The Lipschitzianity of convex vector and set-valued functions

Vu Anh Tuan (), Christiane Tammer () and Constantin Zălinescu ()
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Vu Anh Tuan: Martin-Luther-University Halle-Wittenberg
Christiane Tammer: Martin-Luther-University Halle-Wittenberg
Constantin Zălinescu: University Al.I.Cuza Iaşi

TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, 2016, vol. 24, issue 1, No 15, 273-299

Abstract: Abstract It is well known that every scalar convex function is locally Lipschitz on the interior of its domain in finite dimensional spaces. The aim of this paper is to extend this result for both vector functions and set-valued mappings acting between infinite dimensional spaces with an order generated by a proper convex cone C. Under the additional assumption that the ordering cone C is normal, we prove that a locally C-bounded C-convex vector function is Lipschitz on the interior of its domain by two different ways. Moreover, we derive necessary conditions for Pareto minimal points of vector-valued optimization problems where the objective function is C-convex and C-bounded. Corresponding results are derived for set-valued optimization problems.

Keywords: Lipschitz property; Vector-valued convex functions; Set-valued convex functions; Optimality conditions; 46A40; 49J53; 52A41; 90C25; 90C30; 90C46 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s11750-015-0401-0

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