{"paper":{"title":"Tractable approximations of sets defined with quantifiers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Jean B. Lasserre","submitted_at":"2014-10-27T11:20:16Z","abstract_excerpt":"Given a compact basic semi-algebraic set $K\\subset R^n\\times R^m$, a simple set $B$ (box or ellipsoid), and some semi-algebraic function $f$, we consider sets defined with quantifiers, of the form $R_f:=\\{x\\in B: \\mbox{$f(x,y)\\leq 0$ for all $y$ such that $(x,y)\\in K$}\\}$ and $D_f:=\\{x\\in B: \\mbox{$f(x,y)\\geq 0$ for some $y$ such that $(x,y)\\in K$}\\}$. The former set $R_f$ is particularly useful to qualify \"robust\" decisions $x$ versus noise parameter $y$ (e.g. in robust optimization on some set $\\mathbf{\\Omega}\\subset B$) whereas the latter set $D_f$ is useful (e.g. in optimization) when one "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7187","kind":"arxiv","version":1},"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"}