The limited memory steepest descent method (Fletcher, 2012) for unconstrained optimization problems stores a few past gradients to compute multiple stepsizes at once. We review this method and propose new variants. For strictly convex quadratic objective functions, we study the numerical behavior of different techniques to compute new stepsizes. In particular, we introduce a method to improve the use of harmonic Ritz values. We also show the existence of a secant condition associated with LMSD, where the approximating Hessian is projected onto a low-dimensional space. In the general nonlinear case, we propose two new alternatives to Fletcher's method: first, the addition of symmetry constraints to the secant condition valid for the quadratic case; second, a perturbation of the last differences between consecutive gradients, to satisfy multiple secant equations simultaneously. We show that Fletcher's method can also be interpreted from this viewpoint.

翻译：有限记忆最速下降法（Fletcher，2012）针对无约束优化问题，通过存储少量历史梯度以一次性计算多个步长。本文回顾了该方法并提出了新变体。对于严格凸二次目标函数，我们研究了不同步长计算技术的数值行为，并特别引入了一种改进调和Ritz值利用的方法。此外，我们证明了与LMSD相关的割线条件的存在性，其中近似Hessian矩阵被投影到低维空间。针对一般非线性情形，我们提出了两种替代Fletcher方法的新方案：其一，为适用于二次情形的割线条件添加对称性约束；其二，对连续梯度间的末次差分进行扰动，以同时满足多个割线方程。研究表明，Fletcher方法亦可从该视角进行解释。