SIAM Journal on Optimization, Volume 31, Issue 1, Page 307-330, January 2021.
We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on $q$th-order Taylor models (with $q\geq 1$) that have been recently proposed in the literature. The use of high-order models, while decreasing the worst-case complexity bound, makes these methods computationally more expensive. Hence, to counteract this effect, we propose a multilevel strategy that exploits a hierarchy of problems of decreasing dimension, still approximating the original one, to reduce the global cost of the step computation. A theoretical analysis of the family of methods is proposed. Specifically, local and global convergence results are proved, and a worst-case complexity bound to reach first-order stationary points is also derived. A multilevel version of the well-known adaptive regularization by cubics (corresponding to $q=2$ in our setting) has been implemented, as well as a multilevel third-order method ($q=3$). Numerical experiments clearly highlight the relevance of the new multilevel approaches leading to considerable computational savings compared to their one-level counterparts.

中文翻译：

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高阶多级优化策略

SIAM优化杂志，第31卷，第1期，第307-330页，2021年1月。
我们提出了一个新的多级方法家族，用于无约束最小化。所得策略是最近在文献中提出的基于$ q $阶泰勒模型（$ q \ geq 1 $）的高阶优化方法的多级扩展。高阶模型的使用在降低最坏情况下的复杂性界限的同时，使这些方法的计算成本更高。因此，为了抵消这种影响，我们提出了一种多层次的策略，该策略利用了递减维数的问题的层次结构，仍然与原始维数近似，以减少步骤计算的整体成本。提出了一系列方法的理论分析。具体来说，证明了局部和全局收敛的结果，并且还得出了到达一阶平稳点的最坏情况下的复杂度。已经实现了三次方的自适应自适应正则化的多级版本（对应于我们设置中的$ q = 2 $），以及多级三阶方法（$ q = 3 $）。数值实验清楚地表明了新的多级方法的相关性，与一级方法相比，可节省大量计算量。