Abstract
We investigate, in the framework of set optimization, some issues that are well studied in vectorial setting, that is, penalization procedures, properness of solutions and optimality conditions on primal spaces. Therefore, with this study we aim at completing the literature dedicated to set optimization with some results that have well established correspondence in the classical vector optimization.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ansari QH, Hamel AH, Sharma PK (2020) Ekeland’s variational principle with weighted set order relations. Math Methods Oper Res 91:117–136
Aubin JP, Frankowska H (1990) Set-valued analysis. Birkhäuser, Basel
Burlică M, Durea M, Strugariu R (2023) New concepts of directional derivatives for set-valued maps and applications to set optimization. Optimization 72:1069–1091
Chelmuş T, Durea M (2020) Exact penalization and optimality conditions for constrained directional Pareto efficiency, Pure and Applied Functional. Analysis 5:533–553
Chelmuş T, Durea M, Florea E-A (2019) Directional Pareto efficiency: concepts and optimality conditions. J Optim Theory Appl 182:336–365
Durea M, Florea E-A, Maxim D-E, Strugariu R (2022) Approximate efficiency in set-valued optimization with variable order. J Nonlinear Var Anal 6:619–640
Durea M, Strugariu R (2023) Directional derivatives and subdifferentials for set-valued maps applied to set optimization. J Global Optim 85:687–707
Göpfert A, Riahi H, Tammer C, Zălinescu C (2003) Variational Methods in Partially Ordered Spaces. Springer, Berlin
Gutiérrez C, López R, Novo V (2010) Generalized \(\varepsilon \)-quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions. Nonlinear Anal 72:4331–4346
Gutiérrez C, López R, Martínez J (2022) Generalized \(\varepsilon \)-quasi solutions of set optimization problems. J Global Optim 82:559–576
Ha TXD (2020) A new concept of slope for set-valued maps and applications in set optimization studied with Kuroiwa’s set approach. Math Methods Oper Res 91:137–158
Hamel AH, Heyde F, Löhne A, Rudloff B, Schrage C (eds) (2015) Set optimization and applications—the state of the art. Springer, Berlin
Han Y, Zhang K, Huang N-J (2020) The stability and extended well-posedness of the solution sets for set optimization problems via the Painlevé–Kuratowski convergence. Math Methods Oper Res 91:175–196
Hernández E, López R (2019) About asymptotic analysis and set optimization. Set Valued Var Anal 27:643–664
Hernández E, Rodríguez-Marín L (2007) Nonconvex scalarization in set optimization with set-valued maps. J Math Anal Appl 325:1–18
Huerga L, Jiménez B, Novo V, Vílchez A (2021) Six set scalarizations based on the oriented distance: continuity, convexity and application to convex set optimization. Math Methods Oper Res 93:413–436
Ioffe AD (2017) Variational analysis of regular mappings theory and applications. Springer, Cham
Jiménez B, Novo V (2003) A notion of local proper efficiency in the Borwein sense in vector optimization. ANZIAM J 44:75–89
Jiménez B, Novo V (2003) First and second order sufficient conditions for strict minimality in nonsmooth vector optimization. J Math Anal Appl 284:496–510
Jiménez B, Novo V, Vílchez A (2020) Characterization of set relations through extensions of the oriented distance. Math Methods Oper Res 91:89–115
Khan AA, Tammer C, Zălinescu C (2015) Set-valued optimization. An introduction with applications. Springer, Berlin
Kruger AY, Luke DR, Thao NH (2017) About subtransversality of collections of sets. Set Valued Var Anal 25:701–729
Kuroiwa D, Tanaka T, Ha TXD (1997) On cone convexity of set-valued maps. Nonlinear Anal Theory Methods Appl 30:1487–1496
Kuroiwa D, Nuriya T (2006) A generalized embedding vector space in set optimization. In: Proceedings of the fourth international conference on nonlinear and convex analysis, pp 297–303
Loridan P (1982) Necessary conditions for \(\varepsilon \)-optimality. Math Program Stud 19:140–152
Luc DT (1989) Theory of vector optimization. Springer, Berlin
Luc DT, Penot J-P (2001) Convergence of asymptotic directions. Trans Am Math Soc 353:4095–4121
Qiu JH, He F (2020) Ekeland variational principles for set-valued functions with set perturbations. Optimization 69:925–960
Ye JJ (2012) The exact penalty principle. Nonlinear Anal Theory Methods Appl 75:1642–1654
Acknowledgements
The authors thank one anonymous referee for several constructive suggestions that improve the presentation of the paper.
Funding
This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS - UEFISCDI, project number PN-III-P4-PCE-2021-0690, within PNCDI III.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
conflict of interest
No potential conflict of interest was reported by the authors.
Additional information
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
Durea, M., Florea, EA. Directional and approximate efficiency in set optimization. Math Meth Oper Res 98, 435–459 (2023). https://doi.org/10.1007/s00186-023-00840-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-023-00840-1