{"paper":{"title":"Optimal size of linear matrix inequalities in semidefinite approaches to polynomial optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO","math.MG"],"primary_cat":"math.OC","authors_text":"Gennadiy Averkov","submitted_at":"2018-06-22T13:36:55Z","abstract_excerpt":"The abbreviations LMI and SOS stand for `linear matrix inequality' and `sum of squares', respectively. The cone $\\Sigma_{n,2d}$ of SOS polynomials in $n$ variables of degree at most $2d$ is known to have a semidefinite extended formulation with one LMI of size $\\binom{n+d}{n}$. In other words, $\\Sigma_{n,2d}$ is a linear image of a set described by one LMI of size $\\binom{n+d}{n}$. We show that $\\Sigma_{n,2d}$ has no semidefinite extended formulation with finitely many LMIs of size less than $\\binom{n+d}{n}$. Thus, the standard extended formulation of $\\Sigma_{n,2d}$ is optimal in terms of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.08656","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}