Research Project: Simetrik Özdeğer Optimizasyonu ve Dinamik Sistemlere Uygulamaları
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Contributors
Funders
ID
TB.00023
Authors
Mengi, Emre
Faculty Member
Publications
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.
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.
Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity
(Springer, 2011) Mengi, Emre; N/A; Department of Mathematics; Yes; College of Sciences
The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity. We provide a singular value characterization for this generalized Wilkinson distance. Then we outline a numerical technique to solve the derived singular value optimization problems. In particular the numerical technique is applicable to Malyshev's formula to compute the Wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue.
Matrix polynomials with specified eigenvalues
(Elsevier, 2015) Mengi, Emre; Karow, Michael; Department of Mathematics; Yes; College of Sciences
This work concerns the distance in the 2-norm from a given matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Initially, we consider perturbations of the constant coefficient matrix only. A singular value optimization characterization is derived for the associated distance. We also consider the distance in the general setting, when all of the coefficient matrices are perturbed. In this general setting, we obtain a lower bound in terms of another singular value optimization problem. The singular value optimization problems derived facilitate the numerical computation of the distances.
