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.

翻译：本文提出了一种新颖的基于张量的Dinkelbach型方法，用于计算极值张量广义特征值。我们证明极值张量广义特征值可重构为经典Dinkelbach型方法的关键子问题，该子问题可进一步表述为多线性优化问题。该多线性优化问题在球面约束下通过高效的近端交替最小化方法求解，我们严格证明了该方法的全局收敛性。此外，通过将等价的多线性优化问题重构为无约束优化问题，我们分析了其Kurdyka-Lojasiewicz指数，并给出了所提算法收敛速度的显式表达式。本文提供了求解极值张量广义特征值及最小化高阶信赖域子问题的初步数值实验，验证了所提方法的有效性和实用价值。