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This concept helps understanding existing approaches from different schools of thought, developing novel relaxation schemes, and deriving a flexible duality theory, which can be specialized to many concrete situations that have been considered in the literature. We survey different approaches to polynomial optimization including polyhedral approximations, dense semidefinite relaxations, SONC, SAGE, and TSSOS in a self-contained, unifying exposition.<\/jats:p>","DOI":"10.5802\/ojmo.50","type":"journal-article","created":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T15:26:03Z","timestamp":1780586763000},"page":"1-44","source":"Crossref","is-referenced-by-count":0,"title":["Unifying view on sparse convex relaxations in polynomial optimization"],"prefix":"10.5802","volume":"7","author":[{"given":"Gennadiy","family":"Averkov","sequence":"first","affiliation":[{"name":"Fakult\u00e4t 1, Brandenburgische Technische Universit\u00e4t Cottbus-Senftenberg, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Benjamin","family":"Peters","sequence":"additional","affiliation":[{"name":"Otto-von-Guericke Universit\u00e4t Magdeburg, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sebastian","family":"Sager","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Dynamics of Complex Technical Systems Magdeburg, Germany"},{"name":"Otto-von-Guericke Universit\u00e4t Magdeburg, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"3842","published-online":{"date-parts":[[2026,6,4]]},"reference":[{"issue":"9","key":"key2026060516482469269_1","doi-asserted-by":"publisher","first-page":"1159","DOI":"10.1016\/s0098-1354(98)00218-x","article-title":"A global optimization method, <span class=\"mathjax-formula\" data-tex=\"$\\alpha $\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b1<\/mi><\/math><\/span>BB, for general twice-differentiable constrained NLPs \u2013 II. 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