In order to get a higher order accuracy of approximating the Hessian matrix of the objective function, we use the chain rule and propose two modified secant equations. An interesting property of the proposed methods is that these utilize information from two most recent steps where the usual secant equation uses only the latest step information. The other point of interest to one of the proposed methods is that it makes use of both gradient and function value information. We show that the modified BFGS methods based on the new secant equations are globally convergent. The presented experimental results illustrate that the proposed methods are efficient.

中文翻译：

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使用非线性函数近似求解无约束优化问题的新拟牛顿法

为了获得逼近目标函数Hessian矩阵的更高阶精度，我们使用链式规则并提出了两个修正的割线方程。所提出的方法的一个有趣的特性是，这些方法利用了来自两个最新步骤的信息，而通常的割线方程仅使用最新的步骤信息。所提出的方法之一的另一个关注点是它同时使用了梯度和函数值信息。我们表明，基于新割线方程的改进BFGS方法是全局收敛的。提出的实验结果表明，所提出的方法是有效的。