Been Down So Long it Looks Like Up to Me: A Unified Derivation of Conjugate Gradient and Variable Metric Minimization

Abstract

Simple derivations, at a level appropriate for an undergraduate computational physics course, of the most popular methods for finding the minimum of a function of many variables are presented in a unified manner in the context of a general optimization scheme that emphasizes their essential similarities. The derivations in this paper encompass the conjugate gradient methods with and without conditioning, and the variable metric methods. The common variants of these methods including Fletcher-Reeves, Polak-Ribiere, Davidon-Fletcher-Powell, and Broyden-Fletcher-Goldfarb-Shanno are described and motivated.