Solving the minimal least squares problem subject to bounds on the variables

A computational procedure is developed for determining the solution of minimal length to a linear least squares problem subject to bounds on the variables. In the first stage, a solution to the least squares problem is computed and then in the second stage, the solution of minimal length is determined. The objective function in each step is minimized by an active set method adapted to the special structure of the problem.The systems of linear equations satisfied by the descent direction and the Lagrange multipliers in the minimization algorithm are solved by direct methods based on QR decompositions or iterative preconditioned conjugate gradient methods. The direct and the iterative methods are compared in numerical experiments, where the solutions are sought to a sequence of related, minimal least squares problems subject to bounds on the variables. The application of the iterative methods to large, sparse problems is discussed briefly.This work was supported by The National Swedish Board for Technical Development under contract dnr 80-3341.