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In this paper we present a deep-rooted connection between the S-Procedure and the dual cone calculus formula $(K_1\\cap K_2)^*= K_1^*+K_2^*$, which holds for closed convex cones in $\\R^2$. To establish the link with the S-Procedure, we generalize the dual cone calculus formula to a situation where $K_1$ is nonclosed, nonconvex and nonconic but exhibits s"},"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":"1305.2444","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2013-05-10T21:20:00Z","cross_cats_sorted":[],"title_canon_sha256":"6294d5d4df1e27caf83788e78b4a7a87c770126019a8dcffb1ce694b572b6fd0","abstract_canon_sha256":"32192d8d60f22edf67bcf77a2e9c85ae3999f78c7074fc6fcead6d40d379ab70"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:57.847714Z","signature_b64":"jx2XtjpHKauuH1eOuH/iehLK+T4oVSQu6hO38kYqyYGjjTeb9zwNOh7p0DhHN14IaBCW8fVL8SLVbhkvi96tAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4eb916e9b96087cadcd978230c308c0d50737c0a3329e6d126b86248a2019901","last_reissued_at":"2026-05-18T03:25:57.847178Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:57.847178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The S-Procedure via Dual Cone Calculus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Raphael Hauser","submitted_at":"2013-05-10T21:20:00Z","abstract_excerpt":"Given a quadratic function $h$ that satisfies a Slater condition, Yakubovich's S-Procedure (or S-Lemma) gives a characterization of all other quadratic functions that are copositive with $h$ in a form that is amenable to numerical computations. 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