A Riemannian Newton Algorithm for Nonlinear Eigenvalue Problems

Abstract

We give the formulation of a Riemannian Newton algorithm for solving a class of nonlinear eigenvalue problems by minimizing a total energy function subject to the orthogonality constraint. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Moreover, the positive definiteness condition of the Riemannian Hessian of the total energy function at a solution is derived. Some numerical tests are reported to illustrate the efficiency of the proposed method for solving large-scale problems.

A Brief Introduction to Manifold Optimization

“…Moreover, SCF converges from any initial point and enjoys a local linear convergence rate. Related results can be found in [27,114,22,113,6,110].…”

A Riemannian variant of the Fletcher–Reeves conjugate gradient method for stochastic inverse eigenvalue problems with partial eigendata

“…We now verify that Assumption 1 holds for defined in (24), the retraction (26), and the vector transport (28). We first note that the function is bounded below by zero.…”

“…Recently, there exist some Riemannian optimization methods for eigenproblems and IEPs . Sparked by a variant of the Fletcher–Reeves (FR) conjugate gradient method for solving an unconstrained nonlinear optimization problem over , we propose a Riemannian variant of the FR conjugate gradient method for solving the unconstrained minimization problem on a Riemannian manifold.…”

A Riemannian derivative-free Polak–Ribiére–Polyak method for tangent vector field

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