A Subspace Method for Large Scale Eigenvalue Optimization

TL;DR

This paper introduces a subspace method for efficiently optimizing the largest eigenvalues of large-scale Hermitian matrix functions, enabling practical solutions for very large problems with theoretical convergence guarantees.

Contribution

It develops a subspace procedure that reduces large eigenvalue optimization problems to smaller ones, with proven global convergence and superlinear convergence rates.

Findings

Effective for matrices with sizes of tens instead of thousands

Convergence results established in infinite-dimensional setting

Method applicable to large-scale eigenvalue problems

Abstract

We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi et al. (2014, SIAM J. Matrix Anal. Appl., 35, 699-724). 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.