On extending the class of convex functions

TL;DR

This paper explores conditions under which certain functions involving positive definite matrices are convex, extending the class of known convex functions with potential applications in optimization and matrix analysis.

Contribution

It introduces new convex functions involving matrix-weighted terms and provides conditions for their convexity using linear algebra techniques.

Findings

p^TW log(p) is convex if W is a diagonally dominant positive definite M-matrix

p^TW p^2 and (p^k)^TW p^k are convex homogeneous polynomials, with the latter being SOS-convex

p^TW e^p is convex only if W is a non-negative diagonal matrix

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.