This project mainly concerns the numerical solution of singular value optimization problems. In the literature such problems arise in the robust control of linear dynamical systems, and in numerical linear algebra when sensitivity of numerical problems is considered. In a singular value optimization problem a prespecified singular value (e.g. the smallest, the largest) is sought to be minimized or maximized over a space of parametrized matrices. The inherent difficulty in the numerical solution of such problems is the non-convex and non-smooth nature of singular values. The traditional smooth optimization techniques such as Newton's method may not converge at all and, even if they converge, they converge only to a locally optimal point. The three major problems that will be tackled in this project are described below.
(1) The Project Coordinator (PC) aims to introduce a unified optimization algorithm exploiting the Lipschitzness of singular values and their derivatives. The algorithm will be meant for large-scale problems with many unknowns. The rate of convergence and backward error of the algorithm will be analyzed.
(2) Further applications of singular value optimization problems to dynamical systems will also be explored. Specifically the PC will investigate the applicability of singular value optimization in the context of model reduction of state-space representations of linear dynamical systems.
(3) From a theoretical point of view the available numerical techniques for singular value optimization problems can also shed light on the geometric properties of the pseudospectrum. The epsln-pseudospectrum of a matrix A is the set consisting of eigenvalues of all matrices within an epsln neighborhood of A. The PC hopes to prove various conjectures regarding the coalescence of the components of the pseudospectra.
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