Well-Posedness for Set Optimization Problems Involving Set Order Relations

Abstract

In this paper, we investigate set optimization problems with three types of set order relations. Various kinds of well-posedness for these problems and their relationship are concerned. Then, sufficient conditions for set optimization problems to be well-posed are established. Moreover, Kuratowski measure of noncompactness is applied to survey characterizations of well-posedness for set optimization problems. Furthermore, approximating solution maps and their stability are researched to propose the link between stability of the approximating problem and well-posedness of the set optimization problem.

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References

  1. 1.

    Alonso, M., Rodríguez-marín, L.: Set-relations and optimality conditions in set-valued maps. Nonlinear Anal. 63(8), 1167–1179 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Springer, Boston (2009)

    Google Scholar 

  3. 3.

    Bednarczuk, E., Penot, J.P.: Metrically well-set minimization problems. Appl. Math. Optim. 26(3), 273–285 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Chiriaev, A., Walster, G.W.: Interval arithmetic specification. Technical Report (1998)

  5. 5.

    Crespi, G.P., Kuroiwa, D., Rocca, M.: Convexity and global well-posedness in set-optimization. Taiwan. J. Math. 18(6), 1897–1908 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Dhingra, M., Lalitha, C.S.: Well-setness and scalarization in set optimization. Optim. Lett. 10(8), 1657–1667 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics. Springer, Berlin (1993)

    Google Scholar 

  8. 8.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P., Rassias, T., Khan, A. (eds.) Nonlinear Analysis and Variational Problems, pp 305–324. Springer, New York (2010)

  10. 10.

    Ha, T.X.D.: Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124(1), 187–206 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Hamel, A., Heyde, F.: Duality for set-valued measures of risk. SIAM. J. Financial Math. 1(1), 66–95 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Han, Y., Huang, N.: Well-posedness and stability of solutions for set optimization problems. Optimization 66(1), 17–33 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Hernández, E., Rodríguez-marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67(6), 1276–1736 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Hernández, E., Rodríguez-marín, L.: Nonconvex scalarization in set optimization with set-valed maps. J. Math. Anal. Appl. 325(1), 1–18 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Hernández, E., Rodríguez-marín, L.: Lagrangian duality in set-valued optimization. J. Optim. Theory Appl. 134(1), 119–134 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148(2), 209–236 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization, An Introduction with Applications. Springer, Berlin (2015)

  18. 18.

    Köbis, E., Tam, L.T., Tammer, C.: A generalized scalarization method in set optimization with respect to variable domination structures. Vietnam J. Math. 46 (1), 95–125 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Kuratowski, K.: Topology, vol. 2. Academic Press, New York (1968)

    Google Scholar 

  20. 20.

    Kuroiwa, D.: On set-valued optimization. Nonlinear Anal. 47(2), 1395–1400 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Kuroiwa, D.: Some duality theorems of set-valued optimization with natural criteria. In: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, pp 221–228. World Scientific, River Edge (1999)

  22. 22.

    Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal. 30(3), 1487–1496 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Long, X.J., Peng, J.W.: Generalized B-well-posedness for set optimization problems. J. Optim. Theory Appl. 157(3), 612–623 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126(2), 391–409 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Milovanovíc-Arandjelovíc, M.M.: Measures of noncompactness on uniform spacesthe axiomatic approach. Filomat 15, 221–225 (2001)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Nishnianidze, Z.G.: Fixed points of monotone multivalued operators. Soobshch. Akad. Nauk Gruzin SSR. 114, 489–491 (1984)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Sun Microsystems, Inc. Interval Arithmetic Programming Reference. Palo Alto (2000)

  28. 28.

    Neukel, N.: Order relations of sets and its application in socio-economics. Appl. Math. Sci. 7, 5711–5739 (2013)

    MathSciNet  Google Scholar 

  29. 29.

    Tikhonov, A.N.: On the stability of the functional optimization problem. USSR Comput. Math. Math. Phys. 6(4), 28–33 (1966)

    MATH  Article  Google Scholar 

  30. 30.

    Young, R.C.: The algebra of many-valued quantities. Math. Ann. 104(1), 260–290 (1931)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlinear Anal. 71(9), 3769–3778 (2009)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgments

We are very grateful to the referees for the valuable and detailed remarks and suggestions that helped us significantly improve the paper.

Funding

Rabian Wangkeeree was partially supported by the Thailand Research Fund, Grant No. RSA6080077 and Naresuan University. The second author was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED), Grant No. 101.01-2017.18.

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Correspondence to Rabian Wangkeeree.

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Dedicated to Professor Hoang Tuy

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Vui, P.T., Anh, L.Q. & Wangkeeree, R. Well-Posedness for Set Optimization Problems Involving Set Order Relations. Acta Math Vietnam 45, 329–344 (2020). https://doi.org/10.1007/s40306-020-00362-6

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Keywords

  • Set order relation
  • Set optimization problem
  • Well-posedness
  • Stability
  • Measure of noncompactness

Mathematics Subject Classification (2010)

  • 80M50
  • 49K40
  • 54C60