Using generalized simplex methods to approximate derivatives

This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. It is also shown how to approximate a Hessian-vector product with a minimal number of function evaluations. These approximations are obtained using the techniques called generalized simplex Hessian and generalized centred 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.