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Lagrange Duality in Set Optimization

Author

Listed:
  • Andreas H. Hamel

    (Yeshiva University New York)

  • Andreas Löhne

    (Martin-Luther-University Halle-Wittenberg)

Abstract

Based on the complete-lattice approach, a new Lagrangian type duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued versions for the known scalar formulas involving infimum and supremum are obtained. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions: The ordering cone may have an empty interior or may not be pointed. “Saddle sets” replace the usual notion of saddle points for the Lagrangian, and this concept is proven to be sufficient to show the equivalence between the existence of primal/dual solutions and strong duality on the one hand, and the existence of a saddle set for the Lagrangian on the other hand. Applications to set-valued risk measures are indicated.

Suggested Citation

  • Andreas H. Hamel & Andreas Löhne, 2014. "Lagrange Duality in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 368-397, May.
    • Handle: RePEc:spr:joptap:v:161:y:2014:i:2:d:10.1007_s10957-013-0431-4
    DOI: 10.1007/s10957-013-0431-4
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    References listed on IDEAS

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    1. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    2. Andreas H. Hamel & Birgit Rudloff & Mihaela Yankova, 2012. "Set-valued average value at risk and its computation," Papers 1202.5702, arXiv.org, revised Jan 2013.
    3. repec:dau:papers:123456789/353 is not listed on IDEAS
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    Cited by:

    1. Robert Bassett & Khoa Le, 2016. "Multistage Portfolio Optimization: A Duality Result in Conic Market Models," Papers 1601.00712, arXiv.org, revised Jan 2016.
    2. Weixuan Xia, 2023. "Optimal Consumption--Investment Problems under Time-Varying Incomplete Preferences," Papers 2312.00266, arXiv.org.
    3. Fernando García-Castaño & Miguel Ángel Melguizo-Padial, 2022. "Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Processes," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 1052-1066, March.

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