A Generic Branch-and-Bound Algorithm for $\ell_0$-Penalized Problems with Supplementary Material
This work addresses the computational challenge of L0-penalized optimization for researchers and practitioners, offering a more flexible and efficient solver, though it is incremental as it builds on existing Branch-and-Bound techniques.
The paper tackles the problem of solving L0-penalized optimization problems by developing a generic Branch-and-Bound algorithm that accommodates a broad class of loss functions and flexible relaxations, resulting in the El0ps solver achieving state-of-the-art performance and extending feasibility to previously intractable instances.
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.