{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/exterior-point-operator-splitting-for","title":"Exterior-point Optimization for Sparse and Low-rank Optimization","arxiv_id":"2011.04552","date":"2020-11-09","proceeding":null,"authors":["Shuvomoy Das Gupta","Bartolomeo Stellato","Bart P. G. Van Parys"],"abstract":"Many problems of substantial current interest in machine learning, statistics, and data science can be formulated as sparse and low-rank optimization problems. In this paper, we present the nonconvex exterior-point optimization solver NExOS -- a first-order algorithm tailored to sparse and low-rank optimization problems. We consider the problem of minimizing a convex function over a nonconvex constraint set, where the set can be decomposed as the intersection of a compact convex set and a nonconvex set involving sparse or low-rank constraints. Unlike the convex relaxation approaches, NExOS finds a locally optimal point of the original problem by solving a sequence of penalized problems with strictly decreasing penalty parameters by exploiting the nonconvex geometry. NExOS solves each penalized problem by applying a first-order algorithm, which converges linearly to a local minimum of the corresponding penalized formulation under regularity conditions. Furthermore, the local minima of the penalized problems converge to a local minimum of the original problem as the penalty parameter goes to zero. We then implement and test NExOS on many instances from a wide variety of sparse and low-rank optimization problems, empirically demonstrating that our algorithm outperforms specialized methods.","url_abs":"https://arxiv.org/abs/2011.04552v7","url_pdf":"https://arxiv.org/pdf/2011.04552v7.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"exterior-point-operator-splitting-for","repo_url":"https://github.com/Shuvomoy/NExOS.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"exterior-point-operator-splitting-for","repo_url":"https://github.com/shuvomoy/nexos","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"exterior-point-operator-splitting-for","repo_url":"https://github.com/shuvomoy/nexos_numerical_experiments_datasets","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}