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Lagrange Duality in Set Optimization

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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.

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Notes

  1. Note that X and Y in this section denote random variables, not linear spaces.

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Acknowledgements

The authors are very grateful for the detailed comments of the reviewers which helped to improve the paper. Moreover, the inspirational remarks of Prof. Giannessi are highly appreciated.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Yeshiva University New York, 2495, Belfer Hall, Amsterdam Avenue, New York, NY, 10033, USA

    Andreas H. Hamel

  2. Institute for Mathematics, Martin-Luther-University Halle-Wittenberg, 06099, Halle, Germany

    Andreas Löhne

Authors
  1. Andreas H. Hamel
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  2. Andreas Löhne
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Corresponding author

Correspondence to Andreas H. Hamel.

Additional information

Communicated by Fabian Flores-Bazàn.

Appendix

Appendix

The following definition is taken from [20], where references and more material about structural properties of conlinear spaces can be found.

Definition 9.1

A non-empty set W together with two algebraic operations +:W×W→W (called addition) and \(\cdot\colon{\mathbb {R}}_{+} \times W \to W\) (called multiplication with non-negative scalars) is called a conlinear space provided that

  1. (C1)

    (W,+) is a commutative semigroup with neutral element θ.

  2. (C2)
    1. (i)

      ∀w 1,w 2∈W, \(\forall r \in{\mathbb{R}}_{+}\): r⋅(w 1+w 2)=r⋅w 1+r⋅w 2,

    2. (ii)

      ∀w∈W, \(\forall r, s \in{\mathbb{R}}_{+}\): s⋅(r⋅w)=(rs)⋅w,

    3. (iii)

      ∀w∈W: 1⋅w=w,

    4. (iv)

      0⋅θ=θ.

An element w∈W is called a convex element of the conlinear space W iff

$$\forall s, t \geq0 \colon\quad ( s+t ) \cdot w = s \cdot w + t \cdot w. $$

A conlinear space (W,+,⋅) together with a partial order ⪯ on W (a reflexive, antisymmetric and transitive relation) is called ordered conlinear space provided that (iv) w,w 1,w 2∈W, w 1⪯w 2 imply w 1+w⪯w 2+w, (v) w 1,w 2∈W, w 1⪯w 2, \(r \in{\mathbb{R}}_{+}\) imply r⋅w 1⪯r⋅w 2. A non-empty subset V⊆W of the conlinear space (W,+,⋅) is called a conlinear subspace of W if (vi) v 1,v 2∈V implies v 1+v 2∈V and (vii) v∈V and t≥0 imply t⋅v∈V.

It can easily be checked that a conlinear subspace of a conlinear space again is a conlinear space.

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Hamel, A.H., Löhne, A. Lagrange Duality in Set Optimization. J Optim Theory Appl 161, 368–397 (2014). https://doi.org/10.1007/s10957-013-0431-4

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