Residual QPAS subspace (ResQPASS) algorithm for bounded-variable least squares (BVLS) with superlinear Krylov convergence

This paper presents the Residual QPAS Subspace (ResQPASS) method that solves large-scale linear least-squares problems with bound constraints on the variables. The problem is solved by creating a series of small projected problems with increasing size. We project on the basis spanned by the residuals. Each projected problem is solved by the QPAS method that is warm-started with the working set and the solution of the previous problem. The method coincides with conjugate gradients (CG) applied to the normal equations when none of the constraints is active. When only a few constraints are active the method converges, after a few initial iterations, as the CG method. Our analysis links the convergence to Krylov subspaces. We also present an efficient implementation where the matrix factorizations using QR are updated over the inner iterations and Cholesky over the outer iterations.