NNLS

NNLS stands for nonnegative least squares, a form of constrained least squares optimization. The problem seeks a vector x in R^n that minimizes the squared Euclidean norm ||Ax - b||_2^2 subject to x >= 0, where A is an m-by-n matrix and b is in R^m. The objective is the standard L2 error, but the nonnegativity constraint yields a solution with nonnegative components, which is often desirable in interpretable modeling such as abundances or concentrations.

The problem was formalized with notable algorithms by Lawson and Hanson in 1974, introducing an active-set method for efficiently handling the nonnegativity constraints. Since then, numerous methods have been developed, including projected gradient approaches, fast NNLS variants, block-pivoting algorithms, and primal-dual interior-point techniques. The optimization problem is convex, and if A has full column rank, the solution is unique; otherwise, multiple optimal solutions may exist on a face of the nonnegative orthant.

Applications of NNLS span several fields. In spectroscopic analysis and chemometrics, it is used to estimate constituent concentrations. In image and signal processing, NNLS appears in deconvolution and reconstruction tasks. It also serves as a nonnegative regression tool in statistics and is a common subproblem within nonnegative matrix factorization and related decompositions.

Software implementations are widely available. In MATLAB, lsqnonneg implements NNLS; in Python, scipy.optimize.nnls provides a standard routine. Large-scale variants and customization for sparsity or additional constraints are present in various scientific computing libraries.