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, <CitationRef CitationID="CR1">1999</CitationRef>), weak Fenchel conjugate dual problem, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${(D_F^w)}$$</EquationSource> </InlineEquation> , and weak Fenchel Lagrange conjugate dual problem <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${(D_{FL}^w)}$$</EquationSource> </InlineEquation> are constructed. Necessary and sufficient conditions for strong duality for the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${(D_F^w)}$$</EquationSource> </InlineEquation> , <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${(D_{FL}^w)}$$</EquationSource> </InlineEquation> 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, <CitationRef CitationID="CR1">1999</CitationRef>), <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${(D_F^w)}$$</EquationSource> </InlineEquation> , <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${(D_{FL}^w)}$$</EquationSource> </InlineEquation> dual problems and primal problem are examined. Lastly, necessary and sufficient optimality conditions for the primal and the dual problems <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$${(D_F^w)}$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$${(D_{FL}^w)}$$</EquationSource> </InlineEquation> are established. Copyright Springer Science+Business Media, LLC. 2012