A matrix algebra approach to approximate Hessians

Abstract

This work presents a novel matrix-based method for constructing an approximation Hessian using only function evaluations. The method requires less computational power than interpolation-based methods and is easy to implement in matrix-based programming languages such as MATLAB. As only function evaluations are required, the method is suitable for use in derivative-free algorithms. For reasonably structured sample sets, the method is proven to create an order-$1$ accurate approximation of the full Hessian. Under more specialized structures, the method is proved to yield order-$2$ accuracy. The undetermined case, where the number of sample points is less than required for full interpolation, is studied and error bounds are developed for the resulting partial Hessians.