Abstract
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.
Highlights
Because of the impreciseness and randomness of the parameters involved in the different kinds of day-to-day real-life problems, solving the decision-making problems under uncertainty is more challenging for academicians, system analysts, and engineers
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
Both the necessary and sufficient optimality conditions for the nonlinear Type-2 interval-valued unconstrained optimization problem have been derived based on the proposed Type-2 interval order relation
Summary
Because of the impreciseness and randomness of the parameters involved in the different kinds of day-to-day real-life problems (especially decision-making problems), solving the decision-making problems under uncertainty is more challenging for academicians, system analysts, and engineers. Wu [12] established the Karush-KuhnTucker (KKT) conditions of a nonlinear interval-valued constrained optimization problem with crisp-type constraints. He used the Ishibuchi and Tanaka’s [13] partial interval order relations and the gH-differentiability (Stefanin and Bede [14]) to derive the optimality conditions. If we do not overcome these challenges, the optimal solutions to the related problems under such a situation either contain a significant error or deal with considerable uncertainty, which is not an optimistic decision maker’s principle To tackle these challenges, recently, Rahman et al [25,26] introduced an essential generalization of the regular interval, called Type-2 interval. For the first time in the proposed work, the optimality conditions (both necessary and sufficient) for Type-2 interval-valued constrained and unconstrained optimization problems are derived. If the functions F and Gi are convex, necessary KKT conditions are sufficient conditions for optimality
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