Nonnegativity Constraints in Numerical Analysis - CiteSeer

Block principal pivoting algorithm:Input: A ∈ R m×n , b ∈ R mOutput: x ∗ ≥ 0 such that x ∗ = arg min ‖Ax − b‖ 2 .Initialization: F = ø and G = 1, . . . , n, x = 0, y = −A T b, and p = 3, N = ∞.repeat:1. Proceed if (x F , y G ) is an infeasible solution.2. Set n to the number of negative entries in x F and y G .2.1 Proceed if n < N,2.1.1 Set N = n, p = 3,2.1.2 Exchange all infeasible variables between F and G.2.2 Proceed if n ≥ N2.2.1 Proceed if p > 0,2.2.1.1 set p = p − 12.2.1.2 Exchange all infeasible variables between F and G.2.2.2 Proceed if p ≤ 0,2.2.2.1 Exchange only the infeasible variable with largest index.3. Update x F and y G by Equation (10) and (11).4. Set Variables in x F < 10 −12 and y G < 10 −12 to zero.Interior Point Newton-like Method:In addition to the methods above, Interior Point methods can be used to solve NNLSproblems. They generate an infinite sequence of strictly feasible points converging to thesolution and are known to be competitive with active set methods for medium and largeproblems. In the paper [1], the authors present an interior-point approach suited for NNLSproblems. Global and locally fast convergence is guaranteed even if a degenerate solution isapproached and the structure of the given problem is exploited both in the linear algebraphase and in the globalization strategy. Viable approaches for implementation are discussedand numerical results are provided. Here we give an interior algorithm for NNLS, moredetailed discussion could be found in the paper [1].Notation: g(x) is the gradient of the objective function (6), i.e. g(x) = ∇f(x) =A T (Ax − b). Therefore, by the KKT conditions, x ∗ can be found by searching for thepositive solution of the system of nonlinear equationsD(x)g(x) = 0, (12)12