On Davidon's collinear scaling algorithms for optimization

Abstract

In 1980, Davidon [5] presented a class of algorithms for unconstrained minimization. The algorithms of Davidon are related to quasi-Newton methods, and are based on local collinear scalings and local conic approximations, which extend respectively, local affine scalings and local quadratic approximations used in derivations of quasi-Newton methods. The algorithms of Davidon [5] have not been studied theoretically or computationally. Algorithms based on collinear scalings and conic approximations related to quasi-Newton methods that have been investigated todate are all derived based on an approach that Sorensen [12] presented. In this note, we show that all these algorithms derived using the approach of Sorensen [12] are in fact different from the algorithms that Davidon [5] presented. It then follows that fresh investigations are necessary to assess the usefulness of the algorithms of Davidon [5], and of algorithms based on collinear scalings and conic approximations extending quasi-Newton methods.