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Gerstewitz Functionals on Linear Spaces and Functionals with Uniform Sublevel Sets

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  • Petra Weidner

    (HAWK Hochschule für angewandte Wissenschaft und Kunst Hildesheim/Holzminden/Göttingen, University of Applied Sciences and Arts)

Abstract

In this paper, we study Gerstewitz functionals that are defined on an arbitrary linear space without assuming any topology. Extended real-valued functions with uniform sublevel sets turn out to be Gerstewitz functionals if the sublevel sets can be described by a linear shift of a set in a specified direction. Gerstewitz functionals can represent binary relations and thus act as a tool for scalarization. Sets, which are not necessarily convex, can be separated by Gerstewitz functionals. Conditions are given under which a Gerstewitz functional is finite-valued, convex, positively homogeneous, subadditive, sublinear or monotone. The values of each Gerstewitz functional are connected with those of a sublinear function. It is shown that some Minkowski functionals—especially order unit norms—coincide with a Gerstewitz functional on a subset of the space.

Suggested Citation

  • Petra Weidner, 2017. "Gerstewitz Functionals on Linear Spaces and Functionals with Uniform Sublevel Sets," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 812-827, June.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:3:d:10.1007_s10957-017-1098-z
    DOI: 10.1007/s10957-017-1098-z
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    References listed on IDEAS

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    1. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    2. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, June.
    3. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
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    Cited by:

    1. M. Chinaie & F. Fakhar & M. Fakhar & H. R. Hajisharifi, 2019. "Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem," Journal of Global Optimization, Springer, vol. 75(1), pages 131-141, September.
    2. Truong Quang Bao & Christiane Tammer, 2019. "Scalarization Functionals with Uniform Level Sets in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 310-335, July.
    3. Chuang-Liang Zhang & Nan-jing Huang, 2021. "Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 894-914, September.

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