Variational Characterization of Monotone Nonlinear Eigenvector Problems and Geometry of Self-Consistent-Field Iteration

TL;DR

This paper introduces a variational framework for monotone nonlinear eigenvalue problems, providing geometric insights, convergence guarantees for SCF iterations, and demonstrating efficiency through numerical experiments.

Contribution

It offers the first variational characterization of mNEPv and analyzes the geometry and convergence of SCF methods for these problems.

Findings

Global convergence of SCF proved.

Accelerated SCF improves computational efficiency.

Numerical examples validate theoretical results.

Abstract

This paper concerns a class of monotone eigenvalue problems with eigenvector nonlinearities (mNEPv). The mNEPv is encountered in applications such as the computation of joint numerical radius of matrices, best rank-one approximation of third-order partial symmetric tensors, and distance to singularity for dissipative Hamiltonian differential-algebraic equations. We first present a variational characterization of the mNEPv. Based on the variational characterization, we provide a geometric interpretation of the self-consistent-field (SCF) iterations for solving the mNEPv, prove the global convergence of the SCF, and devise an accelerated SCF. Numerical examples from a variety of applications demonstrate the theoretical properties and computational efficiency of the SCF and its acceleration.