Duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on different solution concepts

Under a general setting of partial ordering defined on the set of all fuzzy numbers, the duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on two solution concepts for primal problem and three solution concepts for dual problem are derived in this paper. Those solution concepts are inspired by the nondominated solution concept employed in multiobjective programming problems, since the ordering among fuzzy numbers introduced in this paper is a partial ordering, not a total ordering. We also provide a concept of fuzzy scalar (inner) product for fuzzy numbers. Then the fuzzy-valued Lagrangian function and the fuzzy-valued Lagrangian dual function are proposed via the concept of fuzzy scalar product. Under these settings, the dual problem is formulated, and the objective function of this dual problem is a point-to-set fuzzy-valued function. In this case, three solution concepts for dual problem are proposed. The duality theorems and saddle point optimality conditions can be naturally elicited based on these three solution concepts.