Modulus-Type Inner Outer Iteration Methods for Nonnegative Constrained Least Squares Problems

Abstract

For the solution of large sparse nonnegative constrained linear least squares (NNLS) problems, a new iterative method is proposed which uses the CGLS method for the inner iterations and the modulus iterative method for the outer iterations to solve the linear complementarity problem resulting from the Karush--Kuhn--Tucker conditions of the NNLS problem. Theoretical convergence analysis including the optimal choice of the parameter matrix is presented for the proposed method. In addition, the method can be further enhanced by incorporating the active set strategy, which contains two stages; the first stage consists of modulus iterations to identify the active set, while the second stage solves the reduced unconstrained least squares problems only on the inactive variables, and projects the solution into the nonnegative region. Numerical experiments show the efficiency of the proposed methods compared to projection gradient--type methods with fewer iteration steps and less CPU time.