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.

$\min_{\mathbf{x}\in\mathcal{X}} f(\mathbf{x}) \quad \text{subject to} \quad g(\mathbf{x}) \leq 0$

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.