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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1410.7187","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-10-27T11:20:16Z","cross_cats_sorted":[],"title_canon_sha256":"90e4515972d342d03cadd91fbbff551e39c4ea5f0bde3b645c591cc943f8032a","abstract_canon_sha256":"78c41a95b37c15e378281ce8d308eb6106e582f704b7ce08a7316eb4458fb708"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:17.864700Z","signature_b64":"POPl+z59z9EjC4jKyQ68KX30zkHVzLM2RzWq2TPGfXu+UHn+87A2bteXX6GQgvsHhsOjvyHQe1++rEi2CBonCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2054558289b8893df5e22a1df85a73643bca88254fc25224b6b17ca750f40c72","last_reissued_at":"2026-05-18T02:39:17.864272Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:17.864272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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