A Theoretical Framework for Optimality Conditions of Nonlinear Type-2 Interval-Valued Unconstrained and Constrained Optimization Problems Using Type-2 Interval Order Relations

In the traditional nonlinear optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions for constrained optimization problems with inequality constraints play an essential role. The situation becomes challenging when the theory of traditional optimization is discussed under uncertainty....

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Bibliographic Details
Published in:Mathematics (Basel) 2021-04, Vol.9 (8), p.908
Main Authors: Rahman, Md Sadikur, Shaikh, Ali Akbar, Ali, Irfan, Bhunia, Asoke Kumar, Fügenschuh, Armin
Format: Article
Language:English
Subjects:
Constraints
Decision making
Food science
Fuzzy sets
generalized KKT conditions
Inequality
Linear programming
optimality
Optimization
Optimization techniques
Random variables
type-2 interval
type-2 interval order relations
type-2 interval-valued function
Uncertainty
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Summary:In the traditional nonlinear optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions for constrained optimization problems with inequality constraints play an essential role. The situation becomes challenging when the theory of traditional optimization is discussed under uncertainty. Several researchers have discussed the interval approach to tackle nonlinear optimization uncertainty and derived the optimality conditions. However, there are several realistic situations in which the interval approach is not suitable. This study aims to introduce the Type-2 interval approach to overcome the limitation of the classical interval approach. This study introduces Type-2 interval order relation and Type-2 interval-valued function concepts to derive generalized KKT optimality conditions for constrained optimization problems under uncertain environments. Then, the optimality conditions are discussed for the unconstrained Type-2 interval-valued optimization problem and after that, using these conditions, generalized KKT conditions are derived. Finally, the proposed approach is demonstrated by numerical examples.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9080908