On a characterization of convexity-preserving maps, Davidon's collinear scalings and Karmarkar's projective transformations

In a recent paper, the authors have proved results characterizing convexity -preserving maps defined on a subset of a not-necessarily finite dimensiona l real vector space as projective maps. The purpose of this note is three-f old. First, we state a theorem characterizing continuous, injective, convex ity-preserving maps from a relatively open, connected subset of an affine s ubspace of R-m into R-n as projective maps. This result follows From the mo re general results stated and proved in a coordinate-free manner in the abo ve paper, and is intended to be more accessible to researchers interested i n optimization algorithms. Second, based on that characterization theorem, we offer a characterization theorem for collinear scalings first introduced by Davidon in 1977 for deriving certain algorithms for nonlinear optimizat ion, and a characterization theorem for projective transformations used by Karmarkar in 1984 in his linear programming algorithm. These latter two the orems indicate that Davidon's collinear scalings and Karmarkar's projective transformations are the only continuous, injective, convexity-preserving m aps possessing certain features that Davidon and Karmarkar respectively des ired in the derivation of their algorithms. The proofs of these latter two theorems utilize our characterization of continuous, injective, convexity-p reserving maps in a way that has implications to the choice of scalings and transformations in the derivation of optimization algorithms in general. T he third purpose of this note is to point this out.