Abstract In the theory of non-linear constrained optimization problem, the most familiar Karush Kuhn Tucker (KKT) conditions play an important role. Basically, these conditions are necessary optimality conditions of a constrained optimization problem with equality and inequality constraints. The goal of this paper is to introduce the necessary optimality conditions (named as equivalent KKT conditions) of a constrained optimization problem with interval-valued objective function. Depending on the type of objective function and inequality constraints, all possible cases (i.e., objective function interval-valued and constraints are real-valued, objective function and constraints both are interval-valued, the objective function is real-valued and constraints are interval-valued) have been investigated to find the necessary optimality conditions. For this purpose, this paper deals with the necessary and sufficient conditions of unconstrained optimization problem with interval-valued objective function. Also, with the help of interval order relations the definitions of local and global optima of interval-valued function have been proposed. Finally, in order to illustrate the equivalent KKT conditions of the interval-valued constrained optimization problem and also the necessary as well as sufficient conditions of an unconstrained optimization problem, some numerical examples have been considered and solved.

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具有区间值目标函数的非线性无约束和有约束优化问题的充要最优条件

摘要 在非线性约束优化问题的理论中，最熟悉的Karush Kuhn Tucker (KKT)条件起着重要的作用。基本上，这些条件是具有等式和不等式约束的约束优化问题的必要优化条件。本文的目标是介绍具有区间值目标函数的约束优化问题的必要最优性条件（称为等效 KKT 条件）。根据目标函数的类型和不等式约束，所有可能的情况（即目标函数为区间值且约束为实值，目标函数和约束均为区间值，目标函数为实值且约束为区间-valued) 已被调查以找到必要的最优条件。为此，本文研究了具有区间值目标函数的无约束优化问题的充要条件。同时，借助区间序关系，提出了区间值函数局部最优和全局最优的定义。最后，为了说明区间值约束优化问题的等效KKT条件以及无约束优化问题的充要条件，考虑并求解了一些数值例子。借助区间序关系，提出了区间值函数局部最优和全局最优的定义。最后，为了说明区间值约束优化问题的等效KKT条件以及无约束优化问题的充要条件，考虑并求解了一些数值例子。借助区间序关系，提出了区间值函数局部最优和全局最优的定义。最后，为了说明区间值约束优化问题的等效KKT条件以及无约束优化问题的充要条件，考虑并求解了一些数值例子。