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 step length 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使用。我们重点关注利用三次正则化来改进CGM方向，该改进独立于步长计算。本文提出CGM的混合三次正则化方法，其中选择性使用正则化步。利用Shanno将CGM重构为无记忆BFGS方法的理论框架，我们推导出正则化步方向的新公式。研究表明，正则化步方向每次迭代所需的计算负担量级与其非正则化版本相同。此外，CGM的混合三次正则化在更弱的假设条件下即表现出全局收敛性。数值实验表明，新步方向能够减少迭代次数、提升运行效率并降低步方向重置的需求。总体而言，CGM的混合三次正则化在保持无记忆性和无矩阵特性的同时，在迭代次数和运行时间方面均优于作为无记忆BFGS方法的传统CGM。