Locally Unitarily Invariantizable NEPv and Convergence Analysis of SCF

TL;DR

This paper introduces a method to locally transform a class of eigenvector-dependent nonlinear eigenvalue problems into a unitarily invariant form, enabling convergence analysis of SCF iterations with confirmed numerical results.

Contribution

It reveals conditions for global optimality, introduces the aligned NEPv concept, and provides a convergence rate analysis for SCF-type iterations.

Findings

NEPv can be locally unitarily invariantized near optimal solutions.

A closed-form local convergence rate for SCF iterations is established.

Numerical experiments confirm theoretical convergence results.

Abstract

We consider a class of eigenvector-dependent nonlinear eigenvalue problems (NEPv) without the unitary invariance property. Those NEPv commonly arise as the first-order optimality conditions of a particular type of optimization problems over the Stiefel manifold, and previously, special cases have been studied in the literature. Two necessary conditions, a definiteness condition and a rank-preserving condition, on an eigenbasis matrix of the NEPv that is a global optimizer of the associated optimization problem are revealed, where the definiteness condition has been known for the special cases previously investigated. We show that, locally close to the eigenbasis matrix satisfying both necessary conditions, the NEPv can be reformulated as a unitarily invariant NEPv, the so-called aligned NEPv, through a basis alignment operation -- in other words, the NEPv is locally unitarily invariantizable. Numerically, the NEPv is naturally solved by an SCF-type iteration. By exploiting the differentiability of the coefficient matrix of the aligned NEPv, we establish a closed-form local convergence rate for the SCF-type iteration and analyze its level-shifted variant. Numerical experiments confirm our theoretical results.