Vol.:(0123456789) Optimization and Engineering (2021) 22:2505–2535 https://doi.org/10.1007/s11081-021-09676-2 1 3 RESEARCH ARTICLE Provably optimal sparse solutions to overdetermined linear systems with non‑negativity constraints in a least‑squares sense by implicit enumeration Fatih S. Aktaş 1  · Ömer Ekmekcioglu 1  · Mustafa Ç. Pinar 1 Received: 15 January 2021 / Revised: 10 August 2021 / Accepted: 10 August 2021 / Published online: 28 August 2021 © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract Computing sparse solutions to overdetermined linear systems is a ubiquitous prob- lem in several felds such as regression analysis, signal and image processing, infor- mation theory and machine learning. Additional non-negativity constraints in the solution are useful for interpretability. Most of the previous research eforts aimed at approximating the sparsity constrained linear least squares problem, and/or fnding local solutions by means of descent algorithms. The objective of the present paper is to report on an efcient and modular implicit enumeration algorithm to fnd prov- ably optimal solutions to the NP-hard problem of sparsity-constrained non-negative least squares. We focus on the problem where the system is assumed to be over- determined where the matrix has full column rank. Numerical results with real test data as well as comparisons of competing methods and an application to hyperspec- tral imaging are reported. Finally, we present a Python library implementation of our algorithm. Keywords Inverse problems · Sparse approximation · Overdetermined linear systems · Sparse solutions · Branch and bound · Implicit enumeration · Non- negative least squares Mathematics Subject Classifcation 65F20 · 65F22 · 65K05 · 90C26 Submitted to the editors DATE. * Mustafa Ç. Pinar mustafap@bilkent.edu.tr Fatih S. Aktaş selim.aktas@ug.bilkent.edu.tr Ömer Ekmekcioglu omer.ekmekcioglu@bilkent.edu.tr 1 Department of Industrial Engineering, Bilkent University, Ankara, Turkey