This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. These approximations are obtained using the techniques called \emph{generalized simplex Hessian} and \emph{generalized centered simplex Hessian}. We show how to choose the matrices of directions involved in the computation of these two techniques depending on the entries of the Hessian of interest. We discuss the number of function evaluations required in each case and develop a general formula to approximate all order-$P$ partial derivatives. Since only function evaluations are required to compute the methods discussed in this paper, they are suitable for use in derivative-free optimization methods.

翻译：本文提出了两种仅利用函数求值来逼近海森矩阵部分条目的方法。这些逼近采用了称为广义单纯形海森和广义中心单纯形海森的技术。我们展示了如何根据所关注的海森矩阵条目选择这些技术计算中涉及的矩阵方向。我们讨论了每种情形所需的函数求值次数，并推导出逼近所有P阶偏导数的通用公式。由于本文讨论的方法仅需函数求值，因此适用于无导数优化方法。