Gerstewitz Functionals on Linear Spaces and Functionals with Uniform Sublevel Sets

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

Journal of Optimization Theory and Applications, 2017, vol. 173, issue 3, No 6, 812-827

Abstract: 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.

Keywords: Scalarization; Separation theorems; Vector optimization; Production theory; Mathematical finance; 46A99; 46N10; 90C29; 90B30; 91B99 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-017-1098-z

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