On extending the class of convex functions
Optimization and Control
2025-01-06 v1
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.
Cite
@article{arxiv.2501.01854,
title = {On extending the class of convex functions},
author = {Shravan Mohan},
journal= {arXiv preprint arXiv:2501.01854},
year = {2025}
}