turkmath.org

This talk introduces a subspace framework for the optimization of the jth largest eigenvalue of a large, Hermitian and analytic matrix-valued function depending on several parameters for a prescribed j. The range of the large matrix-valued function is projected onto a small subspace orthogonally, and its domain is restricted to the same subspace. This leads to reduced eigenvalue optimization problems involving small matrix-valued functions. The subspace is expanded with the addition of the eigenvectors for the optimal parameter values of the small problem. In the infinite dimensional setting and for the minimization of the jth largest eigenvalue, we prove that the optimal solution of the reduced problem converges to the optimal solution of the infinite dimensional problem as the subspace dimension grows to infinity. Furthermore, we establish that the rate of convergence is at least superlinear with respect to the subspace dimension for both the minimization and the maximization problems. These theoretical results are in harmony with what we observe in practice, that the problems involving matrices of size on the scale of thousands are approximated very accurately with the reduced problems involving matrices of size on the scale of tens.