Scaling-invariant Functions versus Positively Homogeneous Functions

Cheikh Toure (), Armand Gissler (), Anne Auger () and Nikolaus Hansen ()
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Cheikh Toure: IP Paris
Armand Gissler: IP Paris
Anne Auger: IP Paris
Nikolaus Hansen: IP Paris

Journal of Optimization Theory and Applications, 2021, vol. 191, issue 1, No 14, 363-383

Abstract: 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). Composites 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; 49J52; 54C35 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-021-01943-7

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