Some results about duality and exact penalization

In this paper, we introduce the concept of the “valley at 0” augmenting function and apply it to construct a class of “valley at 0” augmented Lagrangian functions. We establish the existence of a path of optimal solutions generated by “valley at 0” augmented Lagrangian problems and its convergence toward the optimal set of the original problem and obtain the zero duality gap property between the primal problem and the “valley at 0” augmented Lagrangian dual problem. Moreover, we establish the exact penalization representation results in the framework of “valley at 0” augmented Lagrangian.