High-order tensor methods for solving both convex and nonconvex optimization problems have generated significant research interest, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's method. On each iteration, these methods require the unconstrained local minimization of a (potentially nonconvex) multivariate polynomial of degree higher than two, constructed using third-order (or higher) derivative information, and regularized by an appropriate power of regularization. Developing efficient techniques for solving such subproblems is an ongoing topic of research, and this paper addresses the case of the third-order tensor subproblem. We propose the CQR algorithmic framework, for minimizing a nonconvex Cubic multivariate polynomial with Quartic Regularisation, by minimizing a sequence of local quadratic models that incorporate simple cubic and quartic terms. The role of the cubic term is to crudely approximate local tensor information, while the quartic one controls model regularization and progress. We provide necessary and sufficient optimality conditions that fully characterise the global minimizers of these cubic-quartic models. We then turn these conditions into secular equations that can be solved using nonlinear eigenvalue techniques. We show, using our optimality characterisations, that a CQR algorithmic variant has the optimal-order evaluation complexity of $\mathcal{O}(\epsilon^{-3/2})$ when applied to minimizing our quartically-regularised cubic subproblem, which can be further improved in special cases. We propose practical CQR variants that use local tensor information to construct the local cubic-quartic models. We test these variants numerically and observe them to be competitive with ARC and other subproblem solvers on typical instances and even superior on ill-conditioned subproblems with special structure.

翻译：针对凸与非凸优化问题的高阶张量方法已引发广泛研究兴趣，这类方法能达到最优全局收敛率及比牛顿法更快的局部收敛率。每步迭代中，方法需利用三阶（或更高阶）导数信息构造次数高于2的多项式（可能非凸），并通过正则项的适当幂次进行约束，进而求解该多项式的无约束局部极小化问题。开发高效求解此类子问题的技术是当前研究热点，本文重点关注三阶张量子问题。我们提出CQR算法框架，通过最小化一系列融合简单立方项与四次项的局部二次模型，实现具有四次正则化的非凸立方多项式极小化。其中立方项用于粗略近似局部张量信息，四次项则控制模型正则化与优化进程。我们建立了完整刻画立方-四次模型全局极小化子的最优性充要条件，进而将其转化为可通过非线性特征值技术求解的久期方程。基于最优性表征理论，我们证明CQR算法变体在求解四次正则化立方子问题时具有$\mathcal{O}(\epsilon^{-3/2})$的最优阶复杂度，且在特殊情形下可进一步优化。我们提出利用局部张量信息构造立方-四次模型的实用CQR变体，数值实验表明，该变体在典型实例中与ARC及其他子问题求解器性能相当，在具有特殊结构的病态子问题上甚至展现出更优表现。