A Subspace Method for Large Scale Eigenvalue Optimization

We consider the minimization or maximization of the jth largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi et al. (2014). This work addresses the setting when the matrix-valued function involved is very large. We describe how the large scale problem can be converted into a small scale problem by means of orthogonal projections. We present a global convergence result, and rapid convergence results with respect to the size of the small scale problem. 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.