On extending the class of convex functions

TL;DR

This work enlarges the set of known convex functions by deriving sufficient conditions for convexity of entropy-like and polynomial forms parameterized by a matrix $W$. Using Hessian factoring and classical linear-algebra tools, it proves that $f(p)=p^{\top} W \log(p)$ is convex on $\mathbb{R}^n_{+}$ when $W$ is a diagonally dominant positive definite M-matrix, and extends the approach to a family of polynomials including $p^{\top} W p^2$, $(p^2)^{\top} W p^2$ (which is SOS-convex), and $(p^{k})^{\top} W p^{k}$ for even/odd $k$ with domain-dependent convexity. A complementary negative result shows $p^{\top} W e^{p}$ cannot be convex on $\mathbb{R}^n_{+}$ unless $W$ is diagonal. The work employs Hessian decomposition, Schur complements, and Hadamard-product properties to establish these convexity conditions and discusses potential extensions and applications to posynomials and SOS-convexity of broader polynomial classes.

Abstract

In this brief note, it is shown that the function p^TW log(p) is convex in p if W is a diagonally dominant positive definite M-matrix. The techniques used to prove convexity are well-known in linear algebra and essentially involves factoring the Hessian in a way that is amenable to martix analysis. Using similar techniques, two classes of convex homogeneous polynomials is derived - namely, p^TW p2 and (p^k)^TW p^k - the latter also happen to be SOS-convex. Lastly, usign the same techniques, it is also shown that the function p^TW ep is convex over the positive reals only if W is a non-negative diagonal matrix. Discussions regarding the utility of these functions and examples accompany the results presented.