Robust Rayleigh quotient minimization and generalized eigenvalue problems

Abstract

We study the problem of minimizing the non-linear trace quotient trace(V^TG(V)V)/trace(V^TH(V)V) over the Stiefel manifold of all n x p matrices with orthonormal columns. Hereby we assume G(V) and H(V) to be symmetric and positive definite for all V. In this way we generalize the robust Rayleigh quotient optimization introduced by Bai et al. in the article "Robust rayleigh quotient minimization and nonlinear eigenvalue problems". We show a possible way to minimize the nonlinear trace quotient by joining and generalizing different known techniques, e.g. the SCF-iteration, and examine it by testing a small example.