A symmetric grouped and ordered multi-secant Quasi-Newton update formula

Nicolas Boutet, J. Degroote, R. Haelterman
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Abstract

For Quasi-Newton methods, one of the most important challenges is to find an estimate of the Jacobian matrix as close as possible to the real matrix. While in root-finding problems multi-secant methods are regularly used, in optimization, it is the symmetric methods (in particular BFGS) that are popular. Combining multi-secant and symmetric methods in one single update formula would combine their benefits. However, it can be proved that the symmetry and multi-secant property are generally not compatible. In this paper, we try to work around this impossibility and approach the combination of both properties into a single update formula. The novelty of our method is to group secant equations based on their relative importance and to order those groups. This leads to a generic formulation of a symmetric Quasi-Newton method that is as close as possible to satisfying multiple secant equations. Our new update formula is modular and can be used in different applications where multiple secant equations, coming from different sources, are available. The formulation encompasses also different existing Quasi-Newton symmetric update formulas that try to approach the multi-secant property.
一个对称分组有序多割线拟牛顿更新公式
对于拟牛顿方法,最重要的挑战之一是找到一个尽可能接近实际矩阵的雅可比矩阵的估计。虽然在寻根问题中经常使用多割线方法,但在优化中,对称方法(特别是BFGS)更受欢迎。在一个更新公式中结合多割线方法和对称方法将结合它们的优点。然而,可以证明,对称性和多重割线性质一般是不相容的。在本文中,我们试图解决这种不可能性,并将这两个属性组合到一个更新公式中。该方法的新颖之处在于根据割线方程的相对重要性对它们进行分组,并对这些组进行排序。这导致了对称准牛顿方法的一般公式,它尽可能接近于满足多个割线方程。我们的新更新公式是模块化的,可以在不同的应用中使用多个来自不同来源的正割方程。该公式还包含了不同的拟牛顿对称更新公式,这些公式试图接近多重割线性质。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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