Fritz-John optimality condition in fuzzy optimization problems and its application to classification of fuzzy data

TL;DR

This paper develops Fritz-John optimality conditions for fuzzy optimization problems, generalizes Gordan's theorem for fuzzy systems, and applies these results to fuzzy data classification in support vector machines.

Contribution

It introduces first-order optimality conditions for fuzzy optimization and extends Gordan's theorem to fuzzy inequalities, with applications to fuzzy data classification.

Findings

Derived Fritz-John optimality conditions for fuzzy problems

Generalized Gordan's theorem for fuzzy inequalities

Applied conditions to fuzzy SVM classification with numerical examples

Abstract

The main objective of this paper is to derive the optimality conditions for one type of fuzzy optimization problems. At the beginning, we define a cone of descent direction for fuzzy optimization, and prove that its intersection with the cone of feasible directions at an optimal point is an empty set. Then, we present first-order optimality conditions for fuzzy optimization problems. Furthermore, we generalize the Gordan's theorem for fuzzy linear inequality systems and utilize it to deduce the Fritz-John optimality condition for the fuzzy optimization with inequality constraints. Finally, we apply the optimality conditions established in this paper to a binary classification problem for support vector machines with fuzzy data. In the meantime, numerical examples are described to demonstrate the primary findings proposed in the present paper.