{"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/solving-low-rank-semidefinite-programs-via","title":"Solving Low-Rank Semidefinite Programs via Manifold Optimization","arxiv_id":"2303.01722","date":"2023-03-03","proceeding":null,"authors":["Jie Wang","Liangbing Hu"],"abstract":"We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions, with an emphasis on SDP relaxations for polynomial optimization problems. This approach incorporates the inexact augmented Lagrangian method (ALM) and the Burer-Monteiro factorization, and features the self-adaptive strategies for updating the factorization size and the penalty parameter. We establish global convergence of the inexact ALM, despite the non-convexity brought by the Burer-Monteiro factorization. We further provide a practical algorithm building on the inexact ALM, and along with the algorithm we release an open-source SDP solver ManiSDP. Comprehensive numerical experiments demonstrate that ManiSDP achieves state-of-the-art in terms of efficiency, accuracy, and scalability, and is faster than several advanced SDP solvers (MOSEK, SDPLR, SDPNAL+, STRIDE) by up to orders of magnitudes on a variety of linear SDPs. The largest SDP solved by ManiSDP (in about 8.5 hours with maximal KKT residue 3.5e-13) is the second-order moment relaxation of a binary quadratic program with 120 variables, which has matrix dimension 7261 and contains 17,869,161 affine constraints.","url_abs":"https://arxiv.org/abs/2303.01722v4","url_pdf":"https://arxiv.org/pdf/2303.01722v4.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":"solving-low-rank-semidefinite-programs-via","repo_url":"https://github.com/wangjie212/manisdp-matlab","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}