Necessary and sufficient optimality conditions for non-linear unconstrained and constrained optimization problem with interval valued objective function

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.