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Duality

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Set-valued Optimization

Part of the book series: Vector Optimization ((VECTOROPT))

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Abstract

In this chapter we present duality assertions for set-valued optimization problems in infinite dimensional spaces where the solution concept is based on vector approach, on set approach as well as on lattice approach. For set-valued optimization problems where the solution concept is based on vector approach we present conjugate duality statements. The notions of conjugate maps, subdifferential and a perturbation approach used for deriving these duality assertions are given. Furthermore, Lagrange duality for set-valued problems based on vector approach is shown. Moreover, we consider set-valued optimization problems where the solution concept is given by a set order relation introduced by Kuroiwa and derive corresponding saddle point assertions. For set-valued problems where the solution concept is based on lattice structure, we present duality theorems that are based on an consequent usage of infimum and supremum (in the sense greatest lower and least upper bounds with respect to a partial ordering). We derive conjugate duality assertions as well as Lagrange duality statements.

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Authors and Affiliations

  1. Rochester Institute of Technology School of Mathematical Sciences, Rochester, New York, USA

    Akhtar A. Khan

  2. Halle, Germany

    Christiane Tammer

  3. Faculty of Mathematics, University “Al. I. Cuza” Iasi, Iasi, Romania

    Constantin Zălinescu

Authors
  1. Akhtar A. Khan
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  2. Christiane Tammer
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  3. Constantin Zălinescu
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Khan, A.A., Tammer, C., Zălinescu, C. (2015). Duality. In: Set-valued Optimization. Vector Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54265-7_8

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