Weak conjugate duality and nonconvex optimization

In this work, by using the notion weak conjugate function defined in [13] weak Fenchel (D^w_F) and weak Fenchel-Lagrange (D^w_FL) dual problems are constructed for nonconvex constrained scalar optimization problems. Weak duality theorem and necessary and sufficient conditions for strong duality of these problems are presented. Then, relationships among the optimal objective values of primal problem, (D^w_F ), (D^w_FL) and Lagrange dual problem (D^w_L) constructed in [14] are examined and necessary and sufficient optimality conditions for optimality of (D^w_F ) and (D^w_FL) are given. In addition, by using notions supremum, infimum of sets and vectorial norm, weak conjugate map, weak biconjugate map and weak subdifferential of a set valued map are defined, relationships between these notions are examined and necessary and sufficient conditions for weakly subdifferentiability of a set-valued map are given. By using weak conjugate maps, a dual problem is constructed for unconstrained vector optimization problems, weak duality and strong duality theorems are presented. At the end, by using a special perturbation function weak Fenchel dual problem for constrained vector optimization problem is constructed and an example of a nonconvex constrained vector optimization problem which can not be solved by using Lagrange dual problem [28] but can be solved by using weak Fenchel Conjugate dual problem is given.