## Final Report Summary - OPT OF SINGULAR VALS (Lipschitz-based Optimization of Singular Values with Applications to Dynamical Systems)

Dr. Mengi, the researcher in charge, would like to express his gratitude to the Commission for the

support of this project PIRG07-GA-2010-268355 entitled “Lipschitz-based Optimization of Singular Values with Applications to Dynamical Systems”.

State-of-the-Art and Objectives

Optimization of eigenvalues has been an active area of research since 1980s. On the theoretical side, much of the emphasis has been put on non-smooth analysis of these optimization problems. On the practical side, much of the effort focused on the development of the efficient algorithms that are immune to non-smoothness, but mainly in the convex setting. Particularly, semi-definite programs in this convex setting are proved to be very efficient, especially after the emergence of interior point methods in 1990s. Non-convex eigenvalue optimization problems, on the other hand, are deemed computationally difficult and mostly ignored.

This project aims for a general numerical solution of non-convex eigenvalue optimization problems depending on a few variables. Singular value optimization is a special case of this setting. We are particularly interested in two frameworks: (F1) the optimization of the jth largest eigenvalue of a Hermitian and analytic matrix-valued function on a box; (F2) the optimization of a linear objective subject to a constraint on the smallest eigenvalue of a Hermitian and analytic matrix-valued function.

The project also digs into various matrix-nearness problems leading to nonconvex singular value optimization problems. These problems involve locating nearest analytic matrix-valued functions with a set of prescribed eigenvalues. A numerical analyst could view these nearness problems as generalized backward error problems. The motivation here is that the nearness of the spectrum of a matrix-valued function to points of singularity indicates problems. For instance, if a matrix is very close to a matrix with a multiple eigenvalue, one of the eigenvalues of the matrix is very sensitive to perturbations of the entries of the matrix. These problems are also of great interest in control theory.

Scientific Progress and Main Results

An algorithm is derived for framework (F1). The algorithm approximates the eigenvalue function by a piece-wise quadratic model, which is a global under-estimator (for minimization) or a global over-estimator (for maximization). Locating a globally optimal point of this piece-wise quadratic model is shown to be computationally tractable when there are a few unknowns. The algorithm is applied effectively to various non-convex eigenvalue optimization problems, such as the computation of the H-infinity norm of a linear system and the minimization of the largest eigenvalue of a Hermitian and analytic matrix-valued function, which is important in structural design and control theory. The global convergence of the algorithm is proven, while the linear rate of convergence is observed in practice.

Another algorithm is tailored for framework (F2). The algorithm convexifies and smoothens the problem by replacing the eigenvalue constraint with a convex quadratic constraint. Solving such problems repeatedly results in a locally convergent algorithm robust against non-smoothness. We establish local convergence and linear rate of convergence results.

Singular value optimization characterizations are derived for locating a nearest pencil, a nearest matrix polynomial and a nearest analytic matrix-valued function with prescribed eigenvalues. The derivation in each one of these works boils down to a study of the dimension of the kernel of an associated Sylvester operator. The derived singular value optimization characterizations can be solved numerically by means of the algorithms developed for eigenvalue optimization.

Impact

The algorithm derived for the first framework is suitable for general non-convex eigenvalue optimization problems depending on a few parameters. There is need in various engineering fields for global solution of such problems. In structural design, a classical problem is to design the strongest column subject to volume constraints. Finite element or finite difference discretizations of a differential operator lead to problems involving minimization of the largest eigenvalue of a Hermitian matrix-valued function. In control theory, it is desirable to design systems so that the poles of the system are on the left half of the complex plane and away from the imaginary axis as much as possible. We hope that the algorithm will be adopted in such contexts. A robust implementation of the algorithm (EIGOPT) is made available on the web publicly, and accompanied by a guide.

The constrained algorithm for the second framework encompasses problems that are of great interest. For instance, this algorithm can be adopted to perform optimization of a convex objective subject to a nuclear norm constraint on a matrix-valued function. This last problem is convex yet non-smooth, and drew substantial attention lately due to its connection with sparse estimation. We also coupled this constrained algorithm with the global algorithm for the first framework to compute the pseudospectral abscissa regarding a non-linear eigenvalue problem. Potential coupling of these algorithms in other contexts may result in significant impact. An implementation of this constrained algorithm (EIGOPTC) is also available on the web.

The singular value optimization characterizations derived generalizes existing results for standard matrix-nearness problems such as distance to instability and distance to defectiveness. Due to their connections with backward errors, they may be valuable for numerical algorithms computing eigenvalues in the linear, polynomial and non-linear settings. They may be also of interest in control theory to determine nearby systems with poles at specified positions. The results and the techniques used to locate nearest rectangular pencils with specified eigenvalues can be valuable to tackle a long-lasting problem, namely the distance to a nearest singular pencil.

• EIGOPT: software for eigenvalue optimization with box constraints http://home.ku.edu.tr/~emengi/software/eigopt.html

• EIGOPTC: software for optimization of a linear objective subject to an eigenvalue constraint http://home.ku.edu.tr/~emengi/software/eigopt_constrained.tar

Prospects of the Research Career Development and Reintegration of the Fellow

The fellow acquired a concrete research vision for the upcoming years. Especially he targets eigenvalue optimization problems in the large-scale setting with emphasis on engineering applications and sparse estimation in near future. The strengthened research ties the fellow possessed during this project is likely to boost his research productivity. This is evidenced by a joint project on eigenvalue optimization with Karl Meerbergen and Wim Michiels (KU-Leuven), along with active collaborations with Michael Karow (TU-Berlin), Daniel Kressner (EPF-Lausanne), and Alper Yildirim locally at Koc University. Furthermore, the fellow is surrounded by a promising group of students.

The fellow feels secure and adjusted to the research environment at Koc University and in Turkey. He received his tenure from the Turkish government in February 2013, and applied for his tenure from Koc University in May 2013, which he hopes will be resolved positively soon. The research of the fellow is recognized by the Turkish Academy of Science in terms of a BAGEP award in Spring 2013. All of these developments are indicators of successful reintegration of the fellow.

support of this project PIRG07-GA-2010-268355 entitled “Lipschitz-based Optimization of Singular Values with Applications to Dynamical Systems”.

State-of-the-Art and Objectives

Optimization of eigenvalues has been an active area of research since 1980s. On the theoretical side, much of the emphasis has been put on non-smooth analysis of these optimization problems. On the practical side, much of the effort focused on the development of the efficient algorithms that are immune to non-smoothness, but mainly in the convex setting. Particularly, semi-definite programs in this convex setting are proved to be very efficient, especially after the emergence of interior point methods in 1990s. Non-convex eigenvalue optimization problems, on the other hand, are deemed computationally difficult and mostly ignored.

This project aims for a general numerical solution of non-convex eigenvalue optimization problems depending on a few variables. Singular value optimization is a special case of this setting. We are particularly interested in two frameworks: (F1) the optimization of the jth largest eigenvalue of a Hermitian and analytic matrix-valued function on a box; (F2) the optimization of a linear objective subject to a constraint on the smallest eigenvalue of a Hermitian and analytic matrix-valued function.

The project also digs into various matrix-nearness problems leading to nonconvex singular value optimization problems. These problems involve locating nearest analytic matrix-valued functions with a set of prescribed eigenvalues. A numerical analyst could view these nearness problems as generalized backward error problems. The motivation here is that the nearness of the spectrum of a matrix-valued function to points of singularity indicates problems. For instance, if a matrix is very close to a matrix with a multiple eigenvalue, one of the eigenvalues of the matrix is very sensitive to perturbations of the entries of the matrix. These problems are also of great interest in control theory.

Scientific Progress and Main Results

An algorithm is derived for framework (F1). The algorithm approximates the eigenvalue function by a piece-wise quadratic model, which is a global under-estimator (for minimization) or a global over-estimator (for maximization). Locating a globally optimal point of this piece-wise quadratic model is shown to be computationally tractable when there are a few unknowns. The algorithm is applied effectively to various non-convex eigenvalue optimization problems, such as the computation of the H-infinity norm of a linear system and the minimization of the largest eigenvalue of a Hermitian and analytic matrix-valued function, which is important in structural design and control theory. The global convergence of the algorithm is proven, while the linear rate of convergence is observed in practice.

Another algorithm is tailored for framework (F2). The algorithm convexifies and smoothens the problem by replacing the eigenvalue constraint with a convex quadratic constraint. Solving such problems repeatedly results in a locally convergent algorithm robust against non-smoothness. We establish local convergence and linear rate of convergence results.

Singular value optimization characterizations are derived for locating a nearest pencil, a nearest matrix polynomial and a nearest analytic matrix-valued function with prescribed eigenvalues. The derivation in each one of these works boils down to a study of the dimension of the kernel of an associated Sylvester operator. The derived singular value optimization characterizations can be solved numerically by means of the algorithms developed for eigenvalue optimization.

Impact

The algorithm derived for the first framework is suitable for general non-convex eigenvalue optimization problems depending on a few parameters. There is need in various engineering fields for global solution of such problems. In structural design, a classical problem is to design the strongest column subject to volume constraints. Finite element or finite difference discretizations of a differential operator lead to problems involving minimization of the largest eigenvalue of a Hermitian matrix-valued function. In control theory, it is desirable to design systems so that the poles of the system are on the left half of the complex plane and away from the imaginary axis as much as possible. We hope that the algorithm will be adopted in such contexts. A robust implementation of the algorithm (EIGOPT) is made available on the web publicly, and accompanied by a guide.

The constrained algorithm for the second framework encompasses problems that are of great interest. For instance, this algorithm can be adopted to perform optimization of a convex objective subject to a nuclear norm constraint on a matrix-valued function. This last problem is convex yet non-smooth, and drew substantial attention lately due to its connection with sparse estimation. We also coupled this constrained algorithm with the global algorithm for the first framework to compute the pseudospectral abscissa regarding a non-linear eigenvalue problem. Potential coupling of these algorithms in other contexts may result in significant impact. An implementation of this constrained algorithm (EIGOPTC) is also available on the web.

The singular value optimization characterizations derived generalizes existing results for standard matrix-nearness problems such as distance to instability and distance to defectiveness. Due to their connections with backward errors, they may be valuable for numerical algorithms computing eigenvalues in the linear, polynomial and non-linear settings. They may be also of interest in control theory to determine nearby systems with poles at specified positions. The results and the techniques used to locate nearest rectangular pencils with specified eigenvalues can be valuable to tackle a long-lasting problem, namely the distance to a nearest singular pencil.

• EIGOPT: software for eigenvalue optimization with box constraints http://home.ku.edu.tr/~emengi/software/eigopt.html

• EIGOPTC: software for optimization of a linear objective subject to an eigenvalue constraint http://home.ku.edu.tr/~emengi/software/eigopt_constrained.tar

Prospects of the Research Career Development and Reintegration of the Fellow

The fellow acquired a concrete research vision for the upcoming years. Especially he targets eigenvalue optimization problems in the large-scale setting with emphasis on engineering applications and sparse estimation in near future. The strengthened research ties the fellow possessed during this project is likely to boost his research productivity. This is evidenced by a joint project on eigenvalue optimization with Karl Meerbergen and Wim Michiels (KU-Leuven), along with active collaborations with Michael Karow (TU-Berlin), Daniel Kressner (EPF-Lausanne), and Alper Yildirim locally at Koc University. Furthermore, the fellow is surrounded by a promising group of students.

The fellow feels secure and adjusted to the research environment at Koc University and in Turkey. He received his tenure from the Turkish government in February 2013, and applied for his tenure from Koc University in May 2013, which he hopes will be resolved positively soon. The research of the fellow is recognized by the Turkish Academy of Science in terms of a BAGEP award in Spring 2013. All of these developments are indicators of successful reintegration of the fellow.