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.

翻译：本文提出了一种残差QPAS子空间（ResQPASS）方法，用于求解变量受边界约束的大规模线性最小二乘问题。该方法通过构建一系列规模递增的小型投影问题来求解原问题，其投影基由残差向量张成。每个投影问题采用QPAS方法求解，并利用前一个问题的有效集与解进行热启动。当所有约束均未激活时，该方法退化为共轭梯度法（CG）作用于正规方程的形式；当仅有少量约束激活时，该方法在经过若干初始迭代后收敛性等价于CG法。理论分析揭示了该方法与Krylov子空间之间的收敛性关联。本文还提出了一种高效实现方案：内层迭代采用QR分解更新矩阵分解，外层迭代采用Cholesky分解进行矩阵分解更新。