Optimality Conditions of Given Set-valued Optimization Problems with Respect to Set Optimization by Using Embedding Function
Abstract
In
this current study, set-valued optimization problem is considered. There are
some criteria to obtain solutions of this set-valued optimization problem. The two
most commonly used criteria are set and vector approaches in the literature. In
this work, we investigated the solutions of set-valued optimization problems
with respect to set approach. Many methods such as scalarization, vectorization
and directional derivative are used to find the optimality conditions of
set-valued optimization problems with respect to set approach. Apart from these
methods, we used embedding function to obtain optimality conditions in this
study. An embedding space is used in order to obtain optimality conditions by
using embedding function. Some properties of this space and embedding function
are studied. Moreover, an example is given to understand more better of the
study.
References
- A. Chinchuluun, P.M. Pardalos, A. Migdalas, L. Pitsoulis, Pareto Optimality, Game Theory and Equilibria. New York, USA: Springer-Verlag, 2008.
- A.A. Khan, C. Tammer, C. Zalinescu, Set-Valued Optimization: An Introduction with Applications. Berlin: Springer-Verlag, 2015.
- E. Karaman, M. Soyertem, İ. Atasever Güvenç, D. Tozkan, M. Küçük and Y. Küçük, “A Vectorization for nonconvex set-valued optimization,” Turk. J. Math., vol. 42, 2018, pp. 1815-1832.
- E. Karaman, M. Soyertem, İ. Atasever Güvenç, D. Tozkan, M. Küçük and Y. Küçük, “Partial order relations on family of sets and scalarizations for set optimization,” Positivity, vol. 22 (3), 2018, pp. 783-802.
- E. Hernandez and L. Rodriguez-Marin, “Nonconvex scalarization in set optimization with set-valued maps,” J. Math. Anal. Appl., vol. 325, 2007, pp. 1-18.
- D. Kuroiwa, T. Tanaka and T.X.D. Ha, “On cone convexity of set-valued maps,” Nonlinear Anal-Theor., vol. 30(3), 1997, pp. 1487-1496.
- D. Kuroiwa, “The natural criteria in set-valued optimization,” RIMS Kokyuroku, vol. 1031, 1998, pp. 85-90.
- D. Kuroiwa, “On set-valued optimization,” Nonlinear Anal-Theor., vol. 47 (2), 2001, pp. 1395-1400.
- J. Jahn, T.X.D. Ha, “New order relations in set optimization,” J. Optimiz. Theory. App., vol. 148, 2011, pp. 209-236.
- M. Pilecka, “Optimality conditions in set-valued programming using the set criterion,” Thecnical University of Freiberg, vol. 2014 (2), 2014.
- D. Kuroiwa, “On derivative of set-valued maps in set optimization,” Kyoto University Research Information Repository, vol. 1611, 2008, pp. 51-55.
- D. Kuroiwa, “Canonical type DC set optimization,” in Proc. 3th Asian Conference on Nonlinear Analysis and Optimization, Matseu, 2012, pp. 197-204.
- D. Kuroiwa, “Generalized minimality in set optimization,” Set optimization and applications – the state of the art: from set relations to set-valued risk measures, in mathematics & statistics, A.H. Hamel, F. Heyde, A. Löhne, B. Rudloff, C. Schrage, Ed. Berlin: Springer proceedings, vol. 151, 2015, pp. 293–311.
- D. Kuroiwa, “Some duality theorems of set-valued optimization,” RIMS Kokyuroku, vol. 1079, 1999, pp. 15–19.
- D. Pallaschke, R. Urbanski, Pairs of Compact Convex Sets. Dordrecht: Kluwer academic publishers, 2002.
- R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Encyclopedia Math. Appl., 1993.
Gömme Fonksiyonu Kullanılarak Küme Optimizasyonuna Göre Verilen Küme Değerli Optimizasyon Problemlerinin Optimallik Koşulları
Abstract
Bu çalışmada küme
değerli optimizasyon problemi ele alınmıştır. Küme değerli optimizasyon
problemlerinin çözümlerini bulmak için kullanılan bazı kriterler vardır.
Literatürde en çok kullanılan kriterler küme yaklaşımı ve vektör yaklaşımıdır.
Bu çalışmada ise küme değerli optimizasyon problemlerinin küme yaklaşımına göre
çözümleri araştırıldı. Küme değerli optimizasyon problemlerinin küme
yaklaşımına göre optimallik koşullarını elde etmek için skalerizasyon,
vektörizasyon ve yönlü türev gibi yöntemler kullanılır. Bu yöntemlerden farklı
olarak çalışmada problemin optimallik koşullarını elde etmek için gömme
fonksiyonu kullanılmıştır. Gömme fonksiyonu ile optimallik koşullarını elde
edebilmek için bir gömme uzayı kullanılmıştır. Bu uzayın ve gömme fonksiyonunun
bazı özellikleri incelenmiştir. Bunlara ek olarak çalışmanın daha iyi
anlaşılabilmesi için bir örnek verilmiştir.
References
- A. Chinchuluun, P.M. Pardalos, A. Migdalas, L. Pitsoulis, Pareto Optimality, Game Theory and Equilibria. New York, USA: Springer-Verlag, 2008.
- A.A. Khan, C. Tammer, C. Zalinescu, Set-Valued Optimization: An Introduction with Applications. Berlin: Springer-Verlag, 2015.
- E. Karaman, M. Soyertem, İ. Atasever Güvenç, D. Tozkan, M. Küçük and Y. Küçük, “A Vectorization for nonconvex set-valued optimization,” Turk. J. Math., vol. 42, 2018, pp. 1815-1832.
- E. Karaman, M. Soyertem, İ. Atasever Güvenç, D. Tozkan, M. Küçük and Y. Küçük, “Partial order relations on family of sets and scalarizations for set optimization,” Positivity, vol. 22 (3), 2018, pp. 783-802.
- E. Hernandez and L. Rodriguez-Marin, “Nonconvex scalarization in set optimization with set-valued maps,” J. Math. Anal. Appl., vol. 325, 2007, pp. 1-18.
- D. Kuroiwa, T. Tanaka and T.X.D. Ha, “On cone convexity of set-valued maps,” Nonlinear Anal-Theor., vol. 30(3), 1997, pp. 1487-1496.
- D. Kuroiwa, “The natural criteria in set-valued optimization,” RIMS Kokyuroku, vol. 1031, 1998, pp. 85-90.
- D. Kuroiwa, “On set-valued optimization,” Nonlinear Anal-Theor., vol. 47 (2), 2001, pp. 1395-1400.
- J. Jahn, T.X.D. Ha, “New order relations in set optimization,” J. Optimiz. Theory. App., vol. 148, 2011, pp. 209-236.
- M. Pilecka, “Optimality conditions in set-valued programming using the set criterion,” Thecnical University of Freiberg, vol. 2014 (2), 2014.
- D. Kuroiwa, “On derivative of set-valued maps in set optimization,” Kyoto University Research Information Repository, vol. 1611, 2008, pp. 51-55.
- D. Kuroiwa, “Canonical type DC set optimization,” in Proc. 3th Asian Conference on Nonlinear Analysis and Optimization, Matseu, 2012, pp. 197-204.
- D. Kuroiwa, “Generalized minimality in set optimization,” Set optimization and applications – the state of the art: from set relations to set-valued risk measures, in mathematics & statistics, A.H. Hamel, F. Heyde, A. Löhne, B. Rudloff, C. Schrage, Ed. Berlin: Springer proceedings, vol. 151, 2015, pp. 293–311.
- D. Kuroiwa, “Some duality theorems of set-valued optimization,” RIMS Kokyuroku, vol. 1079, 1999, pp. 15–19.
- D. Pallaschke, R. Urbanski, Pairs of Compact Convex Sets. Dordrecht: Kluwer academic publishers, 2002.
- R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Encyclopedia Math. Appl., 1993.
Details
| Primary Language | Turkish |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Makaleler |
| Authors | |
| Publication Date | May 31, 2019 |
| Published in Issue | Year 2019 Volume: 14 Issue: 1 |
