Optimality Conditions of Interval-Valued Optimization Problems by Using Subdifferantials

Abstract

In this study, interval-valued optimization problems (shortly, interval optimization) are considered. In order to obtain optimality conditions of interval optimization, subdifferential and weak subdifferential are defined. After some properties including an existence condition of the subdifferentials are studied, optimality conditions including the necessary and sufficient conditions for interval optimization are obtained.

Keywords

• Optimization,Subdifferential,Optimality condition,Interval optimization

References

1. Hernandez, E. and Rodriguez-Marin, L., “Nonconvex scalarization in set optimization with set-valued maps”, J. Math. Anal. Appl., 325: 1-18, (2007).
2. 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, 22 (3): 783-802, (2018).
3. Xu Y. D. and Li S. J., “A new nonlinear scalarization function and applications”, Optim., 65: 207–231, (2016).
4. Küçük, M., Soyertem, M. and Küçük, Y., “The generalization of total ordering cones and vectorization to separable Hilbert spaces”, J. Math. Anal. Appl., 389: 1344-1351, (2012).
5. Küçük, M., Soyertem, M., Küçük, Y. and Atasever, İ., “Vectorization of set-valued maps with respect to total ordering cones and its applications to set-valued optimization problems”, J. Math. Anal. Appl., 385: 285-292, (2012).
6. Jahn, J., “Vectorization in set optimization”, J. Optimiz. Theory. App., 167: 783-795, (2013).
7. Karaman, E., Soyertem, M., Atasever Güvenç, İ., Tozkan, D., Küçük, M. and Küçük, Y., “A Vectorization for nonconvex set-valued optimization,” Turk. J. Math., 42: 1815-1832, (2018).
8. Jahn, J., “Directional derivatives in set optimization with the set less order relation” Taiwan. J. Math. 19: 737–757, (2015).

9. Karaman, E., Soyertem, M. and Atasever Güvenç, İ., “Optimality conditions in set-valued optimization problem with respect to a partial order relation via directional derivative”, Taiwan. J. Math., (2019). doi: 10.11650/tjm/190604.
10. Hernández, E. and Rodríguez-Marín, L., “Weak and strong subgradients of set-valued maps”, J. Optim. Theory. Appl., 149: 352–365, (2011).
11. Chen, G.Y. and Jahn, J., “Optimality conditions for set-valued optimization problems”, Math. Methods. Oper. Res., 48 (2): 187–200, (1998).
12. Wu, H.C., “Duality Theory for Optimization Problems with Interval-Valued Objective Functions”, J. Optim. Theory. Appl., 144: 615–628, (2010).
13. Bhurjee, A. K. and Padhan, S. K., “Optimality conditions and duality results for non-differentiable interval optimization problems”, J. Appl. Math. Comput., 50: 59–71, (2016).
14. Wu, H.C., “The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function”, Eur. J. Oper. Res., 176: 46-59, (2007).
15. Chalco-Cano, Y., Lodwick, W. A. and Rufian-Lizana, A., “Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative”, Fuzzy. Optim. Decis. Making., 12: 305–322, (2013).
16. R.E., Moore, Method and Application of Interval Analysis, SIAM, Philadelphia, (1979).
17. Ishibuchi, H. and Tanaka, H., “Multiobjective programming in optimization of the interval objective function”, Eur. J. Oper. Res., 48(2): 219–225, (1990).