Abstract
This paper addresses the optimization problems with interval-valued objective function. We consider three types of total order relationships on the interval space. For each total order relationship, we introduce interval-valued convex functions and obtain Karush-Kuhn-Tucker (KKT) optimality conditions in an optimization problem with interval-valued objective function. In order to illustrate these conditions, some numerical examples have been considered and solved.
Similar content being viewed by others
References
Arana-Jiménez, M., & Sánchez-Gil, C. (2020). On generating the set of nondominated solutions of a linear programming problem with parameterized fuzzy numbers. Journal of Global Optimization, 77, 27–52.
Aubin, J. P., & Cellina, A. (1984). Differential inclusions. New York: Springer.
Bhurjee, A. K., & Panda, G. (2012). Efficient solution of interval optimization problem. Mathematical Methods of Operations Research, 76, 273–288.
Bustince, H., Fernandez, J., Kolesárová, A., & Mesiar, R. (2013). Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets and Systems, 220, 69–77.
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.
Cheng, J., Liu, Z., Wu, Z., et al. (2016). Direct optimization of uncertain structures based on degree of interval constraint violation. Computers & Structures, 164, 83–94.
Ezzati, R., Khorram, E., & Enayati, R. (2015). A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Applied Mathematical Modelling, 39, 3183–3193.
Fu, C., Liu, Y., & Xiao, Z. (2019). Interval differential evolution with dimension- reduction interval analysis method for uncertain optimization problems. Applied Mathematical Modelling, 69, 441–452.
Gong, D. W., Ji, X. F., Sun, J., & Sun, X. Y. (2014). Interactive evolutionary algorithms with decision-makers preferences for solving interval multi-objective optimization problems. Neurocomputing, 137, 241–251.
Huang, H. B., Huang, X. R., Ding, W. P., et al. (2022). Uncertainty optimization of pure electric vehicle interior tire/road noise comfort based on data-driven. Mechanical Systems and Signal Processing, 165, 108300.
Ishibuchi, H., & Tanaka, H. (1990). Multiobjective programming in optimization of the interval objective function. European Journal of Operational Research, 48, 219–225.
Jiang, C., Han, X., Liu, G. R., & Liu, G. P. (2008). A nonlinear interval number programming method for uncertain optimization problems. European Journal of Operational Research, 188, 1–13.
Jin, T., Xia, H. X., & Chen, H. (2021a). Optimal control problem of the uncertain second-order circuit based on first hitting criteria. Mathematical Methods in the Applied Sciences, 44, 882–900.
Jin, T., Xia, H. X., Deng, W., et al. (2021b). Uncertain fractional-order multi-objective optimization based on reliability analysis and application to fractional-order circuit with caputo type. Circuits, Systems, and Signal Processing, 40, 5955–5982.
Kaveh, A., Dadras, A., & Geran Malek, N. (2019). Robust design optimization of laminated plates under uncertain bounded buckling loads. Structural and Multidisciplinary Optimization, 59, 877–891.
Kumar, A., Kaur, J., & Singh, P. (2011). A new method for solving fully fuzzy linear programming problems. Applied Mathematical Modelling, 35, 817–823.
Li, D. C., Leung, Y., & Wu, W. Z. (2019). Multiobjective interval linear programming in admissible-order vector space. Information Sciences, 486, 1–19.
Neumaier, A. (1990). Interval methods for systems of equations. Cambridge: Cambridge University Press.
Rahman, M. S., Shaikh, A. A., & Bhunia, A. K. (2020). Necessary and sufficient optimality conditions for non-linear unconstrained and constrained optimization problem with interval valued objective function. Computers & Industrial Engineering, 147, 106634.
Santoro, R., Muscolino, G., & Elishakoff, I. (2015). Optimization and anti-optimization solution of combined parameterized and improved interval analyses for structures with uncertainties. Computers & Structures, 149, 31–42.
Stefanini, L. (2009). Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Analysis: Theory, Methods & Applications, 71, 1311–1328.
Wang, L. Q., Chen, Z. T., Yang, G. L., et al. (2020). An interval uncertain optimization method using back-propagation neural network differentiation. Computer Methods in Applied Mechanics and Engineering, 366, 113065.
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.
Wu, H. C. (2019). Applying the concept of null set to solve the fuzzy optimization problems. Fuzzy Optimization and Decision Making, 18, 279–314.
Xu, Z. S., & Yager, R. R. (2006). Some geometric aggregation operators based on intuitionistic fuzzy sets. International Journal of General Systems, 35, 417–433.
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (Grant Nos.11401469,F0118).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, L. Optimality conditions for nonlinear optimization problems with interval-valued objective function in admissible orders. Fuzzy Optim Decis Making 22, 247–265 (2023). https://doi.org/10.1007/s10700-022-09391-2
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-022-09391-2