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

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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.