Accelerated conjugate direction methods for unconstrained optimization

A family of accelerated conjugate direction methods, corresponding to the Broyden family of quasi-Newton methods, is described. It is shown thatall members of the family generate the same sequence of points approximating the optimum and the same sequence of search directions, provided only that each direction vector is normalized before the stepsize to be taken in that direction is determined.With minimal restrictions on how the stepsize is determined (sufficient only for convergence), the accelerated methods applied to the optimization of a function ofn variables are shown to have an (n+1)-step quadratic rate of convergence. Furthermore, the information needed to generate an accelerating step can be stored in a singlen-vector, rather than the usualn×n symmetric matrix, without changing the theoretical order of convergence.The relationships between this family of methods and existing conjugate direction methods are discussed, and numerical experience with two members of the family is presented.This research was sponsored by the United States Army under Contract No. DAAG29-75-C-0024.The author gratefully acknowledges the valuable assistance of Julia H. Gray, of the Mathematics Research Center, University of Wisconsin, Madison, who painstakingly programmed these methods and obtained the computational results.