Abstract: Augmented Lagrangian Methods and Proximal Point Methods for Convex Optimization

We present a review of the classical proximal point method for finding zeroes of maximal monotone operators, and its application to augmented Lagrangian methods, including a rather complete convergence analysis. Next we discuss the generalized proximal point methods, either with Bregman distances or $\phi$-divergences, which in turn give raise to a family of generalized augmented Lagrangians, as smooth in the primal variables as the data functions are. We give a sketch of the convergence analysis for the case of the proximal point method with Bregman distances for variational inequality problems. The difficulty with these generalized augmented Lagrangians lies in establishing optimality of the cluster points of the primal sequence, which is rather immediate in the classical case. In connection with this issue we present two results. First we prove optimality of such cluster points under a strict complementarity assumption (basically that no tight constraint is redundant at any solution). In the absence of this assumption, we establish an ergodic convergence result, namely optimality of the cluster points of a sequence of weighted averages of the primal sequence given by the method, improving over weaker ergodic results previously known. Finally we discuss similar ergodic results for the augmented Lagrangian method with $\phi$-divergences and give the explicit formulae of generalized augmented Lagrangian methods for different choices of the Bregman distances and the $\phi$-divergences.