{"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-constructive-approach-to-strengthen","title":"A constructive approach to strengthen algebraic descriptions of function and operator classes","arxiv_id":"2504.14377","date":"2025-04-19","proceeding":null,"authors":["Anne Rubbens","Julien M. Hendrickx","Adrien Taylor"],"abstract":"It is well known that functions (resp. operators) satisfying a property~$p$ on a subset $Q\\subset \\mathbb{R}^d$ cannot necessarily be extended to a function (resp. operator) satisfying~$p$ on the whole of~$\\mathbb{R}^d$. Given $Q \\subseteq \\mathbb{R}^d$, this work considers the problem of obtaining necessary and ideally sufficient conditions to be satisfied by a function (resp. operator) on $Q$, ensuring the existence of an extension of this function (resp. operator) satisfying $p$ on $\\mathbb{R}^d$. More precisely, given some property $p$, we present a refinement procedure to obtain stronger necessary conditions to be imposed on $Q$. This procedure can be applied iteratively until the stronger conditions are also sufficient. We illustrate the procedure on a few examples, including the strengthening of existing descriptions for the classes of smooth functions satisfying a \\L{}ojasiewicz condition, convex blockwise smooth functions, Lipschitz monotone operators, strongly monotone cocoercive operators, and uniformly convex functions. In most cases, these strengthened descriptions can be represented, or relaxed, to semi-definite constraints, which can be used to formulate tractable optimization problems on functions (resp. operators) within those classes.","url_abs":"https://arxiv.org/abs/2504.14377v3","url_pdf":"https://arxiv.org/pdf/2504.14377v3.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-constructive-approach-to-strengthen","repo_url":"https://github.com/annerubbens/constraints_strengthening","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}