A generalization of interval-valued optimization problems and optimality conditions by using scalarization and subdifferentials
Keywords:Interval-valued optimization problem, ordering cone, optimality condition, scalarization, subdifferential
In this work, interval-valued optimization problems are considered. Ordering cone is used in order to obtain a generalization of interval-valued optimization problems on real topological vector spaces. Some definitions and their properties are obtained for intervals, defined according to ordering cone. The Gerstewitz’s function is used to derive scalarization for interval-valued optimization problems. Also, two subdifferentials of interval-valued functions are given by using subgradients. Some important optimality conditions are introduced via subdifferentials. An example is given to demonstrate results.
Ansari, Q.H., Köbis, E. & Yao, J.C. (2018) Vector Variational Inequalities and Vector Optimization: Theory Applications. Springer, Berlin.
Bhurjee, A. & Panda, G. (2012) Efficient solution of interval optimization problem. Mathematical Methods of Operations Research, 76: 273-288.
Bhurjee, A.K. & Pandahan, S.K. (2016) Optimality conditions and duality results for non-differentiable interval optimization problems. Journal of Applied Mathematics and Computing, 50: 59-71.
Chalco-Cano, Y., Lodwick, W.A. & Rufian-Lizana A. (2013) Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative. Fuzzy Optimization and Decision Making, 12: 305-322.
Chen G.Y. & Jahn J. (1998) Optimality conditions for set-valued optimization problems. Mathematical Methods of Operations Research, 48: 187–200.
Costa, T.M., Chalco-Cano, Y., Lodwick, W.A. & Silva, G.N. (2015) Generalized interval vector spaces and interval optimization. Information Sciences, 311: 74-85.
Gerth C. & Weidner P. (1990) Nonconvex separation theorems and some applications in vector optimization. Journal of Optimization Theory and Applications, 67: 297-320.
Hernández E. & Rodríguez-Marín, L. (2011) Weak and strongly subgradients of set-valued maps. Journal of Optimization Theory and Applications, 149: 352–365.
Ishibuchi, H. & Tanaka, H. (1990) Multiobjective programming in optimization of the interval objective function. European Journal of Operational Research, 48: 219-225.
Karaman, E., Atasever Güvenç İ., Soyertem, M., Tozkan, D., Küçük, M. & Küçük, M. (2018) A vectorization for nonconvex set-valued optimization. Turkish Journal of Mathematics, 42: 1815-1832.
Karaman, E., Soyertem, M., Atasever Güvenç, I., Tozkan, D., Küçük, M. & Küçük, Y. (2017) Partial order relations on family of sets and scalarizations for set optimization. Positivity, 22: 783–802.
Karmakar, S., Mahato, S.K. & Bhunia, A.K. (2009) Interval oriented multi-section techniques for global optimization. Journal of Computational and Applied Mathematics, 224: 476–491
Khan, A.A., Tammer C. & Zălinescu, C. (2015) Set-valued optimization. Springer, Heidelberg.
Luc, D.T. (1989) Theory of vector optimization. Springer, Berlin.
Moore, R. (1966) Interval analysis, Englewood Cliffs. Prentice-Hall, New Jersey.