2025/01/03 by Shravan Mohan, Mohan, Shravan
Computer Science · Mathematics · #Optimization and Variational Analysis #Functional Equations Stability Results
paper · pdf · doi:10.48550/arxiv.2501.01854
In this brief note, it is shown that the function pTW 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, pTW p2 and (pk)TW pk - the latter also happen to be SOS-convex. Lastly, usign the same techniques, it is also shown that the function pTW 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.