Optimality conditions for fuzzy optimization problems and its application to classification problems with fuzzy data

Abstract

The main objective of this paper is to investigate the KKT optimality condition for fuzzy optimization problems with inequality constraints. To begin with, by proving that the intersection of the cone of descent directions and the cone of feasible directions at the optimal point is an empty set, we establish the first-order optimality condition for unconstrained fuzzy optimization problems. On this basis, the Fritz-John optimality condition for fuzzy optimization problems with inequality constraints is derived through the fuzzy Gordan’s theorem. Furthermore, in order to ensure that the Lagrangian multipliers must satisfy not all zero, we strengthen the assumptions to deduce the KKT optimality condition. Meanwhile, some numerical examples are created to verify the validity of theoretical results. It is particularly worth mentioning that the optimality conditions established in this paper are such that zero belongs to a certain interval, which makes our results computationally superior than in the previous literature, where the optimality conditions are equalities. Finally, the developed optimality conditions are employed to address a binary classification problem related to support vector machines with fuzzy data.