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 on 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 of the previous problem. The method coincides with conjugate gradients (CG) or, equivalently, LSQR 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, like the 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 are updated over the inner iterations and Cholesky over the outer iterations.

翻译：摘要：本文提出残差二次规划有效集子空间（ResQPASS）方法，用于求解带变量界约束的大规模线性最小二乘问题。该方法通过基于残差投影构建一系列规模递增的小型子问题来求解原问题。每个投影子问题采用凸二次规划的有效集法求解，并利用前一个子问题的工作集与解进行热启动。当无约束激活时，该方法退化为共轭梯度法（CG），等价于应用于正规方程的LSQR算法。当仅有少量约束激活时，该方法经过少量初始迭代后，收敛行为与CG和LSQR类似。理论分析将收敛性与渐近Krylov子空间联系起来。本文还提出高效实现方案：在内部迭代中更新投影子问题的QR分解，在外部迭代中更新Cholesky分解。