从标量到矩阵设置的戴-廖参数
From a Scalar to a Matrix Setting for the Dai--Liao Parameter
AI总结：
研究针对戴-廖共轭梯度算法关键参数设置，先强化其标量自适应设置最优性，后从标量设置转向矩阵设置，使三项共轭梯度算法成为扩展类算法一员，经实验验证该框架兼具理论合理性与计算可靠性。
AI中文摘要：
众所周知，戴-廖共轭梯度算法的数值性能和理论性质高度依赖其关键参数的调整。本文首先利用著名的调和-几何-算术-二次均值不等式强化了之前提出的戴-廖参数的两个标量自适应设置的最优性，展示了这些参数选择如何通过缩小包含其奇异值的区间来增强戴-廖搜索方向矩阵的良态性。对缩放无记忆拟牛顿更新公式也进行了类似分析。然后，作为主要贡献，从戴-廖参数的经典标量设置转向矩阵设置以增强优化方法的灵活性和多样性。在参数的矩阵形式下，一个著名的三项共轭梯度算法成为扩展的戴-廖算法类的一员，并具有一些计算上吸引人的性质。最后通过在标准基准测试问题上进行数值实验提供支持证据，结果表明所提出的框架兼具理论合理性和计算可靠性。
英文摘要：
As is well known, both the numerical performance and the theoretical properties of the Dai--Liao conjugate gradient algorithm are highly dependent on the adjustment of its key parameter. Here, we first employ the well-known Harmonic--Geometric--Arithmetic--Quadratic mean inequality to reinforce the optimality of two previously proposed scalar adaptive settings of the Dai--Liao parameter. In other words, we show how these parameter choices are capable of enhancing the well-conditioning of the Dai--Liao search direction matrix by shrinking the intervals containing its singular values. A similar analysis is also carried out for scaled memoryless quasi--Newton updating formulas in order to further justify the optimality of two classical scaling parameters associated with these updates. Then, as the main contribution of this work, we move from the classical scalar setting of the Dai--Liao parameter to a matrix setting aimed at enhancing flexibility and diversity within optimization methods. In particular, we show that, under such a matrix formulation of the parameter, a well-known three-term conjugate gradient algorithm emerges as a member of the proposed extended Dai--Liao class of algorithms while enjoying several computationally attractive properties. Among these, the well-conditioning of the associated search direction matrix is especially noteworthy, as well as the ability to make more explicit use of the second-order information of the model. Finally, to provide practical evidence supporting the proposed matrix setting of the Dai--Liao parameter, we conduct a series of numerical experiments on standard benchmark test problems and report the results in detail. Generally speaking, the proposed framework is shown to retain both theoretical soundness and computational reliability.
