Distance Majorization and Its Applications

TL;DR

This paper introduces a scalable algorithm for convex optimization problems involving intersections of convex sets, leveraging majorization-minimization, penalty methods, and quasi-Newton acceleration, suitable for high-dimensional applications.

Contribution

It proposes a novel distance majorization algorithm that efficiently handles large-scale convex programming problems with intersection constraints.

Findings

Algorithm scales well with high dimensionality

Effective in applications in statistics, engineering, and machine learning

Demonstrated improved performance over traditional methods

Abstract

The problem of minimizing a continuously differentiable convex function over an intersection of closed convex sets is ubiquitous in applied mathematics. It is particularly interesting when it is easy to project onto each separate set, but nontrivial to project onto their intersection. Algorithms based on Newton's method such as the interior point method are viable for small to medium-scale problems. However, modern applications in statistics, engineering, and machine learning are posing problems with potentially tens of thousands of parameters or more. We revisit this convex programming problem and propose an algorithm that scales well with dimensionality. Our proposal is an instance of a sequential unconstrained minimization technique and revolves around three ideas: the majorization-minimization (MM) principle, the classical penalty method for constrained optimization, and quasi-Newton acceleration of fixed-point algorithms. The performance of our distance majorization algorithms is illustrated in several applications.