Research Project: Büyük ve Doğrusal Olmayan Özdeğer optimizasyonu Problemleri için Destek Fonksiyon Tabanlı Algoritmalar
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Contributors
Funders
ID
TB.00142
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.
Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems
(Oxford University Press (OUP), 2017) Mengi, Emre; Meerbergen, Karl; Michiels, Wim; Van Beeumen, Roel; Department of Mathematics; Yes; College of Sciences
We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of nonlinear eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available.
A support function based algorithm for optimization with eigenvalue constraints
(Society for Industrial and Applied Mathematics (SIAM) Publications, 2017) Mengi, Emre; N/A; Department of Mathematics; Yes; College of Sciences
Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytic and Hermitian matrix-valued function. We propose a numerical approach based on quadratic support functions that overestimate the smallest eigenvalue function globally. the quadratic support functions are derived by employing variational properties of the smallest eigenvalue function over a set of Hermitian matrices. We establish the local convergence of the algorithm under mild assumptions and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration. the convergence analysis reveals that the algorithm is immune to the nonsmooth nature of the smallest eigenvalue. We illustrate the practical applicability of the algorithm on the pseudospectral functions.
