On the Relative Efficiencies of Gradient Methods

A comparison is made among various gradient methods for maximizing a function, based on a characterization by Crockett and Chernoff of the class of these methods. By defining the "efficiency" of a gradient step in a certain way, it becomes easy to compare the efficiencies of different schemes with that of Newton's method, which can be regarded as a particular gradient scheme. For quadratic functions, it is shown that Newton's method is the most efficient (a conclusion which may be approximately true for nonquadratic functions). For functions which are not concave (downward), it is shown that the Newton direction may be just the opposite of the most desirable one. A simple way of correcting this is explained. In trying to maximize a function/ of the N variables [xi, Xt, • ■ • ,xn\, iterative techniques of the class known as gradient methods have proved of great utility [1], [2]. For general functions, the analysis usually begins with an expansion of / in a Taylor series around some point x (= [xi, Xt, • • •, xk}):