A smoothed augmented Lagrangian framework for nonsmooth convex optimization

We focus on developing an Augmented Lagrangian Method (ALM) frame- work for resolving nonsmooth convex optimization problems. The problem of interest is formulated as follows.

$\underset{\mathbf{x}\in \mathcal{X}}{min}f(\mathbf{x})\text{subject to}g(\mathbf{x})\le 0$

where $f$ and $g$ are nonsmooth convex functions and $\mathcal{X}\subset {\mathbb{R}}^n$ is closed and convex.

The presence of nonsmoothness introduces additional challenges to the solution methods. However, by leveraging smoothing techniques, we are able to propose a comprehensive ALM framework that can contend with nonsmoothness.