Minimization over the l1-ball using an active-set non-monotone projected gradient

TL;DR

This paper introduces an active-set method combined with a non-monotone projected gradient approach to efficiently solve minimization problems over the l1-ball, ensuring convergence and demonstrating effectiveness through numerical experiments.

Contribution

It presents a novel active-set strategy integrated into a non-monotone gradient algorithm for l1-ball minimization, with proven convergence and practical efficiency.

Findings

Algorithm converges globally to stationary points.

Numerical experiments show high efficiency on test instances.

Method effectively handles sparsity constraints in practice.

Abstract

The l1-ball is a nicely structured feasible set that is widely used in many fields (e.g., machine learning, statistics and signal analysis) to enforce some sparsity in the model solutions. In this paper, we devise an active-set strategy for efficiently dealing with minimization problems over the l1-ball and embed it into a tailored algorithmic scheme that makes use of a non-monotone first-order approach to explore the given subspace at each iteration. We prove global convergence to stationary points. Finally, we report numerical experiments, on two different classes of instances, showing the effectiveness of the algorithm.