Nonnegative least squares
In mathematical optimization, the problem of nonnegative 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]
 subject to x ≥ 0.
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Here x ≥ 0 means that each component of the vector x should be nonnegative, and ‖·‖₂ denotes the Euclidean norm.
Nonnegative least squares problems turn up as subproblems in matrix decomposition, e.g. in algorithms for PARAFAC[2] and nonnegative matrix/tensor factorization.[3][4] The latter can be considered a generalization of NNLS.[1]
Another generalization of NNLS is boundedvariable least squares (BVLS), with simultaneous upper and lower bounds αᵢ ≤ xᵢ ≤ βᵢ.[5]^{:291}[6]
Quadratic programming version
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 nonnegativity constraints form a convex feasible set.[7]
Algorithms
The first widely used algorithm for solving this problem is an activeset 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]
 Inputs:
 a realvalued matrix A of dimension m × n,
 a realvalued vector y of dimension m,
 a real value ε, the tolerance for the stopping criterion.
 Initialize:
 Set P = ∅.
 Set R = {1, ..., n}.
 Set x to an allzero vector of dimension n.
 Set w = Aᵀ(y − Ax).
 Main loop: while R ≠ ∅ and max(w) > ε:
 Let j in R be the index of max(w) in w.
 Add j to P.
 Remove j from R.
 Let A^{P} be A restricted to the variables included in P.
 Let s be vector of same length as x. Let s^{P} denote the subvector with indexes from P, and let s^{R} denote the subvector with indexes from R.
 Set s^{P} = ((A^{P})ᵀ A^{P})^{−1} (A^{P})ᵀy
 Set s^{R} to zero
 While min(s^{P}) ≤ 0:
 Let α = min x_{i}/x_{i} − s_{i} for i in P where s_{i} ≤ 0.
 Set x to x + α(s − x).
 Move to R all indices j in P such that x_{j} = 0.
 Set s^{P} = ((A^{P})ᵀ A^{P})^{−1} (A^{P})ᵀy
 Set s^{R} to zero.
 Set x to s.
 Set w to Aᵀ(y − Ax).
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[1] and in SciPy as optimize.nnls.[8]
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[9] and coordinatewise optimization based on the quadratic programming problem above.[7]
See also
References
 Chen, Donghui; Plemmons, Robert J. (2009). Nonnegativity constraints in numerical analysis. Symposium on the Birth of Numerical Analysis. CiteSeerX 10.1.1.157.9203.
 Bro, Rasmus; De Jong, Sijmen (1997). "A fast nonnegativityconstrained least squares algorithm". Journal of Chemometrics. 11 (5): 393. doi:10.1002/(SICI)1099128X(199709/10)11:5<393::AIDCEM483>3.0.CO;2L.
 Lin, ChihJen (2007). "Projected Gradient Methods for Nonnegative Matrix Factorization" (PDF). Neural Computation. 19 (10): 2756–2779. CiteSeerX 10.1.1.308.9135. doi:10.1162/neco.2007.19.10.2756. PMID 17716011.
 Boutsidis, Christos; Drineas, Petros (2009). "Random projections for the nonnegative leastsquares problem". Linear Algebra and Its Applications. 431 (5–7): 760–771. arXiv:0812.4547. doi:10.1016/j.laa.2009.03.026.
 Lawson, Charles L.; Hanson, Richard J. (1995). Solving Least Squares Problems. SIAM.
 Stark, Philip B.; Parker, Robert L. (1995). "Boundedvariable leastsquares: an algorithm and applications" (PDF). Computational Statistics. 10: 129.
 Franc, Vojtěch; Hlaváč, Václav; Navara, Mirko (2005). Sequential CoordinateWise Algorithm for the Nonnegative Least Squares Problem. Computer Analysis of Images and Patterns. Lecture Notes in Computer Science. 3691. pp. 407–414. doi:10.1007/11556121_50. ISBN 9783540289692.
 "scipy.optimize.nnls". SciPy v0.13.0 Reference Guide. Retrieved 25 January 2014.
 Johansson, B. R.; Elfving, T.; Kozlov, V.; Censor, Y.; Forssén, P. E.; Granlund, G. S. (2006). "The application of an obliqueprojected Landweber method to a model of supervised learning". Mathematical and Computer Modelling. 43 (7–8): 892. doi:10.1016/j.mcm.2005.12.010.