Krylov iterative methods for linear least squares problems with linear equality constraints

TL;DR

This paper introduces Krylov iterative methods for efficiently solving large-scale linear least squares problems with linear equality constraints, avoiding matrix factorizations and null space computations.

Contribution

It presents a novel operator-based analysis and two Krylov subspace iterative algorithms for the LSE problem, enhancing scalability and applicability.

Findings

Algorithms effectively solve large-scale LSE problems.

Numerical tests confirm the efficiency and accuracy of the methods.

Proposed methods outperform classical approaches in scalability.

Abstract

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.