Abstract
In this study, proper minimal definition is given through an expanded cone to preserve the glazing structure for the solution of set-valued optimization problems. Thanks to this specially created expanded cone, some definitions of minimal described by the order relation ≼_(m_1 ) are reminded. It has been emphasized that the definition of minimalism given according to the order relation ≼_(m_1 ) can be found depending on the existence of an extended C cone such that K/ {0}⊂ intC. Then, m_1-maximal definition was made using the m_1-minimal definition. Additionally, it has been seen that the m_1-maximal definition is equivalent to the maximal definition defined for vectors. Finally, the connection between m_1-proper minimal and m_1-minimal is explained with examples to better understand.
References
- Henig MI, Proper efficiency with respect to cones. Journal of Optimization Theory and Applicaiton, 1982;36:387-407.
- Benson HP, (2007). Matthias Ehrgott, Multicriteria Optimization, Springer (2005) ISBN 3-540-21398-8 323 pages. European Journal of Operational Research, 176(3):1961-1964.
- Geoffrion AM, Proper efficiency and the theory of vector maximization. Journal of mathematical analysis and applications, 1968;22(3):618-630.
- Guerraggıo A, Molho E, Zaffaronı A, On the notion of proper efficiency in vector optimization. Journal of Optimization Theory and Applications, 1994;82:1-21.
- Hamel AH, Heyde F, Duality for set-valued measures of risk. SIAM Journal on Financial Mathematics, 2010;1(1):66-95.
- Maeda T, On characterization of Nash equilibrium strategy in bi-matrix games with set payoffs. In: Set Optimization and Applications-The State of the Art: From Set Relations to Set-Valued Risk Measures. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015;313-331.
- Kreps DM, A representation theorem for preference for flexibility. Econometrica: Journal of the Econometric Society, 1979;565-577.
- Young RC, The algebra of many-valued quantities. Mathematische Annalen, 1931;104(1):260-290.
- Kuhn HW, Tucker AW, Nonlinear programming. In: Traces and emergence of nonlinear programming. Basel: Springer Basel, 2013;247-258.
- Kuroiwa D, Some Criteria in Set-Valued Optimization (Nonlinear Analysis and Convex Analysis). 数理解析研究所講究録, 1997;985:171-176.
- Huerga L, Jıménez B, Novo V, New notions of proper efficiency in set optimization with the set criterion. Journal of Optimization Theory and Applications, 2022;195(3):878-902.
- Karaman E, Soyertem M, Atasever Güvenç İ, Tozkan D, Küçük M and Küçük Y, Partial order relations on family of sets and scalarizations for set optimization, Positivity, vol. 22 (3), 2018, pp. 783-802.
- Huerga L, et al. On proper minimality in set optimization. Optimization Letters, 2023; 1-16.
- Jahn J (Ed.), (2009). Vector optimization (pp. 2327-4697). Berlin: Springer.
Abstract
Bu çalışmada küme değerli optimizasyon problemlerinin çözümü için sırlama yapısını koruyacak şekilde genişletilmiş bir koni üzerinden has minimallik tanımı verilmiştir. Özel olarak oluşturulmuş olan, bu genişletilmiş koni sayesinde ≼_(m_1 ) sıralama bağıntısı ile tarif edilen bazı minimallik tanımları hatırlatılmıştır. ≼_(m_1 ) sıralama bağıntısına göre verilmiş olan minimallik tanımını K/ {0}⊂ intC olacak şekilde bir genişletilmiş C konisinin varlığına bağlı olarak bulunabileceği üzerine durulmuştur. Daha sonra ise, m_1-minimallik tanımı kullanılarak m_1-maksimal tanımı yapılmıştır. Ayrıca, m_1-maksimal tanımının vektörler için tanımlanan maksimal tanımına denk olduğu görülmüştür. Son olarak, m_1-hasminimallik ile m_1-minimallik arasında bağlantıyı daha iyi anlayabilmek için örneklerle açıklanmıştır.
References
- Henig MI, Proper efficiency with respect to cones. Journal of Optimization Theory and Applicaiton, 1982;36:387-407.
- Benson HP, (2007). Matthias Ehrgott, Multicriteria Optimization, Springer (2005) ISBN 3-540-21398-8 323 pages. European Journal of Operational Research, 176(3):1961-1964.
- Geoffrion AM, Proper efficiency and the theory of vector maximization. Journal of mathematical analysis and applications, 1968;22(3):618-630.
- Guerraggıo A, Molho E, Zaffaronı A, On the notion of proper efficiency in vector optimization. Journal of Optimization Theory and Applications, 1994;82:1-21.
- Hamel AH, Heyde F, Duality for set-valued measures of risk. SIAM Journal on Financial Mathematics, 2010;1(1):66-95.
- Maeda T, On characterization of Nash equilibrium strategy in bi-matrix games with set payoffs. In: Set Optimization and Applications-The State of the Art: From Set Relations to Set-Valued Risk Measures. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015;313-331.
- Kreps DM, A representation theorem for preference for flexibility. Econometrica: Journal of the Econometric Society, 1979;565-577.
- Young RC, The algebra of many-valued quantities. Mathematische Annalen, 1931;104(1):260-290.
- Kuhn HW, Tucker AW, Nonlinear programming. In: Traces and emergence of nonlinear programming. Basel: Springer Basel, 2013;247-258.
- Kuroiwa D, Some Criteria in Set-Valued Optimization (Nonlinear Analysis and Convex Analysis). 数理解析研究所講究録, 1997;985:171-176.
- Huerga L, Jıménez B, Novo V, New notions of proper efficiency in set optimization with the set criterion. Journal of Optimization Theory and Applications, 2022;195(3):878-902.
- Karaman E, Soyertem M, Atasever Güvenç İ, Tozkan D, Küçük M and Küçük Y, Partial order relations on family of sets and scalarizations for set optimization, Positivity, vol. 22 (3), 2018, pp. 783-802.
- Huerga L, et al. On proper minimality in set optimization. Optimization Letters, 2023; 1-16.
- Jahn J (Ed.), (2009). Vector optimization (pp. 2327-4697). Berlin: Springer.
Details
| Primary Language | Turkish |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Research Article |
| Authors | |
| Publication Date | June 30, 2024 |
| Submission Date | January 25, 2024 |
| Acceptance Date | May 3, 2024 |
| Published in Issue | Year 2024 Volume: 8 Issue: 1 |