{"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/pnkh-b-a-projected-newton-krylov-method-for","title":"PNKH-B: A Projected Newton-Krylov Method for Large-Scale Bound-Constrained Optimization","arxiv_id":"2005.13639","date":"2020-05-27","proceeding":null,"authors":["Kelvin Kan","Samy Wu Fung","Lars Ruthotto"],"abstract":"We present PNKH-B, a projected Newton-Krylov method for iteratively solving large-scale optimization problems with bound constraints. PNKH-B is geared toward situations in which function and gradient evaluations are expensive, and the (approximate) Hessian is only available through matrix-vector products. This is commonly the case in large-scale parameter estimation, machine learning, and image processing. In each iteration, PNKH-B uses a low-rank approximation of the (approximate) Hessian to determine the search direction and construct the metric used in a projected line search. The key feature of the metric is its consistency with the low-rank approximation of the Hessian on the Krylov subspace. This renders PNKH-B similar to a projected variable metric method. We present an interior point method to solve the quadratic projection problem efficiently. Since the interior point method effectively exploits the low-rank structure, its computational cost only scales linearly with respect to the number of variables, and it only adds negligible computational time. We also experiment with variants of PNKH-B that incorporate estimates of the active set into the Hessian approximation. We prove the global convergence to a stationary point under standard assumptions. Using three numerical experiments motivated by parameter estimation, machine learning, and image reconstruction, we show that the consistent use of the Hessian metric in PNKH-B leads to fast convergence, particularly in the first few iterations. We provide our MATLAB implementation at https://github.com/EmoryMLIP/PNKH-B.","url_abs":"https://arxiv.org/abs/2005.13639v1","url_pdf":"https://arxiv.org/pdf/2005.13639v1.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":"pnkh-b-a-projected-newton-krylov-method-for","repo_url":"https://github.com/EmoryMLIP/PNKH-B","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":"https://app.syntology.ai/?focus=2005.13639","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}