An Inexact Proximal Path-Following Algorithm for Constrained Convex Minimization

TL;DR

This paper introduces an inexact path-following algorithm for solving constrained convex minimization problems with nonsmooth objectives, leveraging proximal operators and self-concordant barriers for efficiency without dimension lifting.

Contribution

It presents a novel joint approach combining proximal methods and self-concordant barriers, with a theoretical complexity analysis and practical advantages over interior point methods.

Findings

Efficiently solves nonsmooth convex problems with convex constraints.

Demonstrates advantages over standard interior point methods in applications.

Provides a tuning-free approach to obtain Pareto optimal points.

Abstract

Many scientific and engineering applications feature nonsmooth convex minimization problems over convex sets. In this paper, we address an important instance of this broad class where we assume that the nonsmooth objective is equipped with a tractable proximity operator and that the convex constraint set affords a self-concordant barrier. We provide a new joint treatment of proximal and self-concordant barrier concepts and illustrate that such problems can be efficiently solved, without the need of lifting the problem dimensions, as in disciplined convex optimization approach. We propose an inexact path-following algorithmic framework and theoretically characterize the worst-case analytical complexity of this framework when the proximal subproblems are solved inexactly. To show the merits of our framework, we apply its instances to both synthetic and real-world applications, where it shows advantages over standard interior point methods. As a by-product, we describe how our framework can obtain points on the Pareto frontier of regularized problems with self-concordant objectives in a tuning free fashion.