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Set Relations via Families of Scalar Functions and Approximate Solutions in Set Optimization

Author

Listed:
  • Giovanni Paolo Crespi

    (Department of Integrated Business Management, Università Cattaneo–LIUC, 21053 Castellanza, Italy)

  • Andreas H. Hamel

    (Faculty of Economics and Management, Free University of Bozen-Bolzano, I-39031 Bruneck-Brunico, Italy)

  • Matteo Rocca

    (Department of Economics, Università degli Studi dell'Insubria, 21100 Varese, Italy)

  • Carola Schrage

    (Faculty of Economics and Management, Free University of Bozen-Bolzano, I-39031 Bruneck-Brunico, Italy)

Abstract

Via a family of monotone scalar functions, a preorder on a set is extended to its power set and then used to construct a hull operator and a corresponding complete lattice of sets. Functions mapping into the preordered set are extended to complete lattice-valued ones, and concepts for exact and approximate solutions for corresponding set optimization problems are introduced and existence results are given. Well-posedness for complete lattice-valued problems is introduced and characterized. The new approach is compared with existing ones in vector and set optimization. Its relevance is shown by means of many examples from multicriteria decision making, statistics, and mathematical economics and finance.

Suggested Citation

  • Giovanni Paolo Crespi & Andreas H. Hamel & Matteo Rocca & Carola Schrage, 2021. "Set Relations via Families of Scalar Functions and Approximate Solutions in Set Optimization," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 361-381, February.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:1:p:361-381
    DOI: 10.1287/moor.2020.1060
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    References listed on IDEAS

    as
    1. Hamel, Andreas H. & Kostner, Daniel, 2018. "Cone distribution functions and quantiles for multivariate random variables," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 97-113.
    2. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(4), pages 691-705, August.
    3. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(1), pages 225-228, February.
    4. G. P. Crespi & A. Guerraggio & M. Rocca, 2007. "Well Posedness in Vector Optimization Problems and Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 213-226, January.
    5. Dana, Rose-Anne & Riedel, Frank, 2013. "Intertemporal equilibria with Knightian uncertainty," Journal of Economic Theory, Elsevier, vol. 148(4), pages 1582-1605.
    6. X. J. Long & J. W. Peng, 2013. "Generalized B-Well-Posedness for Set Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 612-623, June.
    7. Luca Rigotti & Chris Shannon, 2005. "Uncertainty and Risk in Financial Markets," Econometrica, Econometric Society, vol. 73(1), pages 203-243, January.
    8. Giovanni Crespi & Ivan Ginchev & Matteo Rocca, 2006. "First-order optimality conditions in set-valued optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 87-106, February.
    9. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(5), pages 879-883, October.
    10. Paolo Ghirardato & Fabio Maccheroni & Massimo Marinacci & Marciano Siniscalchi, 2003. "A Subjective Spin on Roulette Wheels," Econometrica, Econometric Society, vol. 71(6), pages 1897-1908, November.
    11. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
    12. Johannes Jahn & Truong Xuan Duc Ha, 2011. "New Order Relations in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 209-236, February.
    13. Evren, Özgür, 2014. "Scalarization methods and expected multi-utility representations," Journal of Economic Theory, Elsevier, vol. 151(C), pages 30-63.
    14. repec:ipg:wpaper:2013-016 is not listed on IDEAS
    15. J. B. Hiriart-Urruty, 1979. "Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 79-97, February.
    16. repec:ipg:wpaper:16 is not listed on IDEAS
    17. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(6), pages 1195-1198, December.
    18. Giovanni P. Crespi & Daishi Kuroiwa & Matteo Rocca, 2017. "Quasiconvexity of set-valued maps assures well-posedness of robust vector optimization," Annals of Operations Research, Springer, vol. 251(1), pages 89-104, April.
    19. repec:dau:papers:123456789/11268 is not listed on IDEAS
    20. E. Miglierina & E. Molho & M. Rocca, 2005. "Well-Posedness and Scalarization in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 126(2), pages 391-409, August.
    21. Evren, Özgür & Ok, Efe A., 2011. "On the multi-utility representation of preference relations," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 554-563.
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    Cited by:

    1. Andreas H. Hamel & Frank Heyde, 2021. "Set-Valued T -Translative Functions and Their Applications in Finance," Mathematics, MDPI, vol. 9(18), pages 1-33, September.

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