A family of variable-metric methods derived by variational means
A family of variable-metric methods derived by variational means
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DOI：
10.1090/s0025-5718-1970-0258249-6
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发表时间：
1970
影响因子：
2
通讯作者：
D. Goldfarb
中科院分区：
文献类型：
--
作者：
D. Goldfarb
利用Greenstadt的变分方法，导出了一种新的二级变度量方法[数学]。比较一下这个问题]。与大卫顿-弗莱彻-鲍威尔（DFP）变度量方法一样，该方法保留了近似矩阵的正定性。该方法与Greenstadt方法一起，形成了一个单参数变度量方法族，其中DFP方法和秩一方法是特例。它相当于Broyden的单参数族[数学]。《论文集》，1967年第21期，第368-381页。给出了变分方法中权重矩阵逆的选择，从而直接推导出DFP和秩一方法。在之前的论文[6]中，Greenstadt使用经典变分方法导出了两种变度量方法。具体来说，开发了两个迭代公式来更新矩阵Hk（即变量度量的逆），其中Hk是最小化函数的逆Hessian G-'（Xk）的近似值。* Greenstadt使用迭代公式Hk+1 = Hk+ Ek对每一步的逆Hessian进行修正估计，根据条件求解使范数N(Ek) = Tr （WEkWEkJ）最小的校正项Ek
A new rank-two variable-metric method is derived using Greenstadt's variational approach [Math. Comp., this issue]. Like the Davidon-Fletcher-Powell (DFP) variable-metric method, the new method preserves the positive-definiteness of the approximating matrix. Together with Greenstadt's method, the new method gives rise to a one-parameter family of variable-metric methods that includes the DFP and rank-one methods as special cases. It is equivalent to Broyden's one-parameter family [Math. Comp., v. 21, 1967, pp. 368-381]. Choices for the inverse of the weighting matrix in the variational approach are given that lead to the derivation of the DFP and rank-one methods directly. In the preceding paper [6], Greenstadt derives two variable-metric methods, using a classical variational approach. Specifically, two iterative formulas are developed for updating the matrix Hk, (i.e., the inverse of the variable metric), where Hk is an approximation to the inverse Hessian G-'(Xk) of the function being minimized.* Using the iteration formula Hk+1 = Hk + Ek to provide revised estimates to the inverse Hessian at each step, Greenstadt solves for the correction term Ek that minimizes the norm N(Ek) = Tr (WEkWEkJ) subject to the conditions