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

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.

Automated summary

This paper applies the generic Riemannian trust-region method of Absil et al. to solve generalized eigenvalue problems for nonsquare matrices, achieving global and local convergence rates. Numerical experiments demonstrate the efficiency of the proposed method, with detailed comparisons to existing results highlighting the algorithm's advantages in different scenarios.