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.
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Notes
Trying to answer a question of one of the anonymous referees, we realized that the first equivalence in Proposition 1 is known for a long time.
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Acknowledgments
The authors thank the anonymous referees for their very helpful comments. C. Zălinescu’s research was supported by the grant PN-II-ID-PCE-2011-3-0084, CNCS-UEFISCDI, Romania.
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Anh Tuan, V., Tammer, C. & Zălinescu, C. The Lipschitzianity of convex vector and set-valued functions. TOP 24, 273–299 (2016). https://doi.org/10.1007/s11750-015-0401-0
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DOI: https://doi.org/10.1007/s11750-015-0401-0