{"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/interior-point-methods-for-second-order","title":"Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems","arxiv_id":"2103.14320","date":"2021-03-26","proceeding":null,"authors":["Shun Arahata","Takayuki Okuno","Akiko Takeda"],"abstract":"We propose a primal-dual interior-point method (IPM) with convergence to second-order stationary points (SOSPs) of nonlinear semidefinite optimization problems, abbreviated as NSDPs. As far as we know, the current algorithms for NSDPs only ensure convergence to first-order stationary points such as Karush-Kuhn-Tucker points, but without a worst-case iteration complexity. The proposed method generates a sequence approximating SOSPs while minimizing a primal-dual merit function for NSDPs by using scaled gradient directions and directions of negative curvature. Under some assumptions, the generated sequence accumulates at an SOSP with a worst-case iteration complexity. This result is also obtained for a primal IPM with a slight modification. Finally, our numerical experiments show the benefits of using directions of negative curvature in the proposed method.","url_abs":"https://arxiv.org/abs/2103.14320v2","url_pdf":"https://arxiv.org/pdf/2103.14320v2.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":"interior-point-methods-for-second-order","repo_url":"https://github.com/mathematical-informatics-5th-lab/decreasing-nc-pdipm","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"jax","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}