Abstract
This paper is devoted to the study of scalarization and Ekeland’s variational principle for a partial set order relation, which is defined by Minkowski difference. Firstly, a kind of scalarization function with the properties of order representing and order preserving is introduced, and it is used to establish an unified scheme for the minimal element of set-valued maps; As two special cases of the unified scheme, an Oriented distance-type function and a Gerstewitz-type function are provided to illustrate the effectiveness of the scheme. Secondly, an Ekeland’s variational principle related to partial set order relation is formulated under the condition of dynamical closeness. Finally, the obtained results are applied to examine the optimality and existence for an uncertain multi-objective optimization problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data availability is not applicable to this article as no date were created or analyzed in this study.
References
Ansari QH, Sharma PK (2019) Ekeland type variational principle for set-valued maps in quasi-metric spaces with applications. J Nonlinear Convex Anal 20(8):1683–1700
Ansari QH, Sharma PK (2021a) Set order relations, set optimization, and ekeland’s variational principle, in optimization, variational analysis and applications. In: Laha V, Maréchal P, Mishra SK (eds) Springer Proceedings in Mathematics and Statistics, vol 355. Springer Nature Singapore Pvt. Ltd., pp 103–165
Ansari QH, Sharma PK (2021b) Levitin-Polyak well-posedness for set optimization problems. J Nonlinear Convex Anal 22(7):1353–1371
Ansari QH, Sharma PK, Yao JC (2018) Minimal element theorems and Ekeland’s variational principle with new set order relations. J Nonlinear Convex Anal 19:1127–1139
Ansari QH, Hamel AH, Sharma PK (2020) Ekeland’s variational principle with weighted set order relations. Math Methods Oper Res 91:117–136
Elisa C, Lorenzo CB, Elena M (2022) Scalarization and robustness in uncertain vector optimization prob1ems a non componentwise approach. J Global Optim 343(1):1–26
Gupta M, Srivastava M (2020) Approximate solutions and Levitin-Polyak well-posedness for set optimization using weak efficiency. J Optim Theory Appl 186(1):191–208
Gutiérrez C, Jiménez B, Miglierina E, Molho E (2015a) Scalarization in set optimization with solid and nonsolid ordering cones. J Global Optim 61(3):525–552
Gutiérrez C, Jiménez B, Novo V (2015b) Nonlinear scalarizations of set optimization problems with set orderings. In: Set optimization and applications-the state of the art. Springer, Berlin 764(6):43-63
Gutiérrez C, Huerga L, Köbis E, Tammer C (2021) A scalarization scheme for binary relations with applications to set-valued and robust optimization. J Global Optim 79:233–256
Hamel A, Löhne A (2006) Minimal element theorems and Ekeland’s principle with set relations. J Nonlinear Convex Anal 7(1):19–37
Han WY, Yu GL (2023a) Directional derivatives in set optimization with the set order defined by Minkowski difference. Optimization. https://doi.org/10.1080/02331934.2023.2252441
Han WY, Yu GL (2023b) Optimality and error bound for set optimization with application to uncertain multi-objective programming. J Glob Optim. https://doi.org/10.1007/s10898-023-01327-3
Han WY, Yu GL (2023c) Ekeland’s variational principle with a scalarization type weighted set order relation. J Nonlinear Var Anal 7:381–396
Jahn J, Ha TXD (2011) New order relations in set optimization. J Optim Theory Appl 148(2):209–236
Karaman E, Soyertem M, Güvenc ÏT, Tozkan D (2018) Partial order relations on family of sets and scalarizations for set optimization. Positivity 22(1):783–802
Khushboo Lalitha CS (2018) Scalarizations for a unified vector optimization problem based on order representing and order preserving properties. J Global Optim 70:903–916
Khushboo Lalitha CS (2019) Scalarizations for a set optimization problem using generalized oriented distance function. Positivity 23(1):1195–1213
Kuroiwa D (2001) On set-valued optimization. Nonlinear Anal Theory Methods Appl 47:1395–400
Qiu JH (2013) Set-valued quasi-metrics and a general Ekeland’s variational principle in vector optimization. SIAM J Control Optim 51:1350–1371
Sharma PK (2022) Some topological properties of solution sets in partially ordered set optimization. J Appl Numer Optim 5(1):163–180
Zaffaroni A (2003) Degrees of efficiency and degrees of minimality. SIAM J Control Optim 42:1071–1086
Zhang CL, Huang NJ (2021) Set relations and weak minimal solutions for nonconvex set optimization problems with applications. J Optim Theory Appl 190:894–914
Acknowledgements
This research was supported by Natural Science Foundation of China under Grant (No. 12361062 and 62366001) and Natural Science Foundation of Ningxia Provincial of China (No.2023 AAC02053).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Carlos Conca.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Han, W., Yu, G. An unified scalarization and Ekeland’s variational principle for partial set order. Comp. Appl. Math. 43, 45 (2024). https://doi.org/10.1007/s40314-023-02567-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-023-02567-5