{"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/frank-wolfe-methods-with-an-unbounded","title":"Frank-Wolfe Methods with an Unbounded Feasible Region and Applications to Structured Learning","arxiv_id":"2012.15361","date":"2020-12-30","proceeding":null,"authors":["Haoyue Wang","Haihao Lu","Rahul Mazumder"],"abstract":"The Frank-Wolfe (FW) method is a popular algorithm for solving large-scale convex optimization problems appearing in structured statistical learning. However, the traditional Frank-Wolfe method can only be applied when the feasible region is bounded, which limits its applicability in practice. Motivated by two applications in statistical learning, the $\\ell_1$ trend filtering problem and matrix optimization problems with generalized nuclear norm constraints, we study a family of convex optimization problems where the unbounded feasible region is the direct sum of an unbounded linear subspace and a bounded constraint set. We propose two new Frank-Wolfe methods: unbounded Frank-Wolfe method (uFW) and unbounded Away-Step Frank-Wolfe method (uAFW), for solving a family of convex optimization problems with this class of unbounded feasible regions. We show that under proper regularity conditions, the unbounded Frank-Wolfe method has a $O(1/k)$ sublinear convergence rate, and unbounded Away-Step Frank-Wolfe method has a linear convergence rate, matching the best-known results for the Frank-Wolfe method when the feasible region is bounded. Furthermore, computational experiments indicate that our proposed methods appear to outperform alternative solvers.","url_abs":"https://arxiv.org/abs/2012.15361v2","url_pdf":"https://arxiv.org/pdf/2012.15361v2.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":"frank-wolfe-methods-with-an-unbounded","repo_url":"https://github.com/wanghaoyue123/frank-wolfe-with-unbounded-constraints","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}