A superlinearly convergent algorithm for minimization without evaluating derivatives

An algorithm for unconstrained minimization of a function of n variables that does not require the evaluation of partial derivatives is presented. It is a second order extension of the method of local variations and it does not require any exact one variable minimizations. This method retains the local variations property of accumulation points being stationary for a continuously differentiable function. Furthermore, because this extension makes the algorithm an approximate Newton method, its convergence is superlinear for a twice continuously differentiable strongly convex function.Research sponsored by National Science Foundation Grant GK-32710 and by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-74-2695.