{"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/the-copositive-way-to-obtain-certificates-of","title":"Reducing non-negativity over general semialgebraic sets to non-negativity over simple sets","arxiv_id":"1909.06689","date":"2019-09-14","proceeding":null,"authors":["Olga Kuryatnikova","Juan C. Vera","Luis F. Zuluaga"],"abstract":"A non-negativity certificate (NNC) is a way to write a polynomial so that its non-negativity on a semialgebraic set becomes evident. Positivstellens\\\"atze (Ps\\\"atze) guarantee the existence of NNCs. Both, NNCs and Ps\\\"atze underlie powerful algorithmic techniques for optimization. This paper proposes a universal approach to derive new Ps\\\"atze for general semialgebraic sets from ones developed for simpler sets, such as a box, a simplex, or the non-negative orthant. We provide several results illustrating the approach. First, by considering Handelman's Positivstellensatz (Psatz) over a box, we construct non-SOS Schm\\\"{u}dgen-type Ps\\\"atze over any compact semialgebraic set. That is, a family of Ps\\\"atze that follow the structure of the fundamental Schm\\\"{u}dgen's Psatz, but where instead of SOS polynomials, any class of polynomials containing the non-negative constants can be used, such as SONC, DSOS/SDSOS, hyperbolic or sums of AM/GM polynomials. Secondly, by considering the simplex as the simple set, we derive a sparse Psatz over general compact sets, which does not require any structural assumptions of the set. Finally, by considering P\\'olya's Psatz over the non-negative orthant, we derive a new non-SOS Psatz over unbounded sets which satisfy some generic conditions. All these results contribute to the literature regarding the use of non-SOS polynomials and sparse NNCs to derive Ps\\\"atze over compact and unbounded sets. Throughout the article, we illustrate our results with relevant examples and numerical experiments.","url_abs":"https://arxiv.org/abs/1909.06689v5","url_pdf":"https://arxiv.org/pdf/1909.06689v5.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":"the-copositive-way-to-obtain-certificates-of","repo_url":"https://github.com/olgakuryatnikova/polylift","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}