We propose an infeasible interior-point algorithm for constrained linear least-squares problems based on the primal-dual regularization of convex programs of Friedlander and Orban. Regularization allows us to dispense with the assumption that the active gradients are linearly independent. At each iteration, a linear system with a symmetric quasi-definite (SQD) matrix is solved. This matrix is shown to be uniformly bounded and nonsingular. While the linear system may be solved using sparse LDL${}^{\mathrm{T}}$ factorization, we observe that other approaches may be used. In particular, we build on the connection between SQD linear systems and unconstrained linear least-squares problems to solve the linear system with sparse QR factorization. We establish conditions under which a solution of the original, constrained least-squares problem is recovered. We report computational experience with the sparse QR factorization and illustrate the potential advantages of our approach.

我们基于Friedlander和Orban凸程序的原始对偶正则化提出了约束线性最小二乘问题的不可行内点算法。正则化使我们可以免除有效梯度是线性独立的假设。在每次迭代中，求解具有对称拟定（SQD）矩阵的线性系统。该矩阵显示为有界且非奇异的。虽然可以使用稀疏LDL解决线性系统${}^{\mathrm{Ť}}$因数分解，我们注意到可以使用其他方法。特别是，我们建立在SQD线性系统与无约束线性最小二乘问题之间的联系上，以解决具有稀疏QR分解的线性系统。我们建立条件，在该条件下可以恢复原始的受约束的最小二乘问题。我们报告了稀疏QR分解的计算经验，并说明了该方法的潜在优势。