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Saddle representations of positively homogeneous functions by linear functions

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Abstract

We say that a positively homogeneous function admits a saddle representation by linear functions iff it admits both an inf-sup-representation and a sup-inf-representation with the same two-index family of linear functions. In the paper we show that each continuous positively homogeneous function can be associated with a two-index family of linear functions which provides its saddle representation. We also establish characteristic properties of those two-index families of linear functions which provides saddle representations of functions belonging to the subspace of Lipschitz continuous positively homogeneous functions as well as the subspaces of difference sublinear and piecewise linear functions.

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Acknowledgements

The research was supported by the Belarussian State Research Programm (Grant “Convergence – 1.04.01”). The authors thank the anonymous referees for careful and thorough reading of the paper and their valuable comments which led to an improved presentation of the results.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, ul. Surganova, 11, 220072, Minsk, Belarus

    Valentin V. Gorokhovik

  2. Sukhoi State Technical University of Gomel, Pr. Octiabria, 48, 246746, Gomel, Belarus

    Marina Trafimovich

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  1. Valentin V. Gorokhovik
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  2. Marina Trafimovich
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Correspondence to Valentin V. Gorokhovik.

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Gorokhovik, V.V., Trafimovich, M. Saddle representations of positively homogeneous functions by linear functions. Optim Lett 12, 1971–1980 (2018). https://doi.org/10.1007/s11590-018-1260-z

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