Well-posedness for set-valued equilibrium problems

Mihaela Miholca

Abstract


In this paper we extend a concept of well-posedness for vector equilibrium problems to the more general framework of set-valued equilibrium problems in topological vector spaces using an appropriate reformulation of the concept of minimality for sets. Sufficient conditions for well-posedness are given in the generalized convex settings and we are able to single out classes of well-posed set-valued equilibrium problems.
On the other hand, in order to relax some conditions, we introduce a concept of minimizing sequences for a set-valued problem, in the set criterion sense, and further we will have a concept of well-posedness for the set-valued equilibrium problem we are interested in. Sufficient results are also given for this well-posedness concept.


Keywords


set-valued equilibrium problems, well-posedness, maximizing sequences, minimizing sequences

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References


Aubin, J.P., Frankowska, H., Set-Valued Analysis, Modern Birkhauser Classics. Birkhauser Boston Inc., Boston, 2009.

Bao, T.Q., Mordukhovich, B.S., Set-valued optimization in welfare economics, Adv. Math. Econ., 13(2010), 113-153.

Bianchi, M., Kassay, G., Pini, R., Well-posedness for vector equilibrium problems, Math. Meth. Oper. Res., 70(2009), 171-182.

Crespi, G.P., Dhingra, M., Lalitha C.S., Pointwise and global well-posedness in set optimization: a direct approach, Ann. Oper. Res., 269(2018), 149-166.

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

Flores-Bazan, F., Hernandez, E., Novo, V., Characterizing eciency without linear structure: a unied approach., J. Glob. Optim., 41(2008), 43-60.

Gopfert, A., Riahi, H., Tammer, C., Zalinescu, C., Variational methods in partially ordered spaces, Springer, CMS Books in Mathematics, 2003.

Hamel, A.H., Heyde, F., Lohne A., Rudlo, B., Schrage, C., Set-optimization-a rather short introduction. In: Hamel AH (ed), Set Optimization and Applications-The State of Art, pp 65-141, Springer, Berlin, 2015.

Han, Y., Huang, N., Well-posedness and stability of solutions for set optimization problems, Optimization, 66(2017), no.1, 17-33.

Kassay, G., Radulescu, V.D., Equilibrium Problems with Applications, Academic Press, Elsevier Science, 2018.

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

Khoshkhabar-amiranloo, S., Khorram, E., Pointwise well-posedness and scalarization in set optimization, Math. Meth. Oper. Res., 82(2015), 195-210.

Khushboo, Lalitha, C.S., A unied minimal solution in set optimization, J. Glob. Optim. 74(2019), 195-211.

Kuroiwa, D., Convexity for set-valued maps and optimization [PhD disertation], Niigata, Niigata University, 1996.

Kuroiwa, D., On set-valued optimization, Nonlinear Analysis, 47(2001), 1395-1400.

Long, X.J., Peng, J.W., Peng, Z.Y., Scalarization and pointwise well-posedness for set optimization problems, J. Global Optim., 62(2015), 763-773.

Miglierina, E., Molho, E.,Well-posedness and convexity in vector optimization, Math. Meth. Oper. Res., 58(2003), 375-385.

Seto, K., Kuroiwa, D., Popovici, N., A systematization of convexity and quasiconvexity concepts for set-valued maps dened by l-type and u-type preorder relations, Optimization, 67(2018), no. 7, 1077-1094.

Tykhonov A.N., On the stability of the functional optimization problems, USSR Comput. Math. Phys. 6(1993), 28-33.




DOI: http://dx.doi.org/10.24193/subbmath.2022.1.07

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