[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Global well-posedness of set-valued optimization with application to uncertain problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Well-posedness for optimization problems is a well-known notion and has been studied extensively for scalar, vector and set-valued optimization problems. There is a broad classification in terms of pointwise and global well-posedness notions in vector and set-valued optimization problems. We have focused on global well-posedness for set-valued optimization problems in this paper. A number of notions of global well-posedness for set-valued optimization problems already exist in the literature. However, we found equivalence between some existing notions of global well-posedness for set-valued optimization problems and also found scope of improving and extending the research in that field. That has been the first aim of this paper. On the other hand, robust approach towards uncertain optimization problems is another growing area of research. The well-posedness for the robust counterparts have been explored in very few papers, and that too only in the scalar and vector cases (see (Anh et al. in Ann Oper Res 295(2):517–533, 2020), (Crespi et al. in Ann Oper Res 251(1–2):89–104, 2017)). Therefore, the second aim of this paper is to study some global well-posedness properties of the robust formulation of uncertain set-valued optimization problems that generalize the concept of the well-posedness of robust formulation of uncertain vector optimization problems as discussed in Anh et al. (Ann Oper Res 295(2):517–533, 2020), Crespi et al. (Ann Oper Res 251(1–2):89–104, 2017).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alonso, M., Rodríguez-Marín, L.: Set-relations and optimality conditions in set-valued maps. Nonlinear Anal. 63(8), 1167–1179 (2005). https://doi.org/10.1016/j.na.2005.06.002

    Article  MATH  Google Scholar 

  2. Anh, L.Q., Duy, T.Q., Hien, D.V.: Well-posedness for the optimistic counterpart of uncertain vector optimization problems. Ann. Oper. Res. 295(2), 517–533 (2020). https://doi.org/10.1007/s10479-020-03840-0

    Article  MATH  Google Scholar 

  3. Beck, A., Ben-Tal, A.: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett. 37(1), 1–6 (2009). https://doi.org/10.1016/j.orl.2008.09.010

    Article  MATH  Google Scholar 

  4. Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ (2009). https://doi.org/10.1515/9781400831050

  5. Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998). https://doi.org/10.1287/moor.23.4.769

    Article  MATH  Google Scholar 

  6. Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1), 1–13 (1999). https://doi.org/10.1016/S0167-6377(99)00016-4

    Article  MATH  Google Scholar 

  7. Crespi, G.P., Dhingra, M., Lalitha, C.S.: Pointwise and global well-posedness in set optimization: a direct approach. Ann. Oper. Res. 269(1–2), 149–166 (2018). https://doi.org/10.1007/s10479-017-2709-7

    Article  MATH  Google Scholar 

  8. Crespi, G.P., Kuroiwa, D., Rocca, M.: Convexity and global well-posedness in set-optimization. Taiwanese J. Math. 18(6), 1897–1908 (2014). https://doi.org/10.11650/tjm.18.2014.4120

    Article  MATH  Google Scholar 

  9. Crespi, G.P., Kuroiwa, D., Rocca, M.: Quasiconvexity of set-valued maps assures well-posedness of robust vector optimization. Ann. Oper. Res. 251(1–2), 89–104 (2017). https://doi.org/10.1007/s10479-015-1813-9

    Article  MATH  Google Scholar 

  10. Dhingra, M., Lalitha, C.S.: Well-setness and scalarization in set optimization. Optim. Lett. 10(8), 1657–1667 (2016). https://doi.org/10.1007/s11590-015-0942-z

    Article  MATH  Google Scholar 

  11. Dontchev, A.L., Zolezzi, T.: Well-posed optimization problems. Lecture Notes in Mathematics, vol. 1543. Springer-Verlag, Berlin (1993)

  12. Durea, M.: Scalarization for pointwise well-posed vectorial problems. Math. Methods Oper. Res. 66(3), 409–418 (2007). https://doi.org/10.1007/s00186-007-0162-0

    Article  MATH  Google Scholar 

  13. Ehrgott, M., Ide, J., Schöbel, A.: Minmax robustness for multi-objective optimization problems. European J. Oper. Res. 239(1), 17–31 (2014). https://doi.org/10.1016/j.ejor.2014.03.013

    Article  MATH  Google Scholar 

  14. Goerigk, M., Schöbel, A.: Algorithm engineering in robust optimization (2016). arXiv:1505.04901v3

  15. Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational methods in partially ordered spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 17. Springer-Verlag, New York (2003). https://doi.org/10.1007/b97568. https://doi.org/10.1007/0-387-21743-6

  16. Gupta, M., Srivastava, M.: Well-posedness and scalarization in set optimization involving ordering cones with possibly empty interior. J. Global Optim. 73(2), 447–463 (2019). https://doi.org/10.1007/s10898-018-0695-1

    Article  MATH  Google Scholar 

  17. Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75(4), 1822–1833 (2012). https://doi.org/10.1016/j.na.2011.09.028

    Article  MATH  Google Scholar 

  18. Hamel, A.H., Löhne, A.: A set optimization approach to zero-sum matrix games with multi-dimensional payoffs. Math. Methods Oper. Res. 88(3), 369–397 (2018). https://doi.org/10.1007/s00186-018-0639-z

    Article  MATH  Google Scholar 

  19. Han, Y., Huang, Nj.: Well-posedness and stability of solutions for set optimization problems. Optim. 66(1), 17–33 (2017). https://doi.org/10.1080/02331934.2016.1247270

    Article  MATH  Google Scholar 

  20. Han, Y., Huang, Nj.: Continuity and convexity of a nonlinear scalarizing function in set optimization problems with applications. J. Optim. Theory Appl. 177(3), 679–695 (2018). https://doi.org/10.1007/s10957-017-1080-9

    Article  MATH  Google Scholar 

  21. Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325(1), 1–18 (2007). https://doi.org/10.1016/j.jmaa.2006.01.033

    Article  MATH  Google Scholar 

  22. Ide, J., Köbis, E., Kuroiwa, D., Schöbel, A., Tammer, C.: The relationship between multi-objective robustness concepts and set-valued optimization. Fixed Point Theory Appl. 2014(83), 20 (2014). https://doi.org/10.1186/1687-1812-2014-83

    Article  MATH  Google Scholar 

  23. Ide, J., Schöbel, A.: Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts. OR Spect. 38(1), 235–271 (2016). https://doi.org/10.1007/s00291-015-0418-7

    Article  MATH  Google Scholar 

  24. Klamroth, K., Köbis, E., Schöbel, A., Tammer, C.: A unified approach to uncertain optimization. European J. Oper. Res. 260(2), 403–420 (2017). https://doi.org/10.1016/j.ejor.2016.12.045

    Article  MATH  Google Scholar 

  25. Kuroiwa, D.: On natural criteria in set-valued optimization. RIMS Kokyuroku 1048, 86–92 (1998). http://hdl.handle.net/2433/62183. Dynamic decision systems in uncertain environments (Japanese) (Kyoto, 1998)

  26. Kuroiwa, D., Lee, G.M.: On robust multiobjective optimization. Vietnam J. Math. 40(2–3), 305–317 (2012)

    MATH  Google Scholar 

  27. Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal.: Theory, Methods Appl. 30(3), 1487–1496 (1997). https://doi.org/10.1016/S0362-546X(97)00213-7

  28. Long, X.J., Peng, J.W.: Generalized \(B\)-well-posedness for set optimization problems. J. Optim. Theory Appl. 157(3), 612–623 (2013). https://doi.org/10.1007/s10957-012-0205-4

    Article  MATH  Google Scholar 

  29. Long, X.J., Peng, J.W., Peng, Z.Y.: Scalarization and pointwise well-posedness for set optimization problems. J. Global Optim. 62(4), 763–773 (2015). https://doi.org/10.1007/s10898-014-0265-0

    Article  MATH  Google Scholar 

  30. Seto, K., Kuroiwa, D., Popovici, N.: A systematization of convexity and quasiconvexity concepts for set-valued maps, defined by \(l\)-type and \(u\)-type preorder relations. Optim. 67(7), 1077–1094 (2018). https://doi.org/10.1080/02331934.2018.1454920

    Article  MATH  Google Scholar 

  31. Som, K., Vetrivel, V.: A note on pointwise well-posedness of set-valued optimization problems. Journal of Optimization Theory and Applications 192, 628–647 (2022). https://doi.org/10.1007/s10957-021-01981-1

    Article  MATH  Google Scholar 

  32. Som, K., Vetrivel, V.: On robustness for set-valued optimization problems. J. Global Optim. 79(4), 905–925 (2021). https://doi.org/10.1007/s10898-020-00959-z

    Article  MATH  Google Scholar 

  33. Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research 21(5), 1154–1157 (1973). http://www.jstor.org/stable/168933

  34. Vui, P.T., Anh, L.Q., Wangkeeree, R.: Well-posedness for set optimization problems involving set order relations. Acta Math. Vietnam. 45(2), 329–344 (2020). https://doi.org/10.1007/s40306-020-00362-6

    Article  MATH  Google Scholar 

  35. Zhang, C..l., Huang, N..j.: Well-posedness and stability in set optimization with applications. Positivity 25, 1153–1173 (2021). https://doi.org/10.1007/s11117-020-00807-0

    Article  MATH  Google Scholar 

  36. Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlinear Anal. 71(9), 3769–3778 (2009). https://doi.org/10.1016/j.na.2009.02.036

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors are indebted to the anonymous referees for their valuable comments, suggestions and important corrections that have helped us to improve the paper substantially. The first author thanks National Board for Higher Mathematics, India (Ref No: 2/39(2)/2015/NBHM/R& D-II/7463) for financial assistance. The second author thanks the Department of Science and Technology (SERB), India, for the financial support under the MATRICS scheme (MTR/2017/000128).

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    Kuntal Som & V. Vetrivel

Authors
  1. Kuntal Som
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Vetrivel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kuntal Som.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Som, K., Vetrivel, V. Global well-posedness of set-valued optimization with application to uncertain problems. J Glob Optim 85, 511–539 (2023). https://doi.org/10.1007/s10898-022-01208-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-022-01208-1

Keywords

Search

Navigation