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.

翻译：本文提出残差QPAS子空间方法（ResQPASS），用于求解带有变量边界约束的大规模最小二乘问题。该方法通过在残差基上进行投影，生成一系列规模递增的小型子问题。每个子问题采用QPAS方法求解，并通过工作集与前一子问题的解进行热启动。当无约束激活时，该方法等价于应用于正规方程的共轭梯度法（CG）。仅少量约束激活时，该方法在经过初始若干次迭代后，其收敛性类似于CG方法。我们建立了联系Krylov子空间收敛性的理论体系，并给出基于QR矩阵分解的内外迭代更新高效实现方案。