Strict convex regularizations, proximal points and augmented Lagrangians

Proximal Point Methods (PPM) can be traced to the pioneer works of Moreau, Martinet and Rockafellar who used as regularization function the square of the Euclidean norm. In this work, we study PPM in the context of optimization and we derive a class of such methods which contains Rockafellar's result. We also present a less stringent criterion to the acceptance of an approximate solution to the subproblems that arise in the inner loops of PPM. Moreover, we introduce a new family of augmented Lagrangian methods for convex constrained optimization, that generalizes that generalizes the PE⁺ class.