Conjugate gradient minimization methods (CGM) and their accelerated variants are widely used. We focus on the use of cubic regularization to improve the CGM direction independent of the steplength computation. In this paper, we propose the Hybrid Cubic Regularization of CGM, where regularized steps are used selectively. Using Shanno's reformulation of CGM as a memoryless BFGS method, we derive new formulas for the regularized step direction. We show that the regularized step direction uses the same order of computational burden per iteration as its non-regularized version. Moreover, the Hybrid Cubic Regularization of CGM exhibits global convergence with fewer assumptions. In numerical experiments, the new step directions are shown to require fewer iteration counts, improve runtime, and reduce the need to reset the step direction. Overall, the Hybrid Cubic Regularization of CGM exhibits the same memoryless and matrix-free properties, while outperforming CGM as a memoryless BFGS method in iterations and runtime.

翻译：共轭梯度最小化方法（CGM）及其加速变体被广泛应用。本文聚焦于利用三次正则化改进CGM方向，使其独立于步长计算。我们提出混合三次正则化CGM方法，在其中选择性使用正则化步长。基于Shanno将CGM重述为无记忆BFGS方法的框架，推导了正则化步长方向的新公式。研究表明，正则化步长方向每次迭代的计算复杂度与非正则化版本相同。此外，混合三次正则化CGM在更少假设条件下具有全局收敛性。数值实验表明，新步长方向能减少迭代次数、提升运行效率并降低方向重置需求。总体而言，混合三次正则化CGM在保持无记忆与无矩阵特性的同时，作为无记忆BFGS方法在迭代次数与运行时间上均优于CGM。