# ResQPASS: an algorithm for bounded variable linear least squares with asymptotic Krylov convergence

## TL;DR

ResQPASS is a novel algorithm for large-scale bounded linear least squares problems that combines active-set methods with Krylov subspace techniques, offering efficient convergence and practical implementation strategies.

## Contribution

The paper introduces ResQPASS, a new method that integrates active-set quadratic programming with Krylov subspace methods for improved large-scale constrained least squares solving.

## Key findings

- Converges like CG/LSQR when constraints are inactive
- Efficient implementation with QR and Cholesky updates
- Links convergence to asymptotic Krylov subspace

## Abstract

We present the Residual Quadratic Programming Active-Set 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 problems of increasing size by projecting onto the basis of residuals. Each projected problem is solved by the active-set method for convex quadratic programming, warm-started with a working set and solution from the previous problem. The method coincides with conjugate gradients (CG) or, equivalently, LSQR when none of the constraints is active. When only a few constraints are active the method converges, after a few initial iterations, like CG and LSQR. An analysis links the convergence to an asymptotic Krylov subspace. We also present an efficient implementation where QR factorizations of the projected problems are updated over the inner iterations and Cholesky or Gram-Schmidt over the outer iterations.