Nonlinear Augmented Lagrangian and Duality Theory

In this paper, a unified framework of a nonlinear augmented Lagrangian dual problem is investigated for the primal problem of minimizing an extended real-valued function by virtue of a nonlinear augmenting penalty function. Our framework is more general than the ones in the literature in the sense that our nonlinear augmenting penalty function is defined on an open set and that our assumptions are presented in terms of a substitution of the dual variable, so our scheme includes barrier penalty functions and the weak peak at zero property as special cases. By assuming that the increment of the nonlinear augmenting penalty function with respect to the penalty parameter satisfies a generalized peak at zero property, necessary and sufficient conditions for the zero duality gap property are established and the existence of an exact penalty representation is obtained.