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Tensor-based Dinkelbach method for computing generalized tensor eigenvalues and its applications

Haibin Chen, Wenqi Zhu, Coralia Cartis

TL;DR

This work addresses computing extremal generalized tensor eigenvalues by recasting the problem as a Dinkelbach-type fractional program on the unit sphere and solving the resulting multilinear optimization via a proximal alternating minimization (PAM) framework. It establishes a rigorous equivalence between the original fractional problem and a concave multilinear model (with an augmentation to enforce concavity when needed), provides a globally convergent PAM algorithm with an explicit convergence rate derived from the KL property, and demonstrates applicability to extremal eigenpairs and high-order trust-region subproblems. The approach yields analytic subproblem solutions, leverages a block-coordinate descent interpretation, and is supported by theoretical convergence guarantees and practical numerical results. Overall, the method offers a scalable, convergent procedure for tackling high-order tensor generalized eigenvalue problems with potential impact on related optimization tasks in tensor analysis and higher-order trust-region methods.

Abstract

In this paper, we propose a novel tensor-based Dinkelbach--Type method for computing extremal tensor generalized eigenvalues. We show that the extremal tensor generalized eigenvalue can be reformulated as a critical subproblem of the classical Dinkelbach--Type method, which can subsequently be expressed as a multilinear optimization problem (MOP). The MOP is solved under a spherical constraint using an efficient proximal alternative minimization method, in which we rigorously establish the global convergence. Additionally, the equivalent MOP is reformulated as an unconstrained optimization problem, allowing for the analysis of the Kurdyka-Lojasiewicz (KL) exponent and providing an explicit expression for the convergence rate of the proposed algorithm. Preliminary numerical experiments on solving extremal tensor generalized eigenvalues and minimizing high-order trust-region subproblems are provided, validating the efficacy and practical utility of the proposed method.

Tensor-based Dinkelbach method for computing generalized tensor eigenvalues and its applications

TL;DR

This work addresses computing extremal generalized tensor eigenvalues by recasting the problem as a Dinkelbach-type fractional program on the unit sphere and solving the resulting multilinear optimization via a proximal alternating minimization (PAM) framework. It establishes a rigorous equivalence between the original fractional problem and a concave multilinear model (with an augmentation to enforce concavity when needed), provides a globally convergent PAM algorithm with an explicit convergence rate derived from the KL property, and demonstrates applicability to extremal eigenpairs and high-order trust-region subproblems. The approach yields analytic subproblem solutions, leverages a block-coordinate descent interpretation, and is supported by theoretical convergence guarantees and practical numerical results. Overall, the method offers a scalable, convergent procedure for tackling high-order tensor generalized eigenvalue problems with potential impact on related optimization tasks in tensor analysis and higher-order trust-region methods.

Abstract

In this paper, we propose a novel tensor-based Dinkelbach--Type method for computing extremal tensor generalized eigenvalues. We show that the extremal tensor generalized eigenvalue can be reformulated as a critical subproblem of the classical Dinkelbach--Type method, which can subsequently be expressed as a multilinear optimization problem (MOP). The MOP is solved under a spherical constraint using an efficient proximal alternative minimization method, in which we rigorously establish the global convergence. Additionally, the equivalent MOP is reformulated as an unconstrained optimization problem, allowing for the analysis of the Kurdyka-Lojasiewicz (KL) exponent and providing an explicit expression for the convergence rate of the proposed algorithm. Preliminary numerical experiments on solving extremal tensor generalized eigenvalues and minimizing high-order trust-region subproblems are provided, validating the efficacy and practical utility of the proposed method.
Paper Structure (4 sections, 2 theorems, 9 equations, 1 algorithm)

This paper contains 4 sections, 2 theorems, 9 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. PAM for Tensor Generalized Eigenvalues Problems
  4. Dinkelbach-Type Method and the Relationship between \ref{['e1']} and \ref{['e2']}

Key Result

proposition thmcounterproposition

Dinkelbach1967 Assume that $S\subseteq\mathbb{R}^n$ is compact, and $f({\bf x}), g({\bf x})$ are homogeneous polynomials such that $g({\bf x})\geq 0, {\bf x}\in S$. Then for problems (e1) and (e2), it always holds that $F(\bar{\theta})=0$. Furthermore, ${\bf x}^*$ is an optimal solution of (e1) if a

Theorems & Definitions (3)

  • proposition thmcounterproposition
  • remark thmcounterremark
  • proposition thmcounterproposition