  Large-scale optimization problems that seek sparse solutions have become
ubiquitous. They are routinely solved with various specialized first-order
methods. Although such methods are often fast, they usually struggle with
not-so-well conditioned problems. In this paper, specialized variants of an
interior point-proximal method of multipliers are proposed and analyzed for
problems of this class. Computational experience on a variety of problems,
namely, multi-period portfolio optimization, classification of data coming from
functional Magnetic Resonance Imaging, restoration of images corrupted by
Poisson noise, and classification via regularized logistic regression, provides
substantial evidence that interior point methods, equipped with suitable linear
algebra, can offer a noticeable advantage over first-order approaches.
