The main computational cost per iteration of adaptive cubic regularization methods for solving large-scale nonconvex problems is the computation of the step $s_k$, which requires an approximate minimizer of the cubic model. We propose a new approach in which this minimizer is sought in a low dimensional subspace that, in contrast to classical approaches, is reused for a number of iterations. A regularized Newton step to correct $s_k$ is also incorporated whenever needed. We show that our method increases efficiency while preserving the worst-case complexity of classical cubic regularized methods. We also explore the use of rational Krylov subspaces for the subspace minimization, to overcome some of the issues encountered when using polynomial Krylov subspaces. We provide several experimental results illustrating the gains of the new approach when compared to classic implementations.

翻译：针对大规模非凸问题的自适应三次正则化方法，每次迭代的主要计算代价在于求解步长$s_k$，这需要近似计算三次模型的极小化点。本文提出一种新方法，在低维子空间中寻找该极小化点——与经典方法不同，该子空间可被重复使用多个迭代周期。同时，在需要时引入正则化牛顿步对$s_k$进行修正。理论分析表明，所提方法在保持经典三次正则化方法最坏情形复杂度的前提下，显著提升了计算效率。此外，我们探索了有理Krylov子空间在子空间极小化中的应用，以克服多项式Krylov子空间存在的若干问题。多组实验结果表明，与经典实现相比，新方法在性能提升方面具有明显优势。