A Generic Branch-and-Bound Algorithm for $\ell_0$-Penalized Problems with Supplementary Material

TL;DR

This paper introduces a flexible Branch-and-Bound algorithm for L0-penalized optimization problems that supports various loss functions and penalties, with an open-source solver demonstrating strong empirical performance.

Contribution

It presents a general framework for L0-penalized problems, providing closed-form solutions and an adaptable Python solver, extending beyond existing quadratic-focused methods.

Findings

El0ps achieves state-of-the-art results on classical benchmarks.

The framework supports diverse loss functions and penalties.

The solver extends computational feasibility to complex instances.

Abstract

We present a generic Branch-and-Bound procedure designed to solve L0-penalized optimization problems. Existing approaches primarily focus on quadratic losses and construct relaxations using "Big-M" constraints and/or L2-norm penalties. In contrast, our method accommodates a broader class of loss functions and allows greater flexibility in relaxation design through a general penalty term, encompassing existing techniques as special cases. We establish theoretical results ensuring that all key quantities required for the Branch-and-Bound implementation admit closed-form expressions under the general blanket assumptions considered in our work. Leveraging this framework, we introduce El0ps, an open-source Python solver with a plug-and-play workflow that enables user-defined losses and penalties in L0-penalized problems. Through extensive numerical experiments, we demonstrate that El0ps achieves state-of-the-art performance on classical instances and extends computational feasibility to previously intractable ones.