We consider the linear least squares problem with linear equality constraints (LSE problem) formulated as $\min_{x\in\mathbb{R}^{n}}\|Ax-b\|_2 \ \mathrm{s.t.} \ Cx = d$. Although there are some classical methods available to solve this problem, most of them rely on matrix factorizations or require the null space of $C$, which limits their applicability to large-scale problems. To address this challenge, we present a novel analysis of the LSE problem from the perspective of operator-type least squares (LS) problems, where the linear operators are induced by $\{A,C\}$. We show that the solution of the LSE problem can be decomposed into two components, each corresponding to the solution of an operator-form LS problem. Building on this decomposed-form solution, we propose two Krylov subspace based iterative methods to approximate each component, thereby providing an approximate solution of the LSE problem. Several numerical examples are constructed to test the proposed iterative algorithm for solving the LSE problems, which demonstrate the effectiveness of the algorithms.

翻译：本文研究具有线性等式约束的线性最小二乘问题（LSE问题），其形式为$\min_{x\in\mathbb{R}^{n}}\|Ax-b\|_2 \ \mathrm{s.t.} \ Cx = d$。尽管已有若干经典方法可用于求解该问题，但其中多数依赖于矩阵分解或需要计算$C$的零空间，这限制了其在大规模问题中的应用。为应对这一挑战，我们从算子型最小二乘（LS）问题的视角对LSE问题进行了新颖分析，其中线性算子由$\{A,C\}$诱导生成。我们证明了LSE问题的解可分解为两个分量，每个分量对应一个算子形式LS问题的解。基于此分解形式解，我们提出了两种基于Krylov子空间的迭代方法分别逼近各分量，从而获得LSE问题的近似解。通过构造若干数值算例对所提迭代算法进行测试，结果验证了算法的有效性。