Research Project: Lipschitz-based Optimization of Singular Values with Applications to Dynamical Systems
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Contributors
Funders
ID
EC.00018
Authors
Mengi, Emre
Faculty Member
Publications
A subspace method for large-scale eigenvalue optimization
(Society for Industrial and Applied Mathematics (SIAM), 2018) Kangal, Fatih; Mengi, Emre; Meerbergen, Karl; Michiels, Wim; Department of Mathematics; Graduate School of Sciences and Engineering; Yes; College of Sciences; GRADUATE SCHOOL OF SCIENCES AND ENGINEERING
We consider the minimization or maximization of the Jth largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi, Yildirim, and Kilic [SIAM T. Matrix Anal. Appl., 35, pp. 699-724, 2014]. This work addresses the setting when the matrix-valued function involved is very large. We describe subspace procedures that convert the original problem into a small-scale one by means of orthogonal projections and restrictions to certain subspaces, and that gradually expand these subspaces based on the optimal solutions of small-scale problems. Global convergence and superlinear rate-of-convergence results with respect to the dimensions of the subspaces are presented in the infinite dimensional setting, where the matrix-valued function is replaced by a compact operator depending on parameters. In practice, it suffices to solve eigenvalue optimization problems involving matrices with sizes on the scale of tens, instead of the original problem involving matrices with sizes on the scale of thousands.
Generalized eigenvalue problems with specified eigenvalues
(Oxford University Press (OUP), 2014) Mengi, Emre; Kressner, Daniel; Nakic, Ivica; Truhar, Ninoslav; Department of Mathematics; Yes; College of Sciences
We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in Boutry et al. (2005, SIAM J. Matrix Anal. Appl., 27, 582-601) regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of Broyden-Fletcher-Goldfarb-Shanno and Lipschitz-based global optimization algorithms.
Nonlinear eigenvalue problems with specified eigenvalues
(Society for Industrial and Applied Mathematics (SIAM), 2014) Mengi, Emre; Karow, Michael; Kressner, Daniel; Department of Mathematics; Yes; College of Sciences
This work considers eigenvalue problems that are nonlinear in the eigenvalue parameter. Given such a nonlinear eigenvalue problem T, we are concerned with finding the minimal backward error such that T has a set of prescribed eigenvalues with prescribed algebraic multiplicities. We consider backward errors that only allow constant perturbations, which do not depend on the eigenvalue parameter. While the usual resolvent norm addresses this question for a single eigenvalue of multiplicity one, the general setting involving several eigenvalues is significantly more difficult. Under mild assumptions, we derive a singular value optimization characterization for the minimal perturbation that addresses the general case.
Numerical optimization of eigenvalues of Hermitian matrix functions
(Society for Industrial and Applied Mathematics (SIAM), 2014) Mengi, Emre; Yıldırım, Emre Alper; Kılıç, Mustafa; Department of Mathematics; Yes; College of Engineering; College of Sciences
This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function depending on its parameters analytically in a box. We describe how the analytical properties of eigenvalue functions can be put into use to derive piecewise quadratic functions that underestimate the eigenvalue functions. These piecewise quadratic underestimators lead us to a global minimization algorithm, originally due to Breiman and Cutler. We prove the global convergence of the algorithm and show that it can be effectively used for the minimization of extreme eigenvalues, e.g., the largest eigenvalue or the sum of the largest specified number of eigenvalues. This is particularly facilitated by the analytical formulas for the first derivatives of eigenvalues, as well as analytical lower bounds on the second derivatives that can be deduced for extreme eigenvalue functions. The applications that we have in mind also include the H-infinity-norm of a linear dynamical system, numerical radius, distance to uncontrollability, and various other nonconvex eigenvalue optimization problems, for which, generically, the eigenvalue function involved is simple at all points.
Nearest linear systems with highly deficient reachable subspaces
(Society for Industrial and Applied Mathematics (SIAM), 2012) Mengi, Emre; Department of Mathematics; Yes; College of Sciences
We consider the 2-norm distance tau(r)(A, B) from a linear time-invariant dynamical system (A, B) of order n to the nearest system (A + Delta A(*), B + Delta B-*) whose reachable subspace is of dimension r < n. We first present a characterization to test whether the reachable subspace of the system has dimension r, which resembles and can be considered as a generalization of the Popov-Belevitch-Hautus test for controllability. Then, by exploiting this generalized Popov-Belevitch-Hautus characterization, we derive the main result of this paper, which is a singular value optimization characterization for tau(r)(A, B). A numerical technique to solve the derived singular value optimization problems is described. The numerical results on a few examples illustrate the significance of the derived singular value characterization for computational purposes.
