Scaling-invariant functions versus positively homogeneous functions
Optimization and Control
2021-09-09 v2
Abstract
Scaling-invariant functions preserve the order of points when the points are scaled by the same positive scalar (with respect to a unique reference point). Composites of strictly monotonic functions with positively homogeneous functions are scaling-invariant with respect to zero. We prove in this paper that the reverse is true for large classes of scaling-invariant functions. Specifically, we give necessary and sufficient conditions for scaling-invariant functions to be composites of a strictly monotonic function with a positively homogeneous function. We also study sublevel sets of scaling-invariant functions generalizing well-known properties of positively homogeneous functions.
Cite
@article{arxiv.2101.03755,
title = {Scaling-invariant functions versus positively homogeneous functions},
author = {Cheikh Touré and Armand Gissler and Anne Auger and Nikolaus Hansen},
journal= {arXiv preprint arXiv:2101.03755},
year = {2021}
}