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Journal Articles Journal of Optimization Theory and Applications Year : 2021

Scaling-invariant functions versus positively homogeneous functions

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.
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Dates and versions

hal-03104436 , version 1 (08-01-2021)
hal-03104436 , version 2 (07-09-2021)

Identifiers

Cite

Cheikh Touré, Armand Gissler, Anne Auger, Nikolaus Hansen. Scaling-invariant functions versus positively homogeneous functions. Journal of Optimization Theory and Applications, 2021, 191 (1), pp.363-383. ⟨10.1007/s10957-021-01943-7⟩. ⟨hal-03104436v2⟩
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