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Relations Among Minimal Elements of a Family of Sets with Respect to Various Set Order Relations

Year 2023, , 531 - 537, 29.09.2023

İlknur Atasever Güvenç

https://doi.org/10.17776/csj.1262453

Abstract

In this article, some set order relations given for set-optimization criterion which is one of the solution concept of set-valued optimization problems are considered. Minimal elements of a family of sets with respect to these set order relations are compared in detail. For this comparison relations between set order relations mentioned in this article are used. Also, in cases where a family of minimal sets is not a subset of the other one, examples are given.

Keywords

Set-valued optimization Set order relations Minimal solution

References

  • [1] Nishnianidze Z.G., Fixed Points Of Monotonic Multiple-Valued Operators, Bull Georgian Acad Sci., 114 (1984) 489–491.
  • [2] Young R.C.,The Algebra Of Many-Valued Quantities, Math Ann., 104 (1931) 260–290.
  • [3] Kuroiwa D., The Natural Criteria in Set-Valued Optimization. Research on Nonlinear Analysis and Convex Analysis, Surikaisekikenkyusho Kokyuroku, 1031 (1998) 85–90.
  • [4] Jahn J., Ha T.X.D., New Order Relations in Set Optimization, J. Optimiz Theory App., 148 (2003) 209–236.
  • [5] Karaman E., Soyertem M., Atasever Güvenç İ., Tozkan D., Küçük M., Küçük Y., Partial Order Relations on Family of Sets and Scalarizations for Set Optimization, Positivity, 22 (2018) 783-802.
  • [6] Jahn J., Characterizations of the Set Less Order Relation in Nonconvex Set Optimization, Journal of Optimization Theory and Applications 193 (2022) 523–544.
  • [7] Kuroiwa D., Some Criteria in Set-Valued Optimization, Investigations on Nonlinear Analysis and Convex Analysis. Surikaisekikenkyusho Kokyuroku; 985 (1998) 171–176.
  • [8] Kuroiwa D., Existence Theorems of Set Optimization with Set-Valued Maps, J. Inf. Optim., 24 (2003) 73–84.
  • [9] Kuroiwa D. On Set-Valued Optimization, Nonlinear Anal., 47 (2001) 1395–1400.
  • [10] Kuroiwa D., Existence of Efficient Points of Set Optimization with Weighted Criteria, J. Nonlinear Convex A., 4 (2003) 117–123.
  • [11] Khushboo, Lalitha C.S., Scalarizations for a Set Optimization Problem Using Generalized Oriented Distance Function. Positivity, 23 (2019) 1195–121.
  • [12] Hernández E., Rodríguez-Marín L., Nonconvex Scalarization in Set Optimization with Set-Valued Maps, J. Math. Anal. Appl. 325 (2007) 1–18.
  • [13] Karaman E., Güvenç İ.A., Soyertem M., Optimality Conditions in Set-Valued Optimization Problems with Respect to a Partial Order Relation by Using Subdifferentials, Optimization, 70(3) (2021) 613-650.
  • [14] Karaman E., Soyertem M., Güvenç İ.A., Optimality Conditions in Set-Valued Optimization Problem with Respect to a Partial Order Relation via Directional Derivative, Taiwan J. Math. 24(3) (2020) 709-722.
  • [15] Karaman E., Nonsmooth Set Variational Inequality Problems and Optimality Criteria for Set Optimization, Miskolc Math Notes, 21(1) (2020) 229-240.
  • [16] Khan A.A., Tammer C., Zălinescu C., Set-Valued Optimization: An Introduction with Applications. Berlin, (2015).
  • [17] Küçük M., Soyertem M., Küçük Y., On the Scalarization of Set-Valued Optimization Problems with Respect to Total Ordering Cones. In: Hu, B., Morasch, K., Pickl, S., Siegle, M., (Eds). Operations research proceedings, Heidelberg: Springer (2011) 347–352..
  • [18] Küçük M., Soyertem M., Küçük Y., Atasever İ., Vectorization of Set-Valued Maps with Respect to Total Ordering Cones and its Applications to Set-Valued Optimization Problems, J. Math. Anal. 385 (2012) 285–292.
  • [19] Ha T.X.D., A Unified Scheme for Scalarization in Set Optimization, Optimization, 55 (2021) 3603-3616.
  • [20] Truong Xuan Duc Ha, A unified scheme for scalarization in set optimization, Taylor & Francis Online, (2023)
  • [21] Atasever Güvenç İ., On Weak Minimal Solutions of Set Valued Optimization Problems and Comparison of Some Set Order Relations, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler, 11(1) (2023) 60-69.

Year 2023, , 531 - 537, 29.09.2023

İlknur Atasever Güvenç

https://doi.org/10.17776/csj.1262453

Abstract

References

  • [1] Nishnianidze Z.G., Fixed Points Of Monotonic Multiple-Valued Operators, Bull Georgian Acad Sci., 114 (1984) 489–491.
  • [2] Young R.C.,The Algebra Of Many-Valued Quantities, Math Ann., 104 (1931) 260–290.
  • [3] Kuroiwa D., The Natural Criteria in Set-Valued Optimization. Research on Nonlinear Analysis and Convex Analysis, Surikaisekikenkyusho Kokyuroku, 1031 (1998) 85–90.
  • [4] Jahn J., Ha T.X.D., New Order Relations in Set Optimization, J. Optimiz Theory App., 148 (2003) 209–236.
  • [5] Karaman E., Soyertem M., Atasever Güvenç İ., Tozkan D., Küçük M., Küçük Y., Partial Order Relations on Family of Sets and Scalarizations for Set Optimization, Positivity, 22 (2018) 783-802.
  • [6] Jahn J., Characterizations of the Set Less Order Relation in Nonconvex Set Optimization, Journal of Optimization Theory and Applications 193 (2022) 523–544.
  • [7] Kuroiwa D., Some Criteria in Set-Valued Optimization, Investigations on Nonlinear Analysis and Convex Analysis. Surikaisekikenkyusho Kokyuroku; 985 (1998) 171–176.
  • [8] Kuroiwa D., Existence Theorems of Set Optimization with Set-Valued Maps, J. Inf. Optim., 24 (2003) 73–84.
  • [9] Kuroiwa D. On Set-Valued Optimization, Nonlinear Anal., 47 (2001) 1395–1400.
  • [10] Kuroiwa D., Existence of Efficient Points of Set Optimization with Weighted Criteria, J. Nonlinear Convex A., 4 (2003) 117–123.
  • [11] Khushboo, Lalitha C.S., Scalarizations for a Set Optimization Problem Using Generalized Oriented Distance Function. Positivity, 23 (2019) 1195–121.
  • [12] Hernández E., Rodríguez-Marín L., Nonconvex Scalarization in Set Optimization with Set-Valued Maps, J. Math. Anal. Appl. 325 (2007) 1–18.
  • [13] Karaman E., Güvenç İ.A., Soyertem M., Optimality Conditions in Set-Valued Optimization Problems with Respect to a Partial Order Relation by Using Subdifferentials, Optimization, 70(3) (2021) 613-650.
  • [14] Karaman E., Soyertem M., Güvenç İ.A., Optimality Conditions in Set-Valued Optimization Problem with Respect to a Partial Order Relation via Directional Derivative, Taiwan J. Math. 24(3) (2020) 709-722.
  • [15] Karaman E., Nonsmooth Set Variational Inequality Problems and Optimality Criteria for Set Optimization, Miskolc Math Notes, 21(1) (2020) 229-240.
  • [16] Khan A.A., Tammer C., Zălinescu C., Set-Valued Optimization: An Introduction with Applications. Berlin, (2015).
  • [17] Küçük M., Soyertem M., Küçük Y., On the Scalarization of Set-Valued Optimization Problems with Respect to Total Ordering Cones. In: Hu, B., Morasch, K., Pickl, S., Siegle, M., (Eds). Operations research proceedings, Heidelberg: Springer (2011) 347–352..
  • [18] Küçük M., Soyertem M., Küçük Y., Atasever İ., Vectorization of Set-Valued Maps with Respect to Total Ordering Cones and its Applications to Set-Valued Optimization Problems, J. Math. Anal. 385 (2012) 285–292.
  • [19] Ha T.X.D., A Unified Scheme for Scalarization in Set Optimization, Optimization, 55 (2021) 3603-3616.
  • [20] Truong Xuan Duc Ha, A unified scheme for scalarization in set optimization, Taylor & Francis Online, (2023)
  • [21] Atasever Güvenç İ., On Weak Minimal Solutions of Set Valued Optimization Problems and Comparison of Some Set Order Relations, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler, 11(1) (2023) 60-69.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

İlknur Atasever Güvenç 0000-0003-0952-1024

Publication Date September 29, 2023
Submission Date March 9, 2023
Acceptance Date May 26, 2023
Published in Issue Year 2023

Cite

APA Atasever Güvenç, İ. (2023). Relations Among Minimal Elements of a Family of Sets with Respect to Various Set Order Relations. Cumhuriyet Science Journal, 44(3), 531-537. https://doi.org/10.17776/csj.1262453