The Lipschitzianity of convex vector and set-valued functions

• Vu Anh Tuan ^[1] ; Christiane Tammer ^[1] ; Constantin Zalinescu ^[2]
  1. [1] Martin Luther University Halle-Wittenberg

     Martin Luther University Halle-Wittenberg

     Kreisfreie Stadt Halle, Alemania

     
  2. [2] University Al.I.Cuza Iasi, Rumanía
• Localización: Top, ISSN-e 1863-8279, ISSN 1134-5764, Vol. 24, Nº. 1, 2016, págs. 273-299
• Idioma: inglés
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• Resumen

  • 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.