A Riemannian under-determined BFGS method for least squares inverse eigenvalue problems

This paper is concerned with the parameterized least squares inverse eigenvalue problems for the case that the number of parameters to be constructed is greater than the number of prescribed realizable eigenvalues. Intrinsically, this is a specific problem of finding a zero of an under-determined nonlinear map defined between a Riemannian product manifold and a matrix space. To solve this problem, we propose a Riemannian under-determined BFGS algorithm with a specialized update formula for iterative linear operators, and an Armijo type line search is used. Global convergence properties of this algorithm are established under some mild assumptions. In addition, we also generalize a Riemannian inexact Newton method for solving this problem. Specially, the explicit form of the inverse of the linear operator corresponding to the perturbed normal Riemannian Newton equation is obtained, which improves the efficiency of Riemannian inexact Newton method. Numerical experiments are provided to illustrate the efficiency of the proposed method.