Abstract
In this paper, some properties of the lower (upper) semi-continuity for set-valued maps taking values in an abstract pre-ordered set are showed, which are then applied to study the existence of solutions for abstract set optimization problems. Moreover, some relationships between minimal solutions for abstract set-valued optimization problems with the vector and set criteria are given under mild conditions. As applications, existence results of solutions for set optimization problems are applied to obtain the existence of saddle points for the set-valued map taking values in the pre-ordered set with the set criterion and of solutions for vector optimization problems whose image space is a real vector space not necessarily endowed with a topology.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alonso, M., Rodríguez-Marín, L.: Set-relations and optimality conditions in set-valued maps. Nonlinear Anal. 63, 1167–1179 (2005)
Browder, F.E.: Remarks on the direct method of the calculus of variations. Arch. Ration. Mech. Anal. 20, 251–258 (1965)
Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis. Lectures Notes in Economics and Mathematical Systems, vol. 514. Springer, Berlin (2005)
Chen, Y.Q., Cho, Y.J., Yang, L.: Note on the results with lower semi-continuity. Bull. Korean Math. Soc. 39, 535–541 (2002)
Chen, Y.Q., Cho, Y.J., Kim, J.K.: Minimization of vector valued convex functions. J. Nonlinear Convex Anal. 16, 2053–2058 (2015)
Chen, Y.Q., Zhang, C.L.: On the weak semi-continuity of vector functions and minimum problems. J. Optim. Theory Appl. 178, 119–130 (2018)
Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht (2000)
Han, Y., Huang, N.J.: Continuity and convexity of a nonlinear scalarizing function in set optimization problems with applications. J. Optim. Theory Appl. 177, 679–695 (2018)
Han, Y., Huang, N.J.: Well-posedness and stability of solutions for set optimization problems. Optimization 66, 17–33 (2017)
Han, Y., Huang, N.J.: Existence and stability of solutions for a class of generalized vector equilibrium problems. Positivity 20, 829–846 (2016)
Han, Y., Wang, S.H., Huang, N.J.: Arcwise connectedness of the solution sets for set optimization problems. Oper. Res. Lett. 47, 168–172 (2019)
Hernández, E., Rodríguez-Marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67, 1726–1736 (2007)
Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)
Hernández, E., Rodríguez-Marín, L., Sama, M.: Some equivalent problems in set optimization. Oper. Res. Lett. 37, 61–64 (2009)
Hernández, E., Rodríguez-Marín, L., Sama, M.: On solutions of set-valued optimization problems. Comput. Math. Appl. 60, 1401–1408 (2010)
Jahn, J.: Vectorization in set optimization. J. Optim. Theory Appl. 167, 783–795 (2015)
Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)
Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization. Springer, New York (2015)
Kuroiwa, D.: The natural criteria in set-valued optimization. RIMS Kokyuroku 1031, 85–90 (1998)
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)
Kuroiwa, D.: Existence of efficent points of set optimization with weighted criteria. J. Nonlinear Convex Anal. 4, 117–123 (2003)
Khanh, P.Q., Quy, D.N.: On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings. J. Glob. Optim. 49, 381–396 (2011)
Lalitha, C.S.: A unified minimal solution in set optimization. J. Glob. Optim. 74, 195–211 (2019)
Long, X.J., Peng, J.W., Peng, Z.Y.: Scalarization and pointwise well-posedness for set optimization problems. J. Glob. Optim. 62, 763–773 (2015)
Luc, D.T.: Theory of Vector Optimization. Lectures Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)
Qiu, J.H.: On Ha’s version of set-valued Ekeland’s variational principle. Acta Math. Sin. Engl. Ser. 28, 717–726 (2012)
Shi, D.S., Ling, C.: Minimax theorem and cone saddle points of uniformly same-order vector-valued functions. J. Optim. Theory Appl. 84, 575–587 (1995)
Xu, Y.D., Li, S.J.: On the solution continuity of parametric set optimization problems. Math. Methods Oper. Res. 84, 223–237 (2016)
Zhang, C.L., Huang, N.J.: On the stability of minimal solutions for parametric set optimization problems. Appl. Anal. (2019). https://doi.org/10.1080/00036811.2019.1652733
Zhang, C.L., Zhou, L.W., Huang, N.J.: Stability of minimal solution mappings for parametric set optimization problems with pre-order relations. Pac. J. Optim. 15, 491–504 (2019)
Acknowledgements
The authors would like to thank Professor Franco Giannessi for his helpful comments and suggestions. We also thank the editor and reviewers for their constructive comments, which helps us to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the National Natural Science Foundation of China (11471230, 11671282)
Rights and permissions
About this article
Cite this article
Zhang, Cl., Huang, Nj. Set optimization problems of generalized semi-continuous set-valued maps with applications. Positivity 25, 353–367 (2021). https://doi.org/10.1007/s11117-020-00766-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-020-00766-6