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A Survey of Set Optimization Problems with Set Solutions

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Set Optimization and Applications - The State of the Art

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 151))

Abstract

This paper presents a state-of-the-art survey on set-valued optimization problems whose solutions are defined by set criteria. It provides a general framework that allows to give an overview about set-valued optimization problems according to decision concepts based on certain set relations. The first part of this paper (Sects. 1 and 2) motivates and describes the set-valued optimization problem (in short, SVOP). The present survey deals with general problems of set-valued optimization and recall its main properties in order to establish the differences between vector set-valued optimization problems (VOP) and set optimization problems (SOP). In this context, in the second part (Sects. 3–5) we focus on those results existing in the literature related with optimality conditions by using a set approach. We list and quote references devoted to (SOP) from the beginning up to now. In Sect. 5, a particular attention is paid to applications of the set relations considered in other fields as fixed point theory. The last section provides some conclusions and suggestions for further study.

Dedicated to the memory of Professor Luis Rodríguez-Marín

This work was partially supported by Ministerio de Ciencia (Spain), project MTM2009-09493 and by UNED, project 2011V/PUNED/0008.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia, E.T.S.I. Industriales, Juan del Rosal 12, 28040, Madrid, Spain

    Elvira Hernández

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  1. Elvira Hernández
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Correspondence to Elvira Hernández .

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Editors and Affiliations

  1. Free University Bozen-Bolzano, Bozen-Bolzano, Italy

    Andreas H Hamel

  2. Freiberg, University of Technology Bergakademie, Freiberg, Germany

    Frank Heyde

  3. Friedrich-Schiller-University Jena, Jena, Germany

    Andreas Löhne

  4. and Business, Vienna University of Economics, Vienna, Austria

    Birgit Rudloff

  5. Free University Bozen-Bolzano, Bozen-Bolzano, Italy

    Carola Schrage

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Hernández, E. (2015). A Survey of Set Optimization Problems with Set Solutions. In: Hamel, A., Heyde, F., Löhne, A., Rudloff, B., Schrage, C. (eds) Set Optimization and Applications - The State of the Art. Springer Proceedings in Mathematics & Statistics, vol 151. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48670-2_4

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