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：求解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成本在于计算步长$s_k$，这需要找到立方模型的近似极小值点。我们提出了一种新方法，该方法在低维子空间中寻找该极小值点——与经典方法不同，该子空间会在多次迭代中复用。此外，在需要时还会引入正则化牛顿步长来修正$s_k$。证明表明，我们的方法在提升效率的同时，保持了经典三次正则化方法的最坏情况复杂度。我们还探索了使用有理Krylov子空间进行子空间最小化，以克服使用多项式Krylov子空间时遇到的部分问题。多个实验结果展示了新方法相较于经典实现方案的优势。