A trust‐region method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

Abstract

The l parameterized generalized eigenvalue problems for the nonsquare matrix pencils, proposed by Chu and Golub [SIAM J. Matrix Anal. Appl., 28(2006), pp. 770-787], can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. Some early proposed algorithms are based on the first-order information of the objective function, and fast convergence could not be expected. In this article, we turn the generic Riemannian trust-region method of Absil et al. into a practical algorithm for solving the underlying problem, which enjoys the global convergence and local superlinear convergence rate. Numerical experiments are provided to illustrate the efficiency of the proposed method. Detailed comparisons with some existing results (l = 1 and l = n) are given first. For the case l = n, the algorithm can yield exactly the same optimal solution as Ito and Murota's algorithm, which is essentially a direct method to solve the problem. The algorithm runs faster than Boutry et al.'s algorithm for the case l = 1. Further comparisons with some early proposed gradient-based algorithms, and some latest infeasible methods for solving manifold optimization problems are also provided to show the merits of the proposed approach.

Least-Squares Spectral Methods for ODE Eigenvalue Problems

“…Such problems are not classically treated in numerical linear algebra, and they usually do not have solutions satisfying the equation exactly. Nonetheless, several practical methods for rectangular eigenvalue problems have been proposed, including [14,34,35,36].…”

A Trust-region Framework for Iteration Solution of the Direct INDSCAL Problem in Metric Multidimensional Scaling

“…However, it is imperative to recognize that while the pure Newton method is restricted to local convergence and lacks the ability to discern between local minima, maxima, and saddle points, the trustregion algorithm demonstrates a more resilient behavior, as evidenced by Absil et al (2007), ensuring convergence to stationary points irrespective of initial conditions. Since optimization on a compact manifold can be conceptualized as a general nonlinear optimization problem with constraints, the extension of the trust-region algorithm from Euclidean space to manifold settings has been a subject of direct pursuit and extensive investigation, yielding successful applications (Absil et al, 2004(Absil et al, , 2007Baker et al, 2008;Boumal and Absil, 2011;Heidel and Schulz, 2018;Ishteva et al, 2011;Li et al, 2021;Sato, 2015;Sato and Sato, 2018;Yang et al, 2019;Zhang, 2012). Particularly noteworthy is the Riemannian trust-region (RTR) approach proposed by Ishteva et al (2011) for achieving the best rank-(R 1 , R 2 , R 3 ) approximation of third-order tensors, a problem framed as the minimization of a cost function over a product of three Grassmann manifolds.…”