Weak Fenchel and weak Fenchel-Lagrange conjugate duality for nonconvex scalar optimization problems

In this work, by using weak conjugate maps given in (Azimov and Gasimov, in Int J Appl Math 1:171–192, 1999), weak Fenchel conjugate dual problem, ${(D_F^w)}$ , and weak Fenchel Lagrange conjugate dual problem ${(D_{FL}^w)}$ are constructed. Necessary and sufficient conditions for strong duality for the ${(D_F^w)}$ , ${(D_{FL}^w)}$ and primal problem are given. Furthermore, relations among the optimal objective values of dual problem constructed by using Augmented Lagrangian in (Azimov and Gasimov, in Int J Appl Math 1:171–192, 1999), ${(D_F^w)}$ , ${(D_{FL}^w)}$ dual problems and primal problem are examined. Lastly, necessary and sufficient optimality conditions for the primal and the dual problems ${(D_F^w)}$ and ${(D_{FL}^w)}$ are established.