We propose a Riemannian limited-memory BFGS method for optimization problems with Euclidean bounds. The method combines a limited-memory quasi-Newton update in the tangent space with a Riemannian adaptation of the generalized Cauchy point strategy from classical L-BFGS-B, enabling efficient handling of Euclidean bounds while exploiting the geometric structure of the optimization domain. This setting is important in several applications, including covariance matrix estimation with bounded variance, neuroimaging, EEG signal classification, and other signal processing or computer-vision tasks that couple manifold variables with constrained Euclidean parameters. We provide a generic algorithmic framework and an implementation of the algorithm in the Manopt.jl library. Numerical experiments on benchmark problems indicate only minor reduction in performance on Euclidean problems compared to the classical L-BFGS-B method, while outperforming interior-point methods. Furthermore, the algorithm was tested on two mixed manifold and bounded Euclidean problems: amplitude-limited blind source separation with Gaussianity penalization and bounded-variance maximum likelihood common principal components analysis. The proposed method outperforms existing methods by several orders of magnitude.

翻译：摘要：本文针对带有欧几里得边界的优化问题，提出了一种黎曼有限内存BFGS方法。该方法将切空间中的有限内存拟牛顿更新与经典L-BFGS-B算法中广义柯西点策略的黎曼适配相结合，在利用优化域几何结构的同时实现对欧几里得边界的高效处理。这一设定在多种应用中具有重要意义，包括有界方差协方差矩阵估计、神经影像学、脑电图信号分类，以及将流形变量与受约束欧几里得参数耦合的其他信号处理或计算机视觉任务。我们提供了通用算法框架及Manopt.jl库中的算法实现。基准问题的数值实验表明，与经典L-BFGS-B方法相比，该方法在处理欧几里得问题时性能仅有小幅下降，同时优于内点法。此外，该算法在两个混合流形与有界欧几里得问题（高斯性惩罚下的振幅受限盲源分离及有界方差极大似然公共主成分分析）上进行了测试。所提方法以数个数量级的优势优于现有方法。