Collaborative Research: Large Scale Regularized Least Squares Problems via Quadratic Eigenvalue Problems

虽然在理论和算法上已经取得了很大的进展，但RLS问题的某些数值性质，如解的敏感性和它的条件数，仍然是一个具有挑战性的任务.本文将进一步努力建立特征值重构作为求解大规模RLS问题的一种数值稳定和高效的方法，并发展稳健的Krylov子空间型方法来求解相关的二次特征值问题.此外，本文还建议研究其他相关的约束二次优化问题的特征值公式，这些工作将有助于加深对RLS问题的理论理解和开发有效的算法。这种通过特征值问题的方法将导致更好地利用特征值算法的未来发展的算法。这项研究的另一个潜在的重要意义是建立了一个框架，用于证明对优化问题使用直接特征值公式是合理的。

While tremendous progresses have been made in theory and algorithms, certain numerical propoerties of the RLS problem such as sensitivity of the solution and its condition number are not well understood, and solving large-scale problems remains a challenging task.This propposal will further efforts to establish the eigenvalue reformulation as a numerically stable and efficient method for solving large-scale RLS problems and to develop robust Krylov subspace type methods for solving the related quadratic eigenvalue problem. It is also proposed to investigate eigenvalue formulations of other related constrained quadratic optimization problems to which this research may be applicable.The proposed work would advance theoretical understanding and develop efficient algorithms for RLS problems. This approach via an eigenvalue problem would lead to algorithms that could better exploit future developments of eigenvalue algorithms. Another potentially important implication of this research is the establishment of a framework for justifying the use of a direct eigenvalue formulation for an optimization problem.