Abstract
This paper aims at studying the generalized well-posedness in the sense of Bednarczuk for set optimization problems with set-valued maps. Three kinds of B-well-posedness for set optimization problems are introduced. Some relations among the three kinds of B-well-posedness are established. Necessary and sufficient conditions of well-posedness for set optimization problems are obtained.
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Aubin, J.P., Cellina, A.: Differential Inclusions, Set-valued Maps and Viability Theory. Grundlehren Math. Wiss., vol. 264. Springer, Berlin (1984)
Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin (2005)
Klein, E., Thompson, A.C.: Theory of Correspondences. Wiley, New York (1984)
Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematics Systems, vol. 319. Springer, New York (1989)
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)
Alonso, M., Rodríguez-Marín, L.: Set-relations and optimality conditions in set-valued maps. Nonlinear Anal. 63, 1167–1179 (2005)
Hernández, E., Rodríguez-Marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67, 1276–1736 (2007)
Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)
Kuroiwa, D.: On set-valued optimization. Nonlinear Anal. 47, 1395–1400 (2001)
Kuroiwa, D.: Existence theorems of set optimization with set-valued maps. J. Inf. Optim. Sci. 24, 73–84 (2003)
Hernández, E., Rodríguez-Marín, L.: Optimality conditions for set-valued maps with set optimization. Nonlinear Anal. 70, 3057–3064 (2009)
Hernández, E., Rodríguez-Marín, L., Sama, M.: On solutions of set-valued optimization problems. Comput. Math. Appl. 60, 1401–1408 (2010)
Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valed maps. J. Math. Anal. Appl. 325, 1–18 (2007)
Hernández, E., Rodríguez-Marín, L.: Lagrangian duality in set-valued optimization. J. Optim. Theory Appl. 134, 119–134 (2007)
Bednarczuck, E.M., Miglierina, E., Molho, E.: A mountain pass-type theorem for vector-valued functions. Set-Valued Var. Anal. 19, 569–587 (2011)
Gutiérrez, C., Jiménez, B., Novo, V., Thibault, L.: Strict approximate solutions in set-valued optimization with applications to the approximate ekeland variational principle. Nonlinear Anal. 73, 3842–3855 (2010)
Ha, T.X.D.: Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124, 187–206 (2005)
Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004)
Tykhonov, A.N.: On the stability of the functional optimization problems. USSR Comput. Math. Math. Phys. 6, 28–33 (1966)
Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993)
Zolezzi, T.: Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal. 25, 437–453 (1995)
Zolezzi, T.: Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91, 257–266 (1996)
Bednarczuck, E.M.: Well posedness of vector optimization problem. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization Problems. Lecture Notes in Economics and Mathematical Systems, vol. 294, pp. 51–61. Springer, Berlin (1987)
Lucchetti, R.: Well posedness, towards vector optimization. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization Problems. Lecture Notes in Economics and Mathematical Systems, vol. 294, pp. 194–207. Springer, Berlin (1987)
Durea, M.: Scalarization for pointwise well-posedness vectorial problems. Math. Methods Oper. Res. 66, 409–418 (2007)
Huang, X.X.: Pointwise well-posedness of perturbed vector optimization problems in a vector-valued variational principle. J. Optim. Theory Appl. 108, 671–687 (2001)
Xiao, G., Xiao, H., Liu, S.Y.: Scalarization and pointwise well-posedness in vector optimization problems. J. Glob. Optim. 49, 561–574 (2011)
Bednarczuck, E.M.: An approach to well-posedness in vector optimization: consequences to stability and parametric optimization. Control Cybern. 23, 107–122 (1994)
Huang, X.X.: Extended well-posedness properties of vector optimization problems. J. Optim. Theory Appl. 106, 165–182 (2000)
Huang, X.X.: Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 53, 101–116 (2001)
Miglierina, E., Molho, E.: Well-posedness and convexity in vector optimization. Math. Methods Oper. Res. 58, 375–385 (2003)
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)
Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391–409 (2005)
Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlinear Anal. 71, 3769–3778 (2009)
Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, 1822–1833 (2011)
Nikodem, K.: Continuity of K-convex set-valued function. Bull. Pol. Acad. Sci., Math. 34, 393–400 (1986)
Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)
Kuratowski, K.: Topology, Vols. 1 and 2. Academic Press, New York (1968)
Acknowledgements
The authors would like to thank the editor and anonymous referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11001287 and 11171363), the Natural Science Foundation Project of Chongqing (CSTC 2010BB9254 and CSTC 2009BB8240), the Education Committee Project Research Foundation of Chongqing (No. KJ100711), and the Special Fund of Chongqing Key Laboratory (CSTC 2011KLORSE01).
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Communicated by Johannes Jahn.
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Long, X.J., Peng, J.W. Generalized B-Well-Posedness for Set Optimization Problems. J Optim Theory Appl 157, 612–623 (2013). https://doi.org/10.1007/s10957-012-0205-4
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DOI: https://doi.org/10.1007/s10957-012-0205-4