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, \({(D_F^w)}\) , 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.