Abstract
The aim of this paper is to study second-order optimality conditions for a set-valued optimization problem with set criterion, where the order relation is induced by a set belong to a class of specific coradiant sets and is not necessarily a preorder. We introduce a notion of generalized second-order radial set, different from the classical second-order radial set, it is defined on a set, not on a point. Using the generalized second-order radial set, we introduce a new type of generalized second-order radial derivatives for set-valued maps and apply them to establish some necessary and sufficient conditions for the lower strict minimal solution of the set optimization problem.
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References
Bao, T.Q., Mordukhovich, B.S.: Set-Valued Optimization in Welfare Economics. Vol. 13, Advances in Mathematical Economics. Springer, Tokyo (2010)
Jahn, J.: Vector Optimization-Theory, Applications and Extensions. Springer, Heidelberg (2011)
Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Springer, Berlin (2015)
Corley, H.W.: Existence and Lagrangian duality for maximizations of set-valued functions. J. Optim. Theory Appl. 54, 489–501 (1987)
Kuroiwa, D.: The natural criteria in set-valued optimization. S\(\bar{u}\)rikaisekikenky\(\bar{u}\)sho k\(\bar{o}\)ky\(\bar{u}\)roku 1031, 85–90 (1998)
Ansari, Q.H., Yao, J.C. (eds.): Recent Advances in Vector Optimization. Springer, Berlin (2012)
Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis. Vol. 541, Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2005)
Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)
Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)
Aubin, J.P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. (ed.) Part A, Mathematical Analysis and Applications. Academic Press, New York (1982)
Taa, A.: Set-valued derivatives of multifunctions and optimality conditions. Numer. Funct. Anal. Optim. 19, 121–140 (1998)
Flores-Bazán, F., Jiménez, B.: Strict efficiency in set-valued optimization. SIAM J. Control. Optim. 48, 881–908 (2009)
Kasimbeyli, R.: Radial epiderivatives and set-valued optimization. Optimization 58, 521–534 (2009)
Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P., Rassias, ThM, Khan, A.A. (eds.) Nonlinear Analysis and Variational Problems, pp. 305–324. Springer, Heidelberg (2010)
Anh, N.L.H., Khanh, P.O.: Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives. J. Glob. Optim. 56, 519–536 (2013)
Anh, N.L.H.: On higher-order mixed duality in set-valued optimization. Bull. Malays. Math. Sci. Soc. 41, 723–739 (2018)
Wang, Q.L., Zhang, X., Li, X., Zeng, J.: New second-order radial epiderivatives and nonconvex set-valued optimization problems. Jpn. J. Ind. Appl. Math. (2019). https://doi.org/10.1007/s13160-019-00370-6
Xu, B.H., Peng, Z.H., Xu, Y.H.: Second-order lower radial tangent derivatives and applications to set-valued optimization. J. Inequal. Appl. 7, 1–19 (2017)
Jahn, J.: Directional derivative in set optimization with the less order relation. Taiwan. J. Math. 19, 737–757 (2015)
Dempe, S., Pilecka, M.: Optimality condtions for set-valued optimisation problems using a modified demyanov difference. J. Optim. Theory Appl. 171, 402–421 (2016)
Yu, G.L.: Optimality conditions in set optimization employing higher-order radial derivatives. Appl. Math. J. Chin. Univ. 32, 225–236 (2017)
Makarov, V.L., Levin, M.J., Rubinov, A.M.: Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms. Elsevier, Amsterdam (1995)
Flores-Bazán, F., Hernández, E.: Optimality conditions for a unified vector optimization problem with not necessarily preordering relations. J. Glob. Optim. 56, 299–315 (2013)
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)
Dhingra, M., Lalitha, C.S.: Set optimization using improvement sets. Yugosl. J. Oper. Res. 27, 153–167 (2017)
Karaman, E., Soyertem, M., Güvenç, I.A., et al.: Partial order relations on family of sets and scalarizations for set optimization. Positivity 22, 783–802 (2018)
Köbis, E., Le, T.T., Tammer, C.: A generalized scalarization method in set optimization with respect to variable domination structures. Vietnam J. Math. 46, 95–125 (2018)
Gutiérrez, C., Jiménez, B., Novo, V.: On approximate effciency in multiobjective programming. Math. Meth. Oper. Res. 64, 165–185 (2006)
Kuroiwa, D.: On derivatives of set-valued maps and optimality conditions for set optimization. J. Nonlinear Convex Anal. 10, 41–50 (2009)
Alonso, M., Rodríguez-Marín, L.: Optimality conditions in set-valued maps with set optimization. Nonlinear Anal. 70, 3057–3064 (2009)
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This research was supported by the National Natural Science Foundation of China (Grant Numbers: 11971078, 11571055, 11171362).
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Yao, B., Li, S. Second-order optimality conditions for set optimization using coradiant sets. Optim Lett 14, 2073–2086 (2020). https://doi.org/10.1007/s11590-019-01531-9
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DOI: https://doi.org/10.1007/s11590-019-01531-9