A Variational Characterization and A Line Search Newton-Noda Method for the unifying spectral problem of nonnegative tensors

Abstract

We study the general $(\boldsymbol{\sigma},\mathbf{p})$-eigenvalue problem of nonnegative tensors introduced by A. Gautier, F. Tudisco, and M. Hein [SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1206--1231], which unifies several well-studied tensor eigenvalue and singular value problems. First, we propose an alternative min-max Collatz--Wielandt formula for the $(\boldsymbol{\sigma},\mathbf{p})$-spectral radius, which bypasses the auxiliary multihomogeneous mapping employed in that work. This variational characterization both recovers several classical results and admits a natural convex reformulation. It arises from an alternative approach that directly connects the $(\boldsymbol{\sigma},\mathbf{p})$-spectral problem to a class of convex programs. We then develop and analyze a line search Newton-Noda method (LS-NNM) for computing the positive $(\boldsymbol{\sigma},\mathbf{p})$-eigenpair of nonnegative tensors. The proposed method integrates Newton method with Noda iteration. The Newton equation is derived from an equivalent nonlinear system, while the eigenvalue sequence is updated by the strategy of the Noda iteration and its variants. To ensure global convergence, we introduce a positivity-preserving line search procedure based on an equivalent constrained optimization problem. The global and quadratic convergence of LS-NNM are established for the class of $(\boldsymbol{\sigma},\mathbf{p})$-spectral problem that admits a unique positive $(\boldsymbol{\sigma},\mathbf{p})$-eigenpair, as guaranteed by the Perron-Frobenius theorem. Finally, numerical experiments are conducted to illustrate the performance of LS-NNM.