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Journal of Optimization Theory and Applications (2021) 191:363–383

https://doi.org/10.1007/s10957-021-01943-7

Scaling-invariant Functions versus Positively Homogeneous

Functions

Cheikh Toure1·Armand Gissler1·Anne Auger1·Nikolaus Hansen1

Received: 8 January 2021 / Accepted: 7 September 2021 / Published online: 23 September 2021

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021

Abstract

Scaling-invariant functions preserve the order of points when the points are scaled

by the same positive scalar (usually with respect to a unique reference point). Com-

posites of strictly monotonic functions with positively homogeneous functions are

scaling-invariant with respect to zero. We prove in this paper that also 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.

Keywords Scaling-invariant function ·Positively homogeneous function ·Compact

level set

Mathematics Subject Classiﬁcation 49J52 ·54C35

Communicated by Juan-Enrique Martinez Legaz.

BAnne Auger

anne.auger@nospam-inria.fr

Cheikh Toure

cheikh.toure@polytechnique.edu

Armand Gissler

armand.gissler@nospam-inria.fr

Nikolaus Hansen

nikolaus.hansen@nospam-inria.fr

1Inria and CMAP, Ecole Polytechnique, IP Paris, Palaiseau, France

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