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Variable Ordering Structures in Vector Optimization

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Recent Developments in Vector Optimization

Part of the book series: Vector Optimization ((VECTOROPT,volume 1))

Abstract

In vector optimization one assumes in general that a partial ordering is given by some nontrivial convex cone K in the considered space Y. But already in 1974 in one of the first publications [37] related to the definition of optimal elements in vector optimization also the idea of variable ordering structures was given: to each element of the space a cone of dominated (or preferred) directions is defined and thus the ordering structure is given by a set-valued map. In [37] a candidate element was defined to be nondominated if it is not dominated by any other reference element w.r.t. the corresponding cone of this other element. Later, also another notion of optimal elements in the case of a variable ordering structure was introduced [7,  8,  9]: a candidate element is called a minimal (or nondominated-like) element if it is not dominated by any other reference element w.r.t. the cone of the candidate element.

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Acknowledgements

This contribution was partially written during the author’s stay at the Institute of Mathematics, Hanoi, Vietnam, which was supported by the Alexander von Humboldt-Foundation and the Institute of Mathematics of Hanoi. The author thanks Prof. Dr. Truong Xuan Duc Ha for hospitality and discussions on the topic of this chapter.

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  1. Department of Mathematics, University of Erlangen-Nuremberg, Martensstr. 3, D-91058, Erlangen, Germany

    Gabriele Eichfelder

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  1. Gabriele Eichfelder
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Correspondence to Gabriele Eichfelder .

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  1. Aligarh, 202001, Uttar Pradesh, India

    Qamrul Hasan Ansari

  2. , Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 80424, Taiwan R.O.C.

    Jen-Chih Yao

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Eichfelder, G. (2012). Variable Ordering Structures in Vector Optimization. In: Ansari, Q., Yao, JC. (eds) Recent Developments in Vector Optimization. Vector Optimization, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21114-0_4

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