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

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., Volume 24, Number 3 (2020), 709-722.

Dates
Received: 30 October 2018
Revised: 22 February 2019
Accepted: 17 June 2019
First available in Project Euclid: 19 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1589875226

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. 24 (2020), no. 3, 709--722. doi:10.11650/tjm/190604. https://projecteuclid.org/euclid.twjm/1589875226


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.