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

This paper presents the Residual QPAS Subspace method (ResQPASS) method that solves large-scale least-squares problem with bound constraints on the variables. The problem is solved by creating a series of small problems with increasing size by projecting on the basis residuals. Each projected problem is solved by the QPAS method that is warm-started with a 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. We develop a convergence theory that links the convergence with Krylov subspaces. We also present an efficient implementation where the matrix factorizations using QR are updated over the inner and outer iterations.