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Mekong Delta Natural Science Conference
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Duong, T. K. N., Tran, T. N. Y., Vo, T. T. M., Bui, N. A. T., & Dinh, N. Q. (2025). Existence results for the interval lexicographic optimization problem. Dong Thap University Journal of Science, 14(04S), 453-466. https://doi.org/10.52714/dthu.14.04S.2025.1614
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Existence results for the interval lexicographic optimization problem
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Abstract
In this paper, we provide two solution concepts for optimization problems with interval-valued objective functions based on lexicographic ordering structures. These are total orderings, so they have many practical applications and significance. Under these settings, we present results on the existence of solutions for the optimization problem of interest.
Keywords
interval optimization, interval-valued function, lexicographic order
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
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Ghosh, D. (2017). Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions. Journal of Applied Mathematics and Computing, 53, 709–731. https://doi.org/10.1007/s12190-016-0990-2
Ghosh, D., Bhuiya, S.K., & Patra, L.K. (2018). A saddle point characterization of efficient solutions for interval optimization problems. Journal of Applied Mathematics and Computing, 58, 193–217. https://doi.org/10.1007/s12190-017-1140-1
Ghosh, D., Chauhan, R.S., Mesiar, R., & Debnath, A.K., (2020a). Generalized Hukuhara Gâteaux and Fréchet derivatives of interval-valued functions and their application in optimization with interval-valued functions. Information Sciences, 510, 317–340. https://doi.org/10.1016/j.ins.2019.09.023
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Ghosh, D., Singh, A., Shukla, K.K., & Manchanda, K. (2019). Extended Karush-Kuhn-Tucker condition for constrained interval optimization problems and its application in support vector machines. Information Sciences, 504, 276–292. https://doi.org/10.1016/j.ins.2019.07.017
Ishibuchi, H., & Tanaka, H. (1990). Multiobjective programming in optimization of the interval objective function. European Journal of Operational Research, 48, 219–225. https://doi.org/10.1016/0377-2217(90)90375-L
Moore, R.E., (1966). Interval Analysis. New Jersey, Englewood Cliffs, Prentice-Hall.
Stefanini, L., & Bede, B. (2009). Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Analysis: Theory, Methods & Applications, 71, 1311–1328. https://doi.org/10.1016/j.na.2008.12.005
Singh, D., Dar, B.A., & Kim, D. (2016). KKT optimality conditions in interval valued multiobjective programming with generalized differentiable functions. European Journal of Operational Research, 254, 29–39. https://doi.org/10.1016/j.ejor.2016.03.042
Wu, H.C., (2007). The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. European Journal of Operational Research, 176, 46–59. https://doi.org/10.1016/j.ejor.2005.09.007
Wu, H.C., (2008a). On interval-valued nonlinear programming problems. Journal of Mathematical Analysis and Applications, 338, 299–316. https://doi.org/10.1016/j.jmaa.2007.05.023
Wu, H.C. (2008b). Wolfe duality for interval-valued optimization. Journal of Optimization Theory and Applications, 138, 497–509. https://doi.org/10.1007/s10957-008-9396-0
Zhang, C., & Huang, N. (2022). On Ekeland’s variational principle for interval-valued functions with applications. Fuzzy Sets and Systems, 436,152–174. https://doi.org/10.1016/j.fss.2021.10.003
Zhang, Z., Wang, X., & Lu, J. (2018). Multi-objective immune genetic algorithm solving nonlinear interval-valued programming. Engineering Applications of Artificial Intelligence, 67, 235–245. https://doi.org/10.1016/j.engappai.2017.10.004
Aubin, J. P., & Ekeland, I. (1984). Applied Nonlinear Analysis. Wiley, NewYork.
Chalco-Cano, Y., Rufían-Lizana, A., Román-Flores, H., & Jiménez-Gamero, M.D. (2013). Calculus for interval-valued functions using generalized Hukuhara derivative and applications. Fuzzy Sets and Systems, 219, 49–6. https://doi.org/10.1016/j.fss.2012.12.004
Ghosh, D. (2017). Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions. Journal of Applied Mathematics and Computing, 53, 709–731. https://doi.org/10.1007/s12190-016-0990-2
Ghosh, D., Bhuiya, S.K., & Patra, L.K. (2018). A saddle point characterization of efficient solutions for interval optimization problems. Journal of Applied Mathematics and Computing, 58, 193–217. https://doi.org/10.1007/s12190-017-1140-1
Ghosh, D., Chauhan, R.S., Mesiar, R., & Debnath, A.K., (2020a). Generalized Hukuhara Gâteaux and Fréchet derivatives of interval-valued functions and their application in optimization with interval-valued functions. Information Sciences, 510, 317–340. https://doi.org/10.1016/j.ins.2019.09.023
Ghosh, D., Debnath, A.K., & Pedrycz, W. (2020b). A variable and a fixed ordering of intervals and their application in optimization with interval-valued functions. International Journal of Approximate Reasoning, 121, 187–205. https://doi.org/10.1016/j.ijar.2020.03.004
Ghosh, D., Singh, A., Shukla, K.K., & Manchanda, K. (2019). Extended Karush-Kuhn-Tucker condition for constrained interval optimization problems and its application in support vector machines. Information Sciences, 504, 276–292. https://doi.org/10.1016/j.ins.2019.07.017
Ishibuchi, H., & Tanaka, H. (1990). Multiobjective programming in optimization of the interval objective function. European Journal of Operational Research, 48, 219–225. https://doi.org/10.1016/0377-2217(90)90375-L
Moore, R.E., (1966). Interval Analysis. New Jersey, Englewood Cliffs, Prentice-Hall.
Stefanini, L., & Bede, B. (2009). Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Analysis: Theory, Methods & Applications, 71, 1311–1328. https://doi.org/10.1016/j.na.2008.12.005
Singh, D., Dar, B.A., & Kim, D. (2016). KKT optimality conditions in interval valued multiobjective programming with generalized differentiable functions. European Journal of Operational Research, 254, 29–39. https://doi.org/10.1016/j.ejor.2016.03.042
Wu, H.C., (2007). The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. European Journal of Operational Research, 176, 46–59. https://doi.org/10.1016/j.ejor.2005.09.007
Wu, H.C., (2008a). On interval-valued nonlinear programming problems. Journal of Mathematical Analysis and Applications, 338, 299–316. https://doi.org/10.1016/j.jmaa.2007.05.023
Wu, H.C. (2008b). Wolfe duality for interval-valued optimization. Journal of Optimization Theory and Applications, 138, 497–509. https://doi.org/10.1007/s10957-008-9396-0
Zhang, C., & Huang, N. (2022). On Ekeland’s variational principle for interval-valued functions with applications. Fuzzy Sets and Systems, 436,152–174. https://doi.org/10.1016/j.fss.2021.10.003
Zhang, Z., Wang, X., & Lu, J. (2018). Multi-objective immune genetic algorithm solving nonlinear interval-valued programming. Engineering Applications of Artificial Intelligence, 67, 235–245. https://doi.org/10.1016/j.engappai.2017.10.004
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