{"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/a-slightly-lifted-convex-relaxation-for","title":"A Slightly Lifted Convex Relaxation for Nonconvex Quadratic Programming with Ball Constraints","arxiv_id":"2303.01624","date":"2023-03-02","proceeding":null,"authors":["Samuel Burer"],"abstract":"Globally optimizing a nonconvex quadratic over the intersection of $m$ balls in $\\mathbb{R}^n$ is known to be polynomial-time solvable for fixed $m$. Moreover, when $m=1$, the standard semidefinite relaxation is exact. When $m=2$, it has been shown recently that an exact relaxation can be constructed using a disjunctive semidefinite formulation based essentially on two copies of the $m=1$ case. However, there is no known explicit, tractable, exact convex representation for $m \\ge 3$. In this paper, we construct a new, polynomially sized semidefinite relaxation for all $m$, which does not employ a disjunctive approach. We show that our relaxation is exact for $m=2$. Then, for $m \\ge 3$, we demonstrate empirically that it is fast and strong compared to existing relaxations. The key idea of the relaxation is a simple lifting of the original problem into dimension $n+1$. Extending this construction: (i) we show that nonconvex quadratic programming over $\\|x\\| \\le \\min \\{ 1, g + h^T x \\}$ has an exact semidefinite representation; and (ii) we construct a new relaxation for quadratic programming over the intersection of two ellipsoids, which globally solves all instances of a benchmark collection from the literature.","url_abs":"https://arxiv.org/abs/2303.01624v2","url_pdf":"https://arxiv.org/pdf/2303.01624v2.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":"a-slightly-lifted-convex-relaxation-for","repo_url":"https://github.com/sburer/ballconstraints","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}