine lsqnonneg[8][1] and in SciPy as optimize.nnls.[9]
Many improved algorithms have been suggested since 1974.[1] Fast NNLS (FNNLS) is an optimized version of the Lawson—Hanson algorithm.[2] Other algorithms include variants of Landweber's gradient descent method[10] and coordinate-wise optimization based on the quadratic programming problem above.[7]