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The Lipschitzianity of convex vector and set-valued functions

Author

Listed:
  • Vu Anh Tuan

    (Martin-Luther-University Halle-Wittenberg)

  • Christiane Tammer

    (Martin-Luther-University Halle-Wittenberg)

  • Constantin Zălinescu

    (University Al.I.Cuza Iaşi
    Institute of Mathematics Octav Mayer (Romanian Academy))

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.

Suggested Citation

  • Vu Anh Tuan & Christiane Tammer & Constantin Zălinescu, 2016. "The Lipschitzianity of convex vector and set-valued functions," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(1), pages 273-299, April.
    • Handle: RePEc:spr:topjnl:v:24:y:2016:i:1:d:10.1007_s11750-015-0401-0
    DOI: 10.1007/s11750-015-0401-0
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    References listed on IDEAS

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    1. Marius Durea & Radu Strugariu & Christiane Tammer, 2013. "Scalarization in Geometric and Functional Vector Optimization Revisited," Journal of Optimization Theory and Applications, Springer, vol. 159(3), pages 635-655, December.
    2. Joydeep Dutta & Christiane Tammer, 2006. "Lagrangian conditions for vector optimization in Banach spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(3), pages 521-540, December.
    3. J. B. Hiriart-Urruty, 1979. "Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 79-97, February.
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