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Scaling-invariant functions versus positively homogeneous functions

Cheikh Touré 1, 2 Armand Gissler 1 Anne Auger 1, 2 Nikolaus Hansen 1, 2 
2 RANDOPT - Randomized Optimisation
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
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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Submitted on : Tuesday, September 7, 2021 - 11:19:06 PM
Last modification on : Friday, April 1, 2022 - 3:57:44 AM


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  • HAL Id : hal-03104436, version 2
  • ARXIV : 2101.03755


Cheikh Touré, Armand Gissler, Anne Auger, Nikolaus Hansen. Scaling-invariant functions versus positively homogeneous functions. Journal of Optimization Theory and Applications, Springer Verlag, 2021. ⟨hal-03104436v2⟩



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