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

Abstract

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, , and weak Fenchel Lagrange conjugate dual problem are constructed. Necessary and sufficient conditions for strong duality for the , 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), , dual problems and primal problem are examined. Lastly, necessary and sufficient optimality conditions for the primal and the dual problems and are established.