For general large-scale optimization problems compact representations exist in which recursive quasi-Newton update formulas are represented as compact matrix factorizations. For problems in which the objective function contains additional structure, recent structured quasi-Newton methods exploit available second-derivative information and approximate unavailable second derivatives. This article develops the compact representations of two structured Broyden-Fletcher-Goldfarb-Shanno update formulas. The compact representations enable efficient limited memory and initialization strategies. Two limited memory line search algorithms are described for which extensive numerical results demonstrate the efficacy of the algorithms, including comparisons to IPOPT on large machine learning problems, and to L-BFGS on a real world large scale ptychographic imaging application.

对于一般的大规模优化问题，存在紧凑表示，其中递归拟牛顿更新公式表示为紧凑矩阵分解。对于目标函数包含附加结构的问题，最近的结构化拟牛顿方法利用可用的二阶导数信息和近似不可用的二阶导数。本文开发了两个结构化 Broyden-Fletcher-Goldfarb-Shanno 更新公式的紧凑表示。紧凑的表示可以实现有效的有限内存和初始化策略。描述了两种有限记忆线搜索算法，其广泛的数值结果证明了算法的有效性，包括与 IPOPT 在大型机器学习问题上的比较，