Second-order information in the adaptive search exploration algorithm

This paper explores the possibility of including second order information on some derivative-free methods that locate the local optimum of a functional, without the computation of derivatives. We note that a superlinear rate of convergence may be obtained at the expense of extra function evaluations, a feature we try to avoid. This paper attempts to capture the essence of second order information that these derivative-free methods can provide with no additional function evaluations. We carried out several tests and observed that, overall, a combination of adaptive search and second order information may improve the efficiency of the algorithm.