Abstract
We consider weak subgradients of a set-valued map and present a new notion of strong subgradient. We study their properties and compare our constructions and results with other developments. We give existence conditions of both types and establish several optimality conditions in terms of set optimization.
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References
Borwein, J.M.: A Lagrange multiplier theorem and a sandwich theorem for convex relations. Math. Scand. 48, 189–204 (1981)
Yang, X.Q.: A Hahn-Banach theorem in ordered linear spaces and its applications. Optimization 25, 1–9 (1992)
Chen, G.Y., Craven, B.D.: A vector variational inequality and optimization over an efficient set. Z. Oper. Res. 34, 1–12 (1990)
Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Set-valued optimization. Math. Methods Oper. Res. 48, 187–200 (1998)
Li, S.J., Guo, X.L.: Weak subdifferential for set-valued mappings and its applications. Nonlinear Anal. 71, 5781–5789 (2009)
Baier, J., Jahn, J.: On subdifferentials of set-valued maps. J. Optim. Theory Appl. 100, 233–240 (1999)
Song, W.: Weak subdifferential of set-valued mappings. Optimization 52, 263–276 (2003)
Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2004)
Bot, R.I., Grad, S.-M., Wanka, G.: Duality in Vector Optimization. Vector Optimization. Springer, Berlin (2009)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I. Basic Theory. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. II. Applications. Springer, Berlin (2006)
Kuroiwa, D.: On set-valued optimization. Nonlinear Anal. 47, 1395–1400 (2001)
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.: Lagrangian duality in set-valued optimization. J. Optim. Theory Appl. 134, 119–134 (2007)
Jameson, G.: Ordered Linear Spaces. Lecture Notes in Mathematics, vol. 141. Springer, Berlin (1970)
Peng, J.W., Lee, H.W.J., Rong, W.D.: Yang, X.M: Hahn-Banach theorems and subgradients of set-valued maps. Math. Methods Oper. Res. 61, 281–297 (2005)
Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Mathematics in Science and Engineering, vol. 176. Academic Press, Orlando (1985)
Bednarczuk, E.M., Song, W.: Contingent epiderivative and its applications to set-valued maps. Control Cybern. 27, 375–386 (1998)
Rodríguez-Marín, L., Sama, M.: Epidifferentiability of the map of infima in Hilbert spaces. J. Math. Anal. Appl. 342, 371–385 (2008)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Systems and Control: Foundations and Applications, vol. 2. Birkhäuser, Boston (1990)
Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Methods Oper. Res. 46, 193–211 (1997)
Rodríguez-Marín, L., Sama, M.: About contingent epiderivatives. J. Math. Anal. Appl. 327, 745–762 (2007)
Rodríguez-Marín, L., Sama, M.: Variational characterization of the contingent epiderivative. J. Math. Anal. Appl. 335, 1374–1382 (2007)
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Communicated by X.Q. Yang.
This work for the first author was supported in part by Ministerio de Ciencia e Innovación (Spain) Project MTM2009-09493. The authors thank both referees for their value suggestions and comments to the previous version.
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Hernández, E., Rodríguez-Marín, L. Weak and Strong Subgradients of Set-Valued Maps. J Optim Theory Appl 149, 352–365 (2011). https://doi.org/10.1007/s10957-010-9787-x
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DOI: https://doi.org/10.1007/s10957-010-9787-x