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Set-valued duality theory for multiple objective linear programs and application to mathematical finance

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  • Frank Heyde
  • Andreas Löhne
  • Christiane Tammer

Abstract

We develop a duality theory for weakly minimal points of multiple objective linear programs which has several advantages in contrast to other theories. For instance, the dual variables are vectors rather than matrices and the dual feasible set is a polyhedron. We use a set-valued dual objective map the values of which have a very simple structure, in fact they are hyperplanes. As in other set-valued (but not in vector-valued) approaches, there is no duality gap in the case that the right-hand side of the linear constraints is zero. Moreover, we show that the whole theory can be developed by working in a complete lattice. Thus the duality theory has a high degree of analogy to its classical counterpart. Another important feature of our theory is that the infimum of the set-valued dual problem is attained in a finite set of vertices of the dual feasible domain. These advantages open the possibility of various applications such as a dual simplex algorithm. Exemplarily, we discuss an application to a Markowitz-type bicriterial portfolio optimization problem where the risk is measured by the Conditional Value at Risk. Copyright Springer-Verlag 2009

Suggested Citation

  • Frank Heyde & Andreas Löhne & Christiane Tammer, 2009. "Set-valued duality theory for multiple objective linear programs and application to mathematical finance," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(1), pages 159-179, March.
    • Handle: RePEc:spr:mathme:v:69:y:2009:i:1:p:159-179
    DOI: 10.1007/s00186-008-0216-y
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    References listed on IDEAS

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    1. Shelby Brumelle, 1981. "Duality for Multiple Objective Convex Programs," Mathematics of Operations Research, INFORMS, vol. 6(2), pages 159-172, May.
    2. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
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    Cited by:

    1. Luis Rodríguez-Marín & Miguel Sama, 2013. "Scalar Lagrange Multiplier Rules for Set-Valued Problems in Infinite-Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 683-700, March.
    2. Luc, Dinh The, 2011. "On duality in multiple objective linear programming," European Journal of Operational Research, Elsevier, vol. 210(2), pages 158-168, April.
    3. Tijani Amahroq & Abdessamad Oussarhan, 2019. "Lagrange Multiplier Rules for Weakly Minimal Solutions of Compact-Valued Set Optimization Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 36(04), pages 1-22, August.
    4. N. Mahdavi-Amiri & F. Salehi Sadaghiani, 2017. "Strictly feasible solutions and strict complementarity in multiple objective linear optimization," 4OR, Springer, vol. 15(3), pages 303-326, September.
    5. B. Jiménez & V. Novo & A. Vílchez, 2020. "Characterization of set relations through extensions of the oriented distance," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 89-115, February.
    6. Andreas H Hamel & Andreas Löhne, 2020. "Choosing sets: preface to the special issue on set optimization and applications," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 1-4, February.

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