Lagrangian dual theory and stability analysis for fuzzy optimization problems

This paper investigates the Lagrangian dual theory for a class of fuzzy optimization problems with equality and inequality constraints. To begin with, the Lagrangian dual problem corresponding to the primal fuzzy optimization problem is established by introducing the Lagrangian fuzzy -valued function, and the weak dual theorem is derived. Subsequently, a necessary and sufficient condition for the existence of saddle-points is formulated, which serves as the fundamental basis for proving the strong dual theorem. The stability of the optimization model when its constraints are perturbed by parameters is then analyzed. Finally, the developed Lagrangian dual theory is applied to support vector machines to address the binary classification problem of triangular fuzzy number datasets.