Sparse Approximations with Interior Point Methods

TL;DR

This paper introduces specialized interior point methods for large-scale sparse optimization problems, demonstrating their advantages over first-order methods through diverse applications and computational experiments.

Contribution

It develops and analyzes interior point-proximal methods tailored for sparse solutions, showing their effectiveness on various real-world problems.

Findings

Interior point methods outperform first-order methods on ill-conditioned problems.

The proposed methods are effective in portfolio optimization, MRI data classification, image restoration, and logistic regression.

Computational experiments show significant performance improvements.

Abstract

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.