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Scaling-invariant Functions versus Positively Homogeneous Functions

Author

Listed:
  • Cheikh Toure

    (IP Paris)

  • Armand Gissler

    (IP Paris)

  • Anne Auger

    (IP Paris)

  • Nikolaus Hansen

    (IP Paris)

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.

Suggested Citation

  • Cheikh Toure & Armand Gissler & Anne Auger & Nikolaus Hansen, 2021. "Scaling-invariant Functions versus Positively Homogeneous Functions," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 363-383, October.
    • Handle: RePEc:spr:joptap:v:191:y:2021:i:1:d:10.1007_s10957-021-01943-7
    DOI: 10.1007/s10957-021-01943-7
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    References listed on IDEAS

    as
    1. Valentin V. Gorokhovik & Marina Trafimovich, 2016. "Positively Homogeneous Functions Revisited," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 481-503, November.
    2. J. B. Lasserre & J. B. Hiriart-Urruty, 2002. "Mathematical Properties of Optimization Problems Defined by Positively Homogeneous Functions," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 31-52, January.
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