Taiwanese Journal of Mathematics

Optimality Conditions in Set-valued Optimization Problem with Respect to a Partial Order Relation via Directional Derivative

Emrah Karaman, Mustafa Soyertem, and İlknur Atasever Güvenç

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

In this study, a new directional derivative is defined by using Minkowski difference. Some properties and existence theorems of this directional derivative are given. Moreover, necessary and sufficient optimality conditions are presented for set-valued optimization problems with respect to $m_1$ order relation via directional derivative.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 14 pages.

Dates
First available in Project Euclid: 10 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1562724021

Digital Object Identifier
doi:10.11650/tjm/190604

Subjects
Primary: 80M50: Optimization 90C26: Nonconvex programming, global optimization

Keywords
set-valued optimization partial order directional derivative optimality conditions

Citation

Karaman, Emrah; Soyertem, Mustafa; Güvenç, İlknur Atasever. Optimality Conditions in Set-valued Optimization Problem with Respect to a Partial Order Relation via Directional Derivative. Taiwanese J. Math., advance publication, 10 July 2019. doi:10.11650/tjm/190604. https://projecteuclid.org/euclid.twjm/1562724021


Export citation

References

  • L. Altangerel, R. I. Boţ and G. Wanka, Conjugate duality in vector optimization and some applications to the vector variational inequality, J. Math. Anal. Appl. 329 (2007), no. 2, 1010–1035.
  • Q. H. Ansari, P. K. Sharma and J.-C. Yao, Minimal element theorems and Ekeland's variational principle with new set order relations, J. Nonlinear Convex Anal. 19 (2018), no. 7, 1127–1139.
  • Q. H. Ansari and J.-C. Yao, Recent Developments in Vector Optimization, Vector Optimization, Springer-Verlag, Berlin, 2012.
  • J.-P. Aubin and A. Cellina, Differential Inclusions: Set-valued maps and viability theory, Grundlehren der Mathematischen Wissenschaften 264, Springer-Verlag, Berlin, 1984.
  • G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (1998), no. 2, 187–200.
  • S. Dempe and M. Pilecka, Optimality conditions for set-valued optimisation problems using a modified Demyanov difference, J. Optim. Theory Appl. 171 (2016), no. 2, 402–421.
  • A. H. Hamel and F. Heyde, Duality for set-valued measures of risk, SIAM J. Financial Math. 1 (2010), no. 1, 66–95.
  • E. Hernández and L. Rodríguez-Marín, Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl. 325 (2007), no. 1, 1–18.
  • J. Jahn, Vector Optimization: Theory, applications, and extensions, Springer-Verlag, Berlin, 2004.
  • ––––, Vectorization in set optimization, J. Optim. Theory Appl. 167 (2013), no. 3, 783–795.
  • ––––, Directional derivatives in set optimization with the set less order relation, Taiwanese J. Math. 19 (2015), no. 3, 737–757.
  • J. Jahn and T. X. D. Ha, New order relations in set optimization, J. Optim. Theory Appl. 148 (2011), no. 2, 209–236.
  • E. Karaman, \.I. Atasever Güvenç, M. Soyertem, D. Tozkan, M. Küçük and Y. Küçük, A vectorization for nonconvex set-valued optimization, Turkish J. Math. 42 (2018), no. 4, 1815–1832.
  • E. Karaman, M. Soyertem, \.I. Atasever Güvenç, D. Tozkan, M. Küçük and Y. Küçük, Partial order relations on family of sets and scalarizations for set optimization, Positivity 22 (2018), no. 3, 783–802.
  • A. A. Khan, C. Tammer and C. Zălinescu, Set-valued Optimization: An introduction with applications, Vector Optimization, Springer-Verlag, Heidelberg, 2015.
  • E. Klein and A. C. Thompson, Theory of Correspondences: Including applications to mathematical economics, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1984.
  • Y. Küçük, \.I. Atasever Güvenç and M. Küçük, Weak conjugate duality for nonconvex vector optimization, Pac. J. Optim. 13 (2017), no. 1, 75–103.
  • D. Kuroiwa, The natural criteria in set-valued optimization, Sūrikaisekikenkyūsho Kōkyūroku 1031 (1998), 85–90.
  • ––––, On set-valued optimization, Nonlinear Anal. 47 (2001), no. 2, 1395–1400.
  • ––––, Existence theorems of set optimization with set-valued maps, J. Inf. Optim. Sci. 24 (2003), no. 1, 73–84.
  • D. Kuroiwa, T. Tanaka and T. X. D. Ha, On cone convexity of set-valued maps, Nonlinear Anal. 30 (1997), no. 3, 1487–1496.
  • A. Löhne and C. Tammer, A new approach to duality in vector optimization, Optimization 56 (2007), no. 1-2, 221–239.
  • D. T. L\duc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems 319, Springer-Verlag, Berlin, 1989.
  • N. Neukel, Order relations of sets and its application in socio-economics, Appl. Math. Sci. (Ruse) 7 (2013), no. 115, 5711–5739.
  • D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets: Fractional arithmetic with convex sets, Mathematics and its Applications 548, Kluwer Academic Publishers, Dordrecht, 2002.
  • M. Pilecka, Optimality conditions in set-valued programming using the set criterion, Technical University of Freiberg, Preprint, 2014.
  • E. Polak, Optimization: Algorithms and consistent approximations, Applied Mathematical Sciences 124, Springer-Verlag, New York, 1997.
  • T. Tanino, Conjugate duality in vector optimization, J. Math. Anal. Appl. 167 (1992), no. 1, 84–97.