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

协作研究：通过二次特征值问题解决大规模正则化最小二乘问题

• 批准号: 0430617
• 负责人: Gene Golub
• 金额: --
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-15 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0430617&HistoricalAwards=false
• 关键词: Collaborative; Research; Large; Scale; Regularized

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.

虽然在理论和算法上取得了巨大的进步，但RLS问题的某些数值性质，如解的灵敏度及其条件数，还没有得到很好的理解，本文的工作将进一步致力于建立一种数值稳定、有效的求解大规模问题的方法--特征值重构法。规模的RLS问题，并发展鲁棒Krylov子空间型方法解决相关的二次特征值问题。本文还提出了研究其他相关约束二次优化问题的特征值公式的方法，这些研究工作将有助于推进对RLS问题的理论理解和发展有效的算法。这种方法通过一个特征值问题将导致算法，可以更好地利用特征值算法的未来发展。这项研究的另一个潜在的重要意义是建立一个框架，证明使用的直接特征值制定的优化问题。