Matrix nearness problems and eigenvalue optimization

Abstract

This book is about solving matrix nearness problems that are related to eigenvalues or singular values or pseudospectra. These problems arise in great diversity in various fields, be they related to dynamics, as in questions of robust stability and robust control, or related to graphs, as in questions of clustering and ranking. Algorithms for such problems work with matrix perturbations that drive eigenvalues or singular values or Rayleigh quotients to desired locations. Remarkably, the optimal perturbation matrices are typically of rank one or are projections of rank-1 matrices onto a linear structure, e.g. a prescribed sparsity pattern. In the approach worked out here, these optimal rank-1 perturbations will be determined in a two-level iteration: In the inner iteration, an eigenvalue optimization problem for a fixed perturbation size is to be solved via gradient-based rank-1 matrix differential equations. This amounts to numerically driving a rank-1 matrix, which is represented by two vectors, into a stationary point, mostly starting nearby. The outer iteration determines the optimal perturbation size by solving a scalar nonlinear equation. A wide variety of matrix nearness problems, as outlined in the introductory Chapter I, will be tackled in Chapters II to VIII by such an approach and its nontrivial extensions.