Extended Karush-Kuhn-Tucker condition for constrained interval optimization problems and its application in support vector machines

Abstract

This paper presents an extended Karush-Kuhn-Tucker condition to characterize efficient solutions to constrained interval optimization problems. We develop the theory from the geometrical fact that at an optimal solution the cone of feasible directions and the set of descent directions have an empty intersection. With the help of this fact, we derive a set of first-order optimality conditions for unconstrained interval optimization problems. In the sequel, we extend Gordan’s theorems of the alternative for the existence of a solution to a system of interval linear inequalities. Using Gordan’s theorem, we derive Fritz John and Karush-Kuhn-Tucker necessary optimality conditions for constrained interval optimization problems. It is observed that these optimality conditions appear with inclusion relations instead of equations. The derived Karush-Kuhn-Tucker condition is applied to the binary classification problem with interval-valued data using support vector machines.