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Journal of Global Optimization

, Volume 73, Issue 2, pp 447–463 | Cite as

Well-posedness and scalarization in set optimization involving ordering cones with possibly empty interior

  • Meenakshi GuptaEmail author
  • Manjari Srivastava
Article
  • 115 Downloads

Abstract

In this paper, we introduce three types of well-posedness for a set optimization problem (u-SOP). Some necessary and sufficient conditions for these well-posedness have been established. Two different scalar optimization problems involving a generalized oriented distance function have been considered. Characterization of u-minimal solutions of (u-SOP) in terms of solutions of these scalar optimization problems have been obtained. Finally, equivalence of well-posedness of (u-SOP) with well-posedness of these scalar optimization problems have been established.

Keywords

Set optimization Well-posedness Hausdorff set-convergence Nonlinear scalarization function 

Mathematics Subject Classification

49J53 49K40 90C48 

References

  1. 1.
    Aubin, J.P., Cellina, A.: Differential Inclusions, Set-Valued Maps and Viability Theory, Grundlehren der mathematischen Wissenschaften, vol. 264. Springer, Berlin (1984)zbMATHGoogle Scholar
  2. 2.
    Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)zbMATHGoogle Scholar
  3. 3.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis, Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston (2009)CrossRefGoogle Scholar
  4. 4.
    Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bao, T.Q., Mordukhovich, B.S.: Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling. Discrete Contin. Dyn. Syst. 31, 1069–1096 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, G.Y., Huang, X., Yang, X.: Vector Optimization-Set Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)Google Scholar
  7. 7.
    Crespi, G.P., Guerraggio, A., Rocca, M.: Well-posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132(1), 213–226 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Crespi, G.P., Dhingra, M., Lalitha, C.S.: Pointwise and global well-posedness in set optimization: a direct approach. Ann. Oper. Res. (2017).  https://doi.org/10.1007/s10479-017-2709-7
  9. 9.
    Crespi, G.P., Papalia, M., Rocca, M.: Extended well-posedness of quasiconvex vector optimization problems. J. Optim. Theory Appl. 141(2), 285–297 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Crespi, G.P., Papalia, M., Rocca, M.: Extended well-posedness of vector optimization problems: the convex case. Taiwan. J. Math. 15(4), 1545–1559 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Crespi, G.P., Kuroiwa, D., Rocca, M.: Convexity and global well-posedness in set optimization. Taiwan. J. Math. 18, 1897–1908 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems, Lecture Notes in Mathematics. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fang, Y.P., Hu, R., Huang, N.J.: Extended B-well-posedness and property (H)- for set-valued vector optimization with convexity. J. Optim. Theory Appl. 135, 445–458 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Göpfert, A., Riahi, H., Tammer, C., et al.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)zbMATHGoogle Scholar
  15. 15.
    Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pontwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, 1822–1833 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B., Schrage, C.: Set optimization—a rather short introduction. In: Hamel, A.H. (ed.) Set Optimization and Applications—The State of the Art, pp. 65–141. Springer, Berlin (2015)CrossRefGoogle Scholar
  17. 17.
    Han, Y., Huang, N.: Well-posedness and stability of solutions for set optimization problems. Optimization 66(1), 17–33 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hiriart-Urruty, J.B.: Tangent cone, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Huang, X.X.: Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 53(1), 101–116 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jahn, J.: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Application. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  22. 22.
    Khoshkhabar-amiranloo, S., Khorram, E.: Scalarization of Levitin–Polyak well-posed set optimization problems. Optimization 66(1), 113–127 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Klein, E., Thompson, A.C.: Theory of Correspondences. Wiley, New York (1984)zbMATHGoogle Scholar
  24. 24.
    Kuratowski, K.: Topology, Volumes 1 and 2. Academic Press, New York (1968)Google Scholar
  25. 25.
    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)Google Scholar
  26. 26.
    Lalitha, C.S., Chatterjee, P.: Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems. J. Glob. Optim. 59, 191–205 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Long, X.J., Peng, J.W.: Generalized B-well-posedness for set optimization problems. J. Optim. Theory Appl. 157, 612–623 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Long, X.J., Peng, J.W., Peng, Z.Y.: Scalarization and pointwise well-posedness for set optimization problems. J. Glob. Optim. 62(4), 763–773 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Loridan, P.: Well-posedness in vector optimization. Math. Appl. 331, 171–192 (1995)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Luc, D.T.: Theory of Vector Optimization: Lecture Notes in Economics and Mathematics Systems, vol. 319. Springer, New York (1989)Google Scholar
  31. 31.
    Lucchetti, R., Revalski, J. (eds.): Recent Development in Well-Posed Variational Problems. Kluwer Academic Publishers, Dordrecht (1995)zbMATHGoogle Scholar
  32. 32.
    Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391–409 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)Google Scholar
  34. 34.
    Tykhonov, A.N.: On the stability of the functional optimization problems. USSR Comput. Math. Phys. 6, 28–33 (1966)CrossRefGoogle Scholar
  35. 35.
    Xu, Y.D., Li, S.J.: A new nonlinear scalarization function and applications. Optimization 65(1), 207–231 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlinear Anal. 71, 3769–3778 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiNew DelhiIndia
  2. 2.Department of Mathematics, Miranda HouseUniversity of DelhiNew DelhiIndia

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