关于具有大二次惩罚项的目标函数最小化的改进增广拉格朗日方法的注记
Note on the Modifed Augmented Lagrangian Method for Minimization of Functions with Large Quadratic Penalties
AI总结：
本文提出改进增广拉格朗日方法用于处理具有大二次惩罚项的目标函数最小化问题，通过重新表述方案实现更稳定的数值性能。
摘要：
在最近的工作（arXiv-DOI: 1804.08072v1）中，我们介绍了改进的增广拉格朗日方法（MALM）用于高效最小化具有大二次惩罚项的目标函数。从MALM推导出的最优性方程系统与原始目标函数的最优性方程系统相关。但数值上表现更好，因为大的惩罚因子被更温和的因子取代。在我们的原始工作中，我们通过内迭代应用拟牛顿迭代来计算多元函数的根来制定MALM。在本注记中，我们展示这种方案可以通过重新表述为一个良好缩放的无约束最小化问题来方便地替代。在本注记中，我们简要回顾了用于最小化等式约束问题的增广拉格朗日方法（ALM）。然后我们提出并推导了新的MALM的公式，用于最小化具有大二次惩罚项的无约束问题。最后，我们讨论了MALM与ALM之间的关系。
英文摘要：
In a recent work (arXiv-DOI: 1804.08072v1) we introduced the Modified Augmented Lagrangian Method (MALM) for the efficient minimization of objective functions with large quadratic penalty terms. From MALM there results an optimality equation system that is related to that of the original objective function. But, it is numerically better behaved, as the large penalty factor is replaced by a milder factor. In our original work, we formulated MALM with an inner iteration that applies a Quasi-Newton iteration to compute the root of a multi-variate function. In this note we show that this formulation of the scheme with a Newton iteration can be replaced conveniently by formulating a well-scaled unconstrained minimization problem. In this note, we briefly review the Augmented Lagrangian Method (ALM) for minimizing equality-constrained problems. Then we motivate and derive the new proposed formulation of MALM for minimizing unconstrained problems with large quadratic penalties. Eventually, we discuss relations between MALM and ALM.
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