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Duality related to approximate proper solutions of vector optimization problems

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  • C. Gutiérrez
  • L. Huerga
  • V. Novo
  • C. Tammer

Abstract

In this work we introduce two approximate duality approaches for vector optimization problems. The first one by means of approximate solutions of a scalar Lagrangian, and the second one by considering $$(C,\varepsilon )$$ ( C , ε ) -proper efficient solutions of a recently introduced set-valued vector Lagrangian. In both approaches we obtain weak and strong duality results for $$(C,\varepsilon )$$ ( C , ε ) -proper efficient solutions of the primal problem, under generalized convexity assumptions. Due to the suitable limit behaviour of the $$(C,\varepsilon )$$ ( C , ε ) -proper efficient solutions when the error $$\varepsilon $$ ε tends to zero, the obtained duality results extend and improve several others in the literature. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • C. Gutiérrez & L. Huerga & V. Novo & C. Tammer, 2016. "Duality related to approximate proper solutions of vector optimization problems," Journal of Global Optimization, Springer, vol. 64(1), pages 117-139, January.
    • Handle: RePEc:spr:jglopt:v:64:y:2016:i:1:p:117-139
    DOI: 10.1007/s10898-015-0366-4
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    References listed on IDEAS

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    1. X. M. Yang & D. Li & S. Y. Wang, 2001. "Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 413-427, August.
    2. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    3. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
    4. Z. F. Li, 1998. "Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 623-649, September.
    5. T. Son & D. Kim, 2013. "ε-Mixed type duality for nonconvex multiobjective programs with an infinite number of constraints," Journal of Global Optimization, Springer, vol. 57(2), pages 447-465, October.
    6. C. Gutiérrez & B. Jiménez & V. Novo, 2011. "A generic approach to approximate efficiency and applications to vector optimization with set-valued maps," Journal of Global Optimization, Springer, vol. 49(2), pages 313-342, February.
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    Cited by:

    1. Christian Günther & Bahareh Khazayel & Christiane Tammer, 2022. "Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 408-442, June.

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