Abstract
This paper combines variable ordering structures with set relations in set optimization. We continue our examinations of set optimization problems and use the optimality notions as well as the set relations introduced in the paper (Eichfelder and Pilecka in Set Approach for Set Optimization with Variable Ordering Structures Part I: Set Relations and Relationship to Vector Approach, 2016). We present linear scalarization results and a new nonlinear scalarization approach for set relations. Moreover, we introduce some additional set relations which allow us to investigate connections between our results, based on the set approach, and others presented in the literature which are based on the vector approach. This gives us the possibility to apply the optimality conditions derived there to our concepts.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Eichfelder, G., Pilecka, M.: Set approach for set optimization with variable ordering structures part I: set relations and relationship to vector approach. J. Optim. Theory Appl. (2016). doi:10.1007/s10957-016-0992-0
Jahn, J.: Vectorization in set optimization. J. Optim. Theory Appl. 167, 783–795 (2015)
Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)
Gutiérrez, C., Jiménez, B., Miglierina, E., Molho, E.: Scalarization in set optimization with solid and nonsolid ordering cones. J. Glob. Optim. 61, 525–552 (2015)
Jahn, J.: Bishop–Phelps cones in optimization. Int. J. Optim.: Theory Methods Appl. 1, 123–139 (2009)
Petschke, M.: On a theorem of Arrow, Barankin, and Blackwell. SIAM J. Control Optim. 28, 395–401 (1990)
Eichfelder, G.: Variable Ordering Structures in Vector Optimization. Springer, Heidelberg (2014)
Eichfelder, G., Ha, T.X.D.: Optimality conditions for vector optimization problems with variable ordering structures. Optimization 62, 597–627 (2013)
Engau, A.: Variable preference modeling with ideal-symmetric convex cones. J. Glob. Optim. 42, 295–311 (2008)
Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal. 30, 1487–1496 (1997)
Durea, M., Strugariu, R., Tammer, C.: On set-valued optimization problems with variable ordering structure. J. Glob. Optim. 61, 745–767 (2015)
Jahn, J.: Vector Optimization. Springer, Berlin (2004)
Eichfelder, G.: Optimal elements in vector optimization with a variable ordering structure. J. Optim. Theory Appl. 151, 217–240 (2011)
Kosmol, P., Müller-Wichards, D.: Optimization in Function Spaces with Stability Considerations in Orlicz Spaces. De Gruyter, Berlin (2011)
Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Birkhäuser, Basel (1983)
Jahn, J.: A derivative-free descent method in set optimization. Comput. Optim. Appl. 60, 393–411 (2015)
Zheng, X.Y.: Pareto solutions of polyhedral-valued vector optimization problems in banach spaces. Set-Valued Anal. 17, 389–408 (2009)
Köbis, E., Köbis, M.: Treatment of Set Order Relations by Means of a Nonlinear Scalarization Functional: A Full Characterization. Report No. 05, University of Halle-Wittenberg (2015)
Eichfelder, G.: Numerical procedures in multiobjective optimization with variable ordering structures. J. Optim. Theory Appl. 162, 489–514 (2014)
Rodríguez-Marín, L., Sama, M.: \((\Lambda, C)\)-contingent derivatives of set-valued maps. Math. Anal. Appl. 335, 974–989 (2007)
Kuroiwa, D.: On derivatives of set-valued maps and optimality conditions for set optimization. J. Nonlinear Convex Anal. 10, 41–50 (2009)
Jahn, J.: Directional derivatives in set optimization with the set less order relation. Taiwan. J. Math. 19, 737–757 (2015)
Pilecka, M.: Set-Valued Optimization and Its Application to Bilevel Optimization. PhD Thesis, TU Bergakademie Freiberg (2016)
Khan, A., Tammer, C., Zǎlinescu, C.: Set-Valued Optimization—An Introduction with Applications. Springer, Heidelberg (2015)
Acknowledgments
The authors are very grateful to the anonymous referees and the editors for valuable suggestions and comments on both parts of the paper that significantly helped to improve the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Nguyen Dong Yen.
Rights and permissions
About this article
Cite this article
Eichfelder, G., Pilecka, M. Set Approach for Set Optimization with Variable Ordering Structures Part II: Scalarization Approaches. J Optim Theory Appl 171, 947–963 (2016). https://doi.org/10.1007/s10957-016-0993-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-0993-z