Hessian approximations

This work introduces the nested-set Hessian approximation, a second-order approximation method that can be used in any derivative-free optimization routine that requires such information. It is built on the foundation of the generalized simplex gradient and proved to have an error bound that is on the order of the maximal radius of the two sets used in its construction. We show that when the points used in the computation of the nested-set Hessian have a favourable structure, (n+1)(n+2)/2 function evaluations are sufficient to approximate the Hessian. However, the nested-set Hessian also allows for evaluation sets with more points without negating the error analysis. Two calculus-based approximation techniques of the Hessian are developed and some advantages of the same are demonstrated.