A matrix algebra approach to approximate Hessians.

This article introduces the generalized simplex Hessian (GSH), a novel matrix-based method for approximating the Hessian of a function using only function evaluations, making it suitable for derivative-free optimization (DFO) algorithms. The GSH and its centered variant, the generalized centered simplex Hessian (GCSH), provide order-1 and order-2 accurate approximations, respectively, of either the full Hessian or a partial Hessian projection, under assumptions on the sampling matrices. The method requires fewer function evaluations and less computational effort than traditional quadratic interpolation, with explicit error bounds established for various cases, including underdetermined sample sets. Additionally, the paper characterizes minimal poised sets of sample points that enable efficient Hessian approximation and explores their relation to quadratic interpolation poisedness, providing explicit formulas for constructing the quadratic interpolation function from these sets.