{"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-proximal-quasi-newton-trust-region-method","title":"A Proximal Quasi-Newton Trust-Region Method for Nonsmooth Regularized Optimization","arxiv_id":"2103.15993","date":"2021-03-29","proceeding":null,"authors":["Aleksandr Y. Aravkin","Robert Baraldi","Dominique Orban"],"abstract":"We develop a trust-region method for minimizing the sum of a smooth term $f$ and a nonsmooth term $h$), both of which can be nonconvex. Each iteration of our method minimizes a possibly nonconvex model of $f + h$ in a trust region. The model coincides with $f + h$ in value and subdifferential at the center. We establish global convergence to a first-order stationary point when $f$ satisfies a smoothness condition that holds, in particular, when it has Lipschitz-continuous gradient, and $h$ is proper and lower semi-continuous. The model of $h$ is required to be proper, lower-semi-continuous and prox-bounded. Under these weak assumptions, we establish a worst-case $O(1/\\epsilon^2)$ iteration complexity bound that matches the best known complexity bound of standard trust-region methods for smooth optimization. We detail a special instance, named TR-PG, in which we use a limited-memory quasi-Newton model of $f$ and compute a step with the proximal gradient method, resulting in a practical proximal quasi-Newton method. We establish similar convergence properties and complexity bound for a quadratic regularization variant, named R2, and provide an interpretation as a proximal gradient method with adaptive step size for nonconvex problems. R2 may also be used to compute steps inside the trust-region method, resulting in an implementation named TR-R2. We describe our Julia implementations and report numerical results on inverse problems from sparse optimization and signal processing. Both TR-PG and TR-R2 exhibit promising performance and compare favorably with two linesearch proximal quasi-Newton methods based on convex models.","url_abs":"https://arxiv.org/abs/2103.15993v3","url_pdf":"https://arxiv.org/pdf/2103.15993v3.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-proximal-quasi-newton-trust-region-method","repo_url":"https://github.com/UW-AMO/TRNC","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"a-proximal-quasi-newton-trust-region-method","repo_url":"https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"a-proximal-quasi-newton-trust-region-method","repo_url":"https://github.com/JuliaSmoothOptimizers/RegularizedProblems.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"a-proximal-quasi-newton-trust-region-method","repo_url":"https://github.com/JuliaSmoothOptimizers/ShiftedProximalOperators.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"a-proximal-quasi-newton-trust-region-method","repo_url":"https://github.com/MaxenceGollier/RegularizedOptimization.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"a-proximal-quasi-newton-trust-region-method","repo_url":"https://github.com/MaxenceGollier/ShiftedProximalOperators.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"a-proximal-quasi-newton-trust-region-method","repo_url":"https://github.com/geoffroyleconte/regularizedoptimization.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"a-proximal-quasi-newton-trust-region-method","repo_url":"https://github.com/rjbaraldi/ShiftedProximalOperators.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2103.15993","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}